














\documentclass[12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{enumerate,mathrsfs}
\usepackage[hyphens]{url}
\usepackage[T1]{fontenc}

\pagestyle{myheadings}\markboth{\sc\small the electronic journal of combinatorics  \#DS8}{\sc\small the electronic journal of combinatorics  \#DS8}


\input{bsg-command-file.tex}


\begin{document}







\setcounter{page}{1}
\pagenumbering{roman}
\thispagestyle{empty}
\baselineskip = 10 pt

\centerline {\large A Mathematical Bibliography of}
\vskip 5 pt
\centerline {\large Signed and Gain Graphs and Allied Areas}

\vskip 30 pt

\centerline {Compiled and Annotated by}
\bigskip

\centerline {\sc Thomas Zaslavsky}
\bigskip


\centerline {Department of Mathematical Sciences}
\centerline {Binghamton University (SUNY)}
\centerline {Binghamton, New York, U.S.A.\ 13902-6000}
\medskip
\centerline {{\it E-mail}: {\tt zaslav@math.binghamton.edu}}

\bigskip\bigskip
\centerline {Submitted: March 19, 1998; Accepted: July 20, 1998.}

\vskip 20 pt

\centerline{\bf Preliminary Tenth Edition}
\centerline{(Working copy as of \today.)} 


\vfill
\bigskip\hrule\bigskip

\noindent Mathematics Subject Classifications (2000):  \\
{\it Primary} 05-00, 05-02, 05C22; \\
{\it Secondary} 05B20, 05B35, 05C07, 05C10, 05C15, 05C17, 05C20, 05C25, 05C30, 05C35, 05C38, 05C40, 05C45, 05C50, 05C60, 05C62, 05C65, 05C70, 05C75, 05C80, 05C83, 05C85, 05C90, 05C99, 05E25, 05E30, 06A07, 15A06, 15A15, 15A39, 15A99, 20B25, 20F55, 34C99, 51D20, 51D35, 51E20, 51M09, 52B12, 52C07, 52C35, 57M27, 68Q15, 68Q25, 68R10, 82B20, 82D30, 90B10, 90C08, 90C27, 90C35, 90C57, 90C60, 91B14, 91C20, 91D30, 91E10, 92D40, 92E10, 94B75.

\bigskip\hrule\bigskip

\def\vertpair#1#2{\hfil\break\noindent{\hbox to .00truein{#1}\textputone{#2}}}

\centerline{\bf Index}

\baselineskip=14pt
\hbox{
\vtop{\hsize = 1.6truein
\vertpair{\hyperlink{LETTER A}{A}}{\pageref*{LETTER A}}
\vertpair{\hyperlink{LETTER B}{B}}{\pageref*{LETTER B}}
\vertpair{\hyperlink{LETTER C}{C}}{\pageref*{LETTER C}}
\vertpair{\hyperlink{LETTER D}{D}}{\pageref*{LETTER D}}
\vertpair{\hyperlink{LETTER E}{E}}{\pageref*{LETTER E}}
\vertpair{\hyperlink{LETTER F}{F}}{\pageref*{LETTER F}}
\vertpair{\hyperlink{LETTER G}{G}}{\pageref*{LETTER G}}
}
\vtop{\hsize = 1.6truein
\vertpair{\hyperlink{LETTER H}{H}}{\pageref*{LETTER H}}
\vertpair{\hyperlink{LETTER I}{I}}{\pageref*{LETTER I}}
\vertpair{\hyperlink{LETTER J}{J}}{\pageref*{LETTER J}}
\vertpair{\hyperlink{LETTER K}{K}}{\pageref*{LETTER K}}
\vertpair{\hyperlink{LETTER L}{L}}{\pageref*{LETTER L}}
\vertpair{\hyperlink{LETTER M}{M}}{\pageref*{LETTER M}}
\vertpair{\hyperlink{LETTER N}{N}}{\pageref*{LETTER N}}
}
\vtop{\hsize = 1.6truein
\vertpair{\hyperlink{LETTER O}{O}}{\pageref*{LETTER O}}
\vertpair{\hyperlink{LETTER P}{P}}{\pageref*{LETTER P}}
\vertpair{\hyperlink{LETTER Q}{Q}}{\pageref*{LETTER Q}}
\vertpair{\hyperlink{LETTER R}{R}}{\pageref*{LETTER R}}
\vertpair{\hyperlink{LETTER S}{S}}{\pageref*{LETTER S}}
\vertpair{\hyperlink{LETTER T}{T}}{\pageref*{LETTER T}}
\vertpair{\hyperlink{LETTER U}{U}}{\pageref*{LETTER U}}
}
\vtop{\hsize = 1.0truein
\vertpair{\hyperlink{LETTER V}{V}}{\pageref*{LETTER V}}
\vertpair{\hyperlink{LETTER W}{W}}{\pageref*{LETTER W}}
\vertpair{\hyperlink{LETTER X}{X}}{\pageref*{LETTER X}}
\vertpair{\hyperlink{LETTER Y}{Y}}{\pageref*{LETTER Y}}
\vertpair{\hyperlink{LETTER Z}{Z}}{\pageref*{LETTER Z}}
}}
\baselineskip=10pt

\bigskip\hrule\bigskip

\centerline {Colleagues:}
\centerline {\it HELP!}
\noindent If you have any suggestions whatever for items to include in this bibliography, or for other changes, please let me hear from you.  Thank you.

\vskip 10 pt

{\small\centerline {Copyright \copyright 1996, 1998, 1999, 2003--2005, 2009--2022 Thomas Zaslavsky}}


\newpage


\centerline {\bf Preface}

\begin{quote}
{\small
[I]t should be borne in mind that incompleteness is a necessary concomitant of every collection of whatever kind.  Much less can completeness be expected in a first collection, made by a single individual, in his leisure hours, and in a field which is already boundless and is yet expanding day by day.
\spitem{}---Robert Edouard Moritz, preface to {\it Memorabilia Mathematica: The Philomath's Quotation Book}, 1914.
}
\end{quote}

\bigskip

A {\it signed graph} is a graph whose edges are labeled by signs.  This is a bibliography of signed graphs and related mathematics.

Several kinds of labelled graph have been called ``signed'' yet are mathematically very different.  I distinguish four types:
\begin{itemize}
\item {\it Group-signed graphs}: the edge labels are elements of a $2$-element group and are multiplied around a circle (or along any walk).
Among the natural generalizations are larger groups and vertex signs.
\item {\it Sign-colored graphs}, in which the edges are labelled from a two-element set that is acted upon by the sign group: $-$ interchanges labels, $+$ leaves them unchanged.  
This is the kind of ``signed graph'' found in knot theory.  The natural generalization is to more colors and more general groups---or no group.
\item {\it Weighted graphs}, in which the edge labels are the elements $+1$ and $-1$ of the integers or another additive domain.  
Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography---except (to some extent) when the author calls the weights ``signs''.
\item Labelled graphs where the labels have no structure or properties but are called ``signs'' for any or no reason.
\end{itemize}

Each of these categories has its own theory or theories, generally very different from the others, so in a logical sense the topic of this bibliography is an accident of terminology.  
However, narrow logic here leads us astray, for the study of true signed graphs, which arise in numerous areas of pure and applied mathematics, forms the great majority of the literature.  
Thus I regard as fundamental for the bibliography the notions of {\it balance} of a circle (sign product equals $+$, the sign group identity) and the vertex-edge incidence matrix (whose column for a negative edge has two $+1$'s or two $-1$'s, for a positive edge one $+1$ and one $-1$, the rest being zero); this has led me to include work on {\it gain graphs} (where the edge labels are taken from any group) and ``consistency'' in {\it vertex-signed graphs}, and generalizable work on two-graphs (the set of unbalanced triangles of a signed complete graph) and on even and odd circles and paths in graphs and digraphs.

Nevertheless, it was not always easy to decide what belongs.  I have employed the following principles:

Only works with mathematical content are entered, except for a few representative purely applied papers and surveys.  I do try to include:

\begin{itemize}
\item  Any (mathematical) work in which signed graphs are mentioned by name or signs are put on the edges of graphs, regardless of whether it makes essential use of signs.  
(However, due to lack of time and in order to maintain ``balance'' in the bibliography, I have included only a limited selection of items concerning binary clutters and postman theory, two-graphs, signed digraphs in qualitative matrix theory, and knot theory.  
For clutters, see \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}; for postman theory, \xref{A.\ Frank}{1996a}{Andras Frank}.  For two-graphs, see any of the review articles by \hyperlink{J.J.\ Seidel}{Seidel}.  For qualitative matrix theory, see e.g.\ \xref{Maybee and Quirk}{1969a}{John Maybee \& James Quirk} and \xref{Brualdi and Shader}{1995a}{Richard A.\ Brualdi \& Bryan L.\ Shader}.  For knot theory there are uncountable books and surveys.)

\item  Any work in which the notion of balance of a circle plays a role.  Example: gain graphs.  (Exception:  purely topological papers concerning ordinary graph embedding.)

\item  Any work in which ideas of signed graph theory are anticipated, or generalized, or transferred to other domains.  Examples:  vertex-signed graphs; signed posets and matroids.

\item  Any mathematical structure that is an example, however disguised, of a signed or gain graph or generalization, and is treated in a way that seems in the spirit of signed graph theory.  Examples:  even-cycle and bicircular matroids; bidirected graphs; binary clutters (which are equivalent to signed binary matroids); some of the literature on two-graphs and double covering graphs.

\item  And some works that have suggested ideas of value for signed graph theory or that have promise of doing so in the future.
\end{itemize}

As for applications, besides works with appropriate mathematical content I include a few (not very carefully) selected representatives of less mathematical papers and surveys, either for their historical importance (e.g., \xref{Heider}{1946a}{Fritz Heider}) or as entrances to the applied literature (e.g., \xref{Taylor}{1970a}{Howard F.\ Taylor} and \xref{Wasserman and Faust}{1994a}{Stanley Wasserman \& Katherine Faust, eds.} for psychosociology and \xref{Trinajsti\'c}{1983a}{Nenad Trinajstic} for chemistry).  Particular difficulty is presented by spin glass theory in statistical physics---that is, Ising models and generalizations.  Here one usually averages random signs and weights over a probability distribution; the problems and methods are rarely graph-theoretic, the topic is very specialized and hard to annotate properly, but it clearly is related to signed (and gain) graphs and suggests some interesting lines of graph-theoretic research.  See \xref{M\'ezard, Parisi, and Virasoro}{1987a}{Marc Mezard, Giorgio Parisi, \& Miguel Angel Virasoro} and citations in its annotation.

Plainly, judgment is required to apply these criteria.  I have employed mine freely, taking account of suggestions from my colleagues.  Still I know that the bibliography is far from complete, due to the quantity and even more the enormous range and dispersion of work in the relevant areas.  I will continue to add both new and old works to future editions and I heartily welcome further suggestions.

There are certainly many errors, some of them egregious.  For these I hand over responsibility to Sloth, Pride, Ambition, Envy, and Confusion.  (Corrections, however, will be gratefully accepted by me.)  And as Diedrich Knickerbocker says:

\vskip 5pt
\begin{quote}
{\small Should any reader find matter of offense in this [bibliography], I should heartily grieve, though I would on no acount question his penetration by telling him he was mistaken, his good nature by telling him he was captious, or his pure conscience by telling him he was startled at a shadow.  Surely when so ingenious in finding offense where none was intended, it were a thousand pities he should not be suffered to enjoy the benefit of his discovery.}
\end{quote}

\newpage

\centerline{\bf Acknowledgement}  
\medskip

I cannot name all the people who have contributed advice and criticism, but many of the annotations have benefited from suggestions by the authors or others and a number of items have been brought to my notice by helpful correspondents.  I am very grateful to you all.  Thanks also to the people who maintain the invaluable MR and Zbl indices (and a special thank-you for creating our own MSC classification: 05C22).  However, I insist on my total responsibility for the final form of all entries, including such things as my restatement of results in signed or gain graphic language and, of course, all the praise and criticism (but not errors; see above) that they contain.

\bigskip

\centerline{\bf Bibliographical Data}  
\medskip

Authors' names are given usually in only one form, even should the name appear in different (but recognizably similar) forms on different publications.  Journal abbreviations follow the style of {\it Mathematical Reviews} (MR) with minor `improvements'.  Reviews and abstracts are cited from MR and its electronic form MathSciNet, from {\it Zentralblatt f\"ur Mathematik} (Zbl) and its electronic version (For early volumes, ``Zbl VVV, PPP'' denotes printed volume and page; the electronic item number is ``(e VVV.PPPNN)''.), and occasionally from {\it Chemical Abstracts} (CA) or {\it Computing Reviews} (CR).  A review marked (\qv) has significance, possibly an insight, a criticism, or a viewpoint orthogonal to mine.

Some---not all---of the most fundamental works are marked with a \dag\dag; some almost as fundamental have a \dag.  This is a personal selection.

\medskip

\newpage

\centerline{\bf Annotations}  
\medskip

I try to describe the relevant content in a consistent terminology and notation, in the language of signed graphs despite occasional clumsiness (hoping that this will suggest generalizations), and sometimes with my [bracketed] editorial comments.  I sometimes try also to explain idiosyncratic terminology, in order to make it easier to read the original item.  Several of the annotations incorporate open problems (of widely varying degrees of importance and difficulty).  

I use these standard symbols:
\smallskip

\spitem{$c(\ )$} is the number of connected components of a graph.

\spitem{$\Gamma$}  is a graph $(V,E)$ of order $n = \#V$, undirected, possibly allowing loops and multiple edges.  It is normally finite unless otherwise indicated.  

\spitem{$\Gamma{:}X$}  is the induced subgraph on $X\subseteq V$; also, $\Gamma{:}\pi := \bigcup_{X\in\pi} \Gamma{:}X$ and $\Gamma(\pi) := \Gamma \setminus E{:}\pi$, for $\pi$ a partition of $V$.

\spitem{$\Gamma^c$}  is the complement of a simple graph.

\spitem{$\Sigma$}  is a signed graph $(V,E,\sigma)$ of order $n$.  $|\Sigma|$  is its underlying graph.  

\spitem{$\sigma$}  is the signature (sign function) of $\Sigma$.

\spitem{$E^+$, $E^-$}  are the sets of positive and negative edges of $\Sigma$.

\spitem{$\Sigma^+$, $\Sigma^-$}  are the corresponding spanning subgraphs (unsigned).

\spitem{$[\Sigma]$}  is the switching class of $\Sigma$.

\spitem{$\widetilde\Sigma$\;}  is the double covering graph of $\Sigma$.

\spitem{$A(\ )$}  is the adjacency matrix.

\spitem{$\incid(\ )$}  (Eta) is the incidence matrix.

\spitem{$\cH[\ ]$}  is the hyperplane arrangement of a graph (ordinary, signed, or gain).

\spitem{$\Lap(\ )$}  is the Laplacian matrix, $= \incid(\ )\incid(\ )\transpose$.

\spitem{$\cL(\ )$}  is the intersection (semi)lattice of a hyperplane arrangement.

\spitem{$\lambda_1$} $= \lambda_{\max}$, the largest eigenvalue of a matrix.

\spitem{$\Phi$}  is a gain graph $(V,E,\phi)$ with gain function $\phi$.  $\|\Phi\|$  is its underlying graph.

\spitem{$\phi$}  is the gain function of a gain graph $\Phi$.

\spitem{$[\Phi]$}  is the switching class of $\Phi$.

\spitem{$\sim$}  means that two signed or gain graphs are switching equivalent (with the same underlying graph and no automorphism).

\spitem{$\simeq$}  means that two signed or gain graphs are switching isomorphic (with isomorphic underlying graphs, which may be the same graph).

\spitem{$\cong$}  denotes isomorphism.

\spitem{$\bgr{\Sigma}, \bgr{\Phi}$}  are the biased graphs of $\Sigma$ and $\Phi$.

\spitem{$\Omega$}  is a biased graph.  $\|\Omega\|$  is its underlying graph.

\spitem{$l(\ )$}  is the frustration index (line index of imbalance).

\spitem{$l_0(\ )$}  is the frustration number (vertex frustration number, vertex elimination number, balanced circle transversal number).

\spitem{$\bfM(\ )$} [formerly written $G(\ )$] is the graphic (cycle) matroid of a graph.

\spitem{$\bfF(\ )$} [formerly written $G(\ )$] is the frame (formerly bias) matroid of a signed, gain, or biased graph.

\spitem{$\bfL(\ ), \bfL_0(\ )$}  are the lift and extended lift matroids.  
[For line graphs see $\Lambda$.  For the Laplacian matrix see $\Lap$.]

\spitem{$\Lambda(\ )$}  is a line graph.  $\Lambda(\Gamma)$ is that of a graph.  For a signed or gain graph, $\Lambda_{BC}$ is that of \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand}; 
$\Lambda_{\times}$ is that of \xref{M.\ Acharya}{2009a}{Mukti Acharya}, 
$\Lambda_\bullet$ is that of M.\ Acharya (\cf\ \xref{B.D.\ Acharya}{2010a}{B.\ Devadas Acharya}), 
$\Lambda$ is that of \xref{Zaslavsky}{1979a}{Thomas Zaslavsky}, \xrefd{1984c}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}.

\medskip

\newpage

Some standard terminology---much more will be found in the {\it Glossary} (\xref{Zaslavsky}{1998c}{Thomas Zaslavsky}):
\smallskip

\spitem{circle:}The graph of a simple closed path, or its edge set.   (Also, ``circuit'', ``cycle'', ``polygon'', ``simple cycle'', etc.)

\spitem{cycle:}In a digraph, a coherently directed circle, i.e., ``dicycle''.  More generally: in an oriented signed, gain, or biased graph, a matroid circuit (usually, of the frame matroid) oriented to have no source or sink.

\bigskip

\newpage

\centerline {\bf Subject Classification Codes}
\medskip

A code in {\it lower case} means the topic appears implicitly but not explicitly.  
A suffix {\bf w} on \CSgnd, \CSG, \CSD, \CVS\ denotes signs used as weights, i.e., treated as the numbers $+1$ and $-1$, added, and (usually) the sum compared to $0$.
A suffix {\bf c} on \CSG, \CSD, \CVS\ denotes signs used as colors (often written as the numbers $+1$ and $-1$), usually permutable by the sign group.
In a string of codes a colon precedes subtopics.
A code may be refined through being suffixed in parentheses, as \CSgnd(\CMatrd) denoting signed matroids while \CSgnd:~\CMatrd\ would indicate matroids of signed objects; thus \CSgnd(\CMatrd):~\CMatrd\ means matroids of signed matroids.  


\smallskip


{

{

\spitem{\CAdj}  Adjacency matrix.
\spitem{\CAlgor}  Algorithms.
\spitem{\CAlgeb}  Algebraic structures upon signed, gain, or biased graphs or digraphs.
\spitem{\CAppl}  Applications other than (\CChem), (\CPhys), (\CBiol), (\CPsS) (partial coverage).
\spitem{\CAut}  Automorphisms, symmetries, group actions.
\spitem{\CBal}  Balance (mathematical), cobalance; harmony, ``consistency'' of vertex signs.
\spitem{\CBic}  Bicircular matroids.
\spitem{\CBiol}  Applications to biology (partial coverage).
\spitem{\CChem}  Applications to chemistry (partial coverage).
\spitem{\CCircles}  Circles.  (\CCycles) for directed circles.
\spitem{\CClu}  Clusterability.
\spitem{\CCol}  Vertex coloring.
\spitem{\CCov}  Covering graphs, double coverings.
\spitem{\CCycles} Directed cycles.  (\CCircles) for undirected.
\spitem{\CDual}  Duality (graphs, matroids, or matrices).
\spitem{\CDyn} Dynamics in (di)graphs.  Predicting aspects, e.g., an edge sign.
\spitem{\CEig} Eigenvalues, eigenvectors, characteristic polynomial, energy.
\spitem{\CEnum}  Enumeration of types of signed graphs, etc.
\spitem{\CECyc}  Even-cycle matroids.	
\spitem{\CECol}  Edge coloring.
\spitem{\CExp}  Expository.
\spitem{\CExr}  Interesting exercises (in an expository work).
\spitem{\CFr}  Frustration (imbalance), esp.\ frustration index (line index of imbalance), other measures; minimum balancing set.
\spitem{\CGeom}  Connections with geometry, e.g., linear programming, complex complement.
\spitem{\CGD}  Digraphs with gains (or voltages).
\spitem{\CGen}  Generalization.
\spitem{\CGG}  Gain, voltage, and biased graphs; includes Dowling lattices (with (\CMatrd)).
\spitem{\CGN}  Generalized or gain networks.  (Multiplicative real gains.)
\spitem{\CGH}  Hypergraphs with gains.
\spitem{\CHom}  Homomorphisms \etal
\spitem{\CInc}  Incidence matrix.
\spitem{\CKG}  Signed complete graphs.
\spitem{\CKnot}  Connections with knot theory (sparse coverage if signs are purely notational).
\spitem{\CLap}  Laplacian matrix $\Lap(\ )$.
\spitem{\CLab} Nonalgebraic labelling of signed, gain, or biased graphs, e.g., gracefulness.
\spitem{\CLG}  Line graphs.  $\CGen$:  Jump graphs, total graphs, \etal
\spitem{\CMatrd}  Matroids and geometric lattices, chain-groups (not signed matroids).
\spitem{\CMtrdF}  Matroidal families.
\spitem{\CInvar}  Numerical and algebraic invariants of signed, gain, biased graphs: polynomials, degree sequences, number of bases, etc.
\spitem{\COri}  Orientations, bidirected graphs.
\spitem{\COG}  Ordered gains.
\spitem{\CPar}  All-negative or antibalanced signed graphs, parity-biased graphs.
\spitem{\Cpar}  Includes results on even or odd paths or circles (partial coverage) that may generalize from antibalanced to all signed graphs.
\spitem{\CPhys}  Applications in physics (limited coverage).
\spitem{\CPsS}  Psychological, sociological, and anthropological applications (partial coverage).
\spitem{\CQM}  Qualitative (sign) matrices: sign patterns, sign stability, sign solvability, etc.: graphical methods.
\spitem{\Cqm}  Qualitative (sign) matrices are implicit or are motivation.
\spitem{\CRand}  Random signs or gains, signed or gain graphs.
\spitem{\CRef}  Many references.
\spitem{\CSgnd}  Signed objects other than graphs and hypergraphs:  mathematical properties.
\spitem{\CSD}  Signed digraphs:  mathematical properties.
\spitem{\CSG}  Signed graphs:  mathematical properties.  See (\CPar) for only all-negative, possibly implicit as with (\Cpar) for signless Laplacian.
\spitem{\CSH}  Signed hypergraphs:  mathematical properties.
\spitem{\CSol}  Sign solvability, sign nonsingularity (partial coverage).
\spitem{\CSta}  Sign stability (partial coverage).
\spitem{\CState}  State space, state space landscape; ground state landscape (very partial coverage).
\spitem{\CStr}  Structure theory.
\spitem{\CSw}  Switching of signs or gains.
\spitem{\CTop}  Topology applied to graphs; surface embeddings.  (Not applications to topology.)
\spitem{\CTG}  Two-graphs, graph (Seidel) switching (partial coverage).
\spitem{\CVS}  Vertex-signed graphs (``marked graphs''); signed vertices and edges.
\spitem{\CWD}  Weighted digraphs.
\spitem{\CWG}  Weighted graphs.
\spitem{\CWH}  Weighted hypergraphs.
\spitem{\CXtreml}  Extremal problems.

}
}




\newpage
\setcounter{page}{1}
\pagenumbering{arabic}
\baselineskip = 10pt

\centerline {\sc A Mathematical Bibliography of}
\smallskip
\centerline{\sc Signed and Gain Graphs and Allied Areas}
\vspace{.75cm}





\label{LETTER A} \hypertarget{LETTER A}{}	% LETTER  A


\author{[Maria Abi Aad]}						%AAD
\authorspace
\authref  See \hyperlink{Maria Abi Aad}{M.~Abi~Aad} (under `Ab').
\itemspace

\authorx{Ahmad Abdi}						%ABDI
\author{Ahmad Abdi \& Bertrand Guenin}
\authorspace
\bsgitem{2014a}{}  The cycling property for the clutter of odd $st$-walks.  In:  \emph{Integer Programming and Combinatorial Optimization} (17th Int.\ IPCO, Bonn, 2014), pp.\ 1--12.  \LNCS 8494.  Springer, Cham, 2014.
\MR{3252290}.  \Zbl{1418.90207}.  
\annot{} Extended abstract of \xrefd{2018a}{Ahmad Abdi \& Bertrand Guenin}.
\codes{\CSG}%
\itemspace

\bsgitem{2017a}{}  The two-point Fano and ideal binary clutters.  In:  Friedrich Eisenbrand and Jochen Koenemann, eds., \emph{Integer Programming and Combinatorial Optimization} (19th Int.\ IPCO, Waterloo, Ont., 2013), pp.\ 1--12.  \LNCS 10328.  Springer, Cham, 2017.
\MR{3678769}.  \Zbl{1418.90208}.  
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{2018a}{}  Packing odd $T$-joins with at most two terminals.  \emph{J.\ Graph Theory} 87 (2018), no.\ 4, 587--652.
\MR{3767184}.  \Zbl{1386.05147}.  \arXiv{1410.7423}.				
\annotctd{} Dictionary:  ``odd'' = negative.
\codes{\CSG}%
\itemspace

\bsgitem{2019a}{}  The minimally non-ideal binary clutters with a triangle.  \emph{Combinatorica} 39 (2019), no.\ 4, 719--752. 
\MR{4015349}.  \Zbl{1438.90393}.  
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{2019b}{}  The two-point Fano and ideal binary clutters.  \emph{Combinatorica} 39 (2019), no.\ 4, 753--777. 
\MR{4015350}.  \Zbl{1449.05041}.  
\codes{\CSG:~\CMatrd}%
\itemspace

\author{Normalah S.\ Abdulcarim}					%ABDULCARIM
\authorspace
\authref  See \hyperlink{Mahid M.\ Mangontarum}{M.M.~Mangontarum}.
\itemspace

\author{Takuro Abe}							%ABE
\authorspace
\bsgitem{2009a}{}  The stability of the family of $B_{2}$-type arrangements.  \emph{Commun.\ Algebra} 37 (2009), no.\ 4, 1193--1215.
\MR{2510979} (2010d:32027).  \Zbl{1194.32014}.  
\annot{} The arrangements are affino-signed-graphic arrangements.  
\par
\codes{\Csg,~\Cgg:~\CGeom}%
\itemspace

\bsgitem{2012a}{}  On a conjecture of Athanasiadis related to freeness of a family of hyperplane arrangements.  \emph{Math.\ Res.\ Lett.}\ 19 (2012), no.\ 2, 469--474.
\MR{2955776}.  \Zbl{272.32024}.  \arXiv{1110.0303}.					
\codes{\CSG:~\CGeom}%
\itemspace

\author{Takuro Abe, Koji Nuida, \& Yasuhide Numata}
\authorspace
\bsgitem{2009a}{}  An edge-signed generalization of chordal graphs, free multiplicities on braid arrangements, and their characterizations.  
In:  Christian Krattenthaler, Volker Strehl, and Manuel Kauers, eds., \emph{21st International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC, Hagenburg, Austria, 2009), pp.\ 1--12.  Discrete Math.\ Theor.\ Computer Sci., Nancy, France, 2009.
\MR{2721497} (2011j:05130).  
\codes{\CSG:~\CStr,~\CGeom}%
\itemspace

\bsgitem{2009b}{}  Signed-eliminable graphs and free multiplicities on the braid arrangement.  \emph{J.\ London Math.\ Soc.}\ (2) 80 (2009), no.\ 1, 121--134. 
\MR{2520381} (2010k:32039).  \Zbl{1177.32017}.  \arXiv{0712.4110 }.			
\codes{\CSG:~\CStr,~\CGeom}%
\itemspace

\author{Takuro Abe, Daisuke Suyama, \& Shuhei Tsujie}
\authorspace
\bsgitem{2015a}{}  The freeness of Ish arrangements.  In:  \emph{Proceedings of the 27th International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC, Daejon, South Korea, 2015), pp.\ 273--284.  Discrete Math.\ Theor.\ Computer Sci., Nancy, France, 2015.
\MR{3470870}.  \Zbl{1335.05189}.  
\annot{} Partial publication of \xrefd{2017a}{Takuro Abe, Daisuke Suyama, \& Shuhei Tsujie}.
\codes{\Cgg:~\CGeom:~\CAlgeb}%
\itemspace

\bsgitem{2017a}{}  The freeness of Ish arrangements.  \emph{J.\ Combin.\ Theory Ser.\ A} 146 (2017), 169--183.
\MR{3574228}.  \Zbl{1351.05224}.  \arXiv{1410.2084}.  
\annot{} Ish arrangements are supersolvable, and more.	
\dated{19 Dec 2020} 
\codes{\Cgg:~\CGeom:~\CAlgeb}%
\itemspace

\author{Toshiki Abe}							%ABE
\authorspace
\authref  See \xref{Wang, Qian, and Abe}{2019a}{Wei Wang, Jianguo Qian, \& Toshiki Abe}.
\itemspace

\author{Peter Abell}							%ABELL
\authorspace
\authref  See also \hyperlink{Hongzhong Deng}{H.Z.~Deng}, \hyperlink{Bernard Kujawski}{B.~Kujawski}, and \hyperlink{Mark Ludwig}{M.~Ludwig}.
\authrefspace

\bsgitem{1968a}{}  Structural balance in dynamic structures.  \emph{Sociology} 2 (1968), no.\ 3, 333--352.
\par
\codes{\CSG,~\CPsS:~\CBal,~\CFr}%
\itemspace

\bsgitem{1969a}{}  Structural balance: a clarification.  \emph{Sociology} 3 (1969), no.\ 3, 421--422.
\annot{} Technical correction to probability estimates in \xrefd{1968a}{Peter Abell}.
\dated{29 Aug 2013} 
\codes{\CSG:~\CFr}%
\itemspace

\author{Peter Abell \& Robin Jenkins}
\authorspace
\bsgitem{1967a}{}  Perception of the structural balance of part of the international system of nations.  \emph{J.\ Peace Res.}\ 4 (1967), no.\ 1, 76--82.
\codes{\CPsS}\codesctd{\CSG:~\CBal:~\CExp}%
\itemspace

\author{Peter Abell \& Mark Ludwig}
\authorspace
\bsgitem{2009a}{}  Structural balance: A dynamic perspective.  \emph{J.\ Math.\ Sociology} 33 (2009), no.\ 2, 129--155.
\Zbl{1169.91434}.  
\annot{} Dynamics of signed graphs in a space of sign probabilities and tolerance of imbalance.  There are three discernibly different domains of dynamical behavior.  
[Continued in \xref{Deng and Abell}{2010a}{Hongzhong Deng \& Peter Abell} and \xref{Kujawski, Ludwig, and Abell}{2010a}{Bernard Kujawski, Mark Ludwig, \& Peter Abell}.]
\dated{10 Sept, 9 Dec 2009} 
\par
\codes{\CSG,~\CPsS:~\CBal,~\CFr,~\CDyn:~\CAlgor}%
\itemspace

\author{Robert P.\ Abelson}						%ABELSON
\authorspace
\authref   See also \hyperlink{Milton J.\ Rosenberg}{M.J.~Rosenberg}.
\authrefspace

\bsgitem{1967a}{}  Mathematical models in social psychology.  
In:  Leonard Berkowitz, ed., \emph{Advances in Experimental Social Psychology}, Vol.\ 3, pp.\ 1--54.  Academic Press, New York, 1967.  
\annot{} \S~II: ``Mathematical models of social structure.''  Part B: ``The balance principle.''  Reviews basic notions of balance and clusterability in signed (di)graphs and measures of degree of balance or clustering.  Notes that signed $K_n$ is balanced iff $I + A = vv\trans$, $v = \pm 1$-vector.  Proposes: degree of balance $= \lambda_{\max}(I + A(\Sigma))/n$.  [\Cf\ \xref{Phillips}{1967a}{J.L.\ Phillips}.]  Part C, 3:  ``Clusterability revisited.''
\codes{\CSG,~\CSD:~\CBal,~\CClu,~\CFr,~\CAdj}%
\itemspace

\author{R.P.\ Abelson, E.\ Aronson, W.J.\ McGuire, T.M.\ Newcomb, M.J.\ Rosenberg, \& P.H.\ Tannenbaum, eds.}
\authorspace
\bsgitem{1968a}{}  \emph{Theories of Cognitive Consistency: A Sourcebook.}  Rand-McNally, Chicago, Ill., 1968.
\codes{\CPsS}%
\itemspace

\author{Robert P.\ Abelson \& Milton J.\ Rosenberg}
\authorspace
\bsgitem{1958a}{\dag}  Symbolic psycho-logic: a model of attitudinal cognition.  \emph{Behavioral Sci.}\ 3 (1958), 1--13.  
\annot{$R(\Sigma)$} They introduce a modified adjacency matrix $R$, called the ``structure matrix'' [I call it the Abelson--Rosenberg adjacency matrix], with entries $o, p, n, a$ for, respectively, nonadjacency [$0$ in the usual adjacency matrix $A$], positive and negative adjacency [$+1$, $-1$ in $A$] and simultaneous positive and negative adjacency [$0$ or indeterminate in $A$].  
They define an algebra (i.e., associative, commutative, and distributive addition and multiplication) of these symbols (p.\ 8): $o$ acts as $0$, $p$ acts as $1$, $pn=n$, $n^2=p$, $a=p+n$, $x+x=x$ and $ax=a$ for $x \neq 0$.  In the algebra one can decide balance of $\Sigma$ via the permanent of $I + R$: $\Sigma$ is balanced if $\per(I+R) = p$ and unbalanced if $\per(I+R) = a$.  (The ``straightforward but space-consuming'' proof is omitted [and the theorem is not completely correct].  
They state that the permanent cannot equal $n$ or $o$ [but that is an error].)  [See \xref{Harary--Norman--Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} for more on this matrix, and \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, Thm.\ 2.1, for a matrix with more precise counting properties.]  
They introduce a clumsy form of switching in terms of the Hadamard product of $R$ with a ``passive $T$-matrix'' [oversimplifying, that is a matrix obtained by switching the square all-$p$'s matrix; the actual definition involves operators $s$ and $c$ and is more interesting].  Thm.\ 11: Switching preserves balance.  [This was never followed up in signed graphs.  Switching signed graphs became significant with \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}.]
\annotctd{} They propose (p.\ 12) ``complexity'' [= frustration index $l(\Sigma)$] as a measure of imbalance.  [\Cf\ \xref{Harary}{1959b}{Frank Harary}.]  Thm.\ 12:  Switching preserves frustration index.  Thm.\ 14:  $\max l(\Sigma)$, taken over all signed graphs $\Sigma$ of order $n$, equals $\lfloor (n - 1)^2/4 \rfloor$.  (Proof omitted.  [Proved by \xref{Petersdorf}{1966a}{M.\ Petersdorf} and \xref{Tomescu}{1973a}{Ioan Tomescu} for signed $K_n$'s and hence for all signed simple graphs of order $n$.])
\codes{\CPsS}\codesctd{\CSG:~\CAdj,~\CBal,~\Csw,~\CFr}%
\itemspace

\author{Maria Abi Aad}						%ABI AAD	%AAD
\authorspace
\authref  See \hyperlink{Amine El Sahili}{A.~El~Sahili}.
\itemspace

\authorx{Aida Abiad}							%ABIAD
\author{Aida Abiad, Francesco Belardo, \& Antonina P.\ Khramova}
\authorspace
\bsgitem{20xxa}{}  A switching method for constructing cospectral gain graphs.  Submitted. \emph{}\ 
\arXiv{2304.03555}.  
\codes{\CGG:~\CAdj:~\CEig,~\CSw(\CGen)}%
\itemspace

\author{Aida Abiad, Raffaella Mulas, \& Dong Zhang}	
\authorspace
\bsgitem{2021a}{}  Coloring the normalized Laplacian for oriented hypergraphs.  \emph{Linear Algebra Appl.}\ 629 (2021), 192--207.
\MR{4295981}.  
\arXiv{2008.03269}.  
\codes{\CSH:~\CLap:~\CEig}%
\itemspace

\authorx{Pierre Aboulker}						%ABOULKER
\author{Pierre Aboulker, Pierre Charbit, Nicolas Trotignon, \& Kristina Vu\v{s}kovi\'c}
\authorx{Pierre Aboulker, Pierre Charbit, Nicolas Trotignon, \& Kristina Vuskovic}
\authorspace
\bsgitem{2015a}{}  Vertex elimination orderings for hereditary graph classes.  \emph{Discrete Math.}\ 338 (2015), 825--834.
\MR{3303861}.  \Zbl{1306.05202}.  \arXiv{1205.2535}.				
\par
\codes{\Csg:~\CStr,~\CAlgor}%
\itemspace

\author{Pierre Aboulker, Marko Radovanovi\'c, Nicolas Trotignon, Th\'eophile Trunck, \& Kristina Vu\v{s}kovi\'c}
\authorx{Pierre Aboulker, Marko Radovanovic, Nicolas Trotignon, Theophile Trunck, \& Kristina Vuskovic}
\authorspace
\bsgitem{2014a}{}  Linear balanceable and subcubic balanceable graphs.  \emph{J.\ Graph Theory} 75 (2014), no.\ 2, 150--166.
\MR{3150570}.  \Zbl{1280.05056}.  
\codes{\CSG:~\CBal(\CGen)}%
\itemspace

\author{Lowell Abrams}						%ABRAMS
\authorspace
\bsgitem{2017a}{}  Families of fixed-point cellular rotations.  \emph{European J.\ Combin.}\ 63 (2017), 197--215.
\MR{3645794}.  \Zbl{1365.05201}.  
\codes{\CTop:~\CGG}%
\itemspace

\author{Lowell Abrams \& Joanna A.\ Ellis-Monaghan}
\authorspace
\bsgitem{2022a}{}  New dualities from old: Generating geometric, Petrie, and Wilson dualities and trialities of ribbon graphs.  \emph{Combin.\ Probab.\ Comput.}\ 31 (2022), no.\ 4, 574--597.
\MR{4439773}.  
\arXiv{1901.03739}.							
\annot{} ``Ribbon graphs'' (\xref{Bollob\'as and Riordan}{2002a}{Bela Bollobas \& Oliver Riordan}) are orientation-embedded signed graphs (\cf\ \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}).
\codes{\Csg:~\CTop}%
\itemspace

\author{Lowell Abrams \& Daniel Slilaty}
\authorspace
\bsgitem{2015a}{}  The minimal $\bbZ_n$-symmetric graphs that are not $\bbZ_n$-spherical.    \emph{European J.\ Combin.}\ 46 (2015), 95--114. 
\MR{3305348}.  \Zbl{1307.05102}.  
\codes{\CTop:~\CGG,~\CSG}%
\itemspace

\bsgitem{2015b}{}  Cellular automorphisms and self-duality.
\emph{Trans.\ Amer.\ Math.\ Soc.}\ 367 (2015), no.\ 11, 7695--7773.
\MR{3391898}.  \Zbl{6479446}.  
\codes{\CTop:~\CGG,~\CSG}%
\itemspace

\authorx{Marien Abreu}						%ABREU M
\author{M.\ Abreu, M.J.\ Funk, D.\ Labbate, \& V.\ Napolitano}
\authorspace
\bsgitem{2013a}{}  On the ubiquity and utility of cyclic schemes.  \emph{Australasian J.\ Combin.}\ 55 (2013), 95--120.
\MR{3058329}.  \Zbl{1278.05246}.  \arXiv{1111.3265}.				
\codes{\CGG:~\CCov}%
\itemspace

\authorx{Nair Maria Maia de Abreu}					%ABREU NMM de
\authorx{Nair Abreu}							%ABREU N
\author{Nair Maria Maia de Abreu [Nair Abreu]}
\authorspace
\authref  See also \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de~Freitas}, \hyperlink{Leonardo Silva de Lima}{L.S.~de~Lima}, \hyperlink{Aroldo Oliveira}{A.~Oliveira}, and \hyperlink{Carla Silva Oliveira}{C.S.~Oliveira}.
\authrefspace

\author{Nair Abreu, Domingos M.\ Cardoso, Ivan Gutman, Enide A.\ Martins, \& Mar{\'\i}a Robbiano}
\authorx{Nair Abreu, Domingos M.\ Cardoso, Ivan Gutman, Enide A.\ Martins, \& Maria Robbiano}
\authorspace
\bsgitem{2011a}{}  Bounds for the signless Laplacian energy.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2365--2374.
\MR{2811121} (2012f:05164).  \Zbl{1222.05143}.  
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Nair M.M.\ Abreu, Domingos M.\ Cardoso, Enide A.\ Martins, Maria Robbiano, \& B.\ San Mart{\'\i}n}
\authorx{Nair M.M.\ Abreu, Domingos M.\ Cardoso, Enide A.\ Martins, Maria Robbiano, \& B.\ San Martin}
\authorspace
\bsgitem{2012a}{}  On the Laplacian and signless Laplacian spectrum of a graph with $k$ pairwise co-neighbor vertices.  \emph{Linear Algebra Appl.}\ 437 (2012), 2308--2316.
\MR{2954492}.  \Zbl{1247.05132}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Nair Maria Maia de Abreu \& Vladimir Nikiforov}
\authorspace
\bsgitem{2012a}{}  Maxima of the $Q$-index: abstract graph properties.  \emph{Electronic J.\ Linear Algebra} 23 (2012), art.\ 55, 782--789.
\MR{2992393}.  \Zbl{1252.05116}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2013a}{}  Maxima of the $Q$-index: graphs with bounded clique number.  \emph{Electronic J.\ Linear Algebra} 26 (2013), art.\ 9, 121--130.
\MR{3065852}.  \Zbl{1282.05165}.  \arXiv{1308.1653}.				
\annot{} The largest eigenvalue of $\Lap(-\Gamma)$ is $\leq 2n(1-1/\omega)$ ($\omega=$ clique number), with equality if $\Gamma$ is complete $\omega$-partite and regular.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Seyed Ebrahim Abtahi}						%ABTAHI
\authorspace
\authref  See \hyperlink{Pouya Esmailian}{P.~Esmailian}.
\itemspace

\author{B.\ Devadas Acharya [Belmannu Devadas Acharya]} 			%ACHARYA BD
\authorx{B.\ Devadas Acharya}
\authorspace
\authref  See also \hyperlink{Mukhtiar Kaur Gill}{\hyperlink{Mukhtiar Kaur Gill}{M.K.~Gill}}, \hyperlink{S.B.\ Rao}{S.B.~Rao}, \hyperlink{Deepa Sinha, Ayushi Dhama, \& B.D.\ Acharya}{D.~Sinha}, and \hyperlink {H.B.\ Walikar} {H.B.~Walikar}.
\authrefspace

\bsgitem{1973a}{} On the product of $p$-balanced and $l$-balanced graphs.  \emph{Graph Theory Newsletter} 2 (Jan., 1973), no.\ 3, Results Announced No.\ 1.
\codes{\CSG,~\CVS:~\CBal}%
\itemspace

\bsgitem{1979a}{}  New directions in the mathematical theory of balance in cognitive organizations.  
MRI Tech.\ Rep.\ No.\ HCS/DST/409/76/BDA (Dec., 1979).  
Mehta Research Institute of Math.\ and Math.\ Physics, Allahabad, 1979.  
\par
\codes{\CSG,~\CSD:~\CBal,~\CAdj,~\CRef}\codesctd{\CPsS:~\CExp,~\CRef}%
\itemspace

\bsgitem{1979b}{}  A programme logic for listing of sigraphs, their characteristic polynomials, and their spectra.  \emph{Graph Theory Newsletter} 9 (1979), no.\ 2, 1.
\annot{} Abstract of a plan for computation.
\codes{\CSG:~\CAdj:~\CAlgor}%
\itemspace

\bsgitem{1980a}{\dag}  Spectral criterion for cycle balance in networks.  \emph{J.\ Graph Theory} 4 (1980), 1--11.  
\MR{0558448} (81e:05097) (\qv).  \Zbl{445.05066}.					
\annot{} A signed simple graph $\Sigma$ with positive edge weights $w$ is balanced iff $A(\Sigma,w)$ has the same spectrum as $A(|\Sigma|,w)$.  A weighted, signed simple digraph $(\vec\Gamma,\sigma,w)$ is cycle balanced (every directed cycle is positive) iff $A(\vec\Gamma,\sigma,w)$ has the same spectrum as $A(\vec\Gamma,w)$.  [Improved for connected, unweighted signed graphs in \xref{Stani\'c}{2019c}{Zoran Stanic}.]
\annotctd{} Proposed measure of imbalance:  the proportion of corresponding coefficients where the characteristic polynomials $p(A(\Sigma);\lambda)$ and $p(A(|\Sigma|);\lambda)$ differ.  [See \xref{M.K.\ Gill}{1981b}{Mukhtiar Kaur Gill}.]
\dated{rev.\ 4 Apr 2012, 30 Nov 2014, 19 Dec 2020]}
\codes{\CSD,~\CSG:~\CBal,~\CAdj}%
\itemspace

\bsgitem{1980b}{}  An extension of the concept of clique graphs and the problem of $K$-convergence to signed graphs.  
\emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 3 (1980), 239--242.  
\Zbl{491.05052}.  
\codes{\CSG:~\CLG,~{\bf Clique~graph}}%
\itemspace

\bsgitem{1980c}{}  Applications of sigraphs in behavioural sciences.  M.R.I.\ Tech.\ Rep.\ No.\ DST/\-HCS/\-409/\-79 (June, 1980).  Mehta Research Institute of Math.\ and Math.\ Physics, Allahabad, 1979.  
\annot{} [Annotation is very incomplete.]  
Let $\Sigma_1 \vee \Sigma_2$ be the join of underlying graphs, with edge signs $\{\pm1\}$ as in $\Sigma_1 \cup \Sigma_2$ and with $\sigma(v_1v_2) := \max(\mu_1(v_1),\mu_2(v_2))$, where $\mu(v) := \prod_{vw \in E} \sigma(vw)$.  
\dated{20 July 2009} 
\codes{\CSG}
\itemspace

\bsgitem{1981a}{} On characterizing graphs switching equivalent to acyclic graphs.  \emph{Indian J.\ Pure Appl.\ Math.}\ 12 (1981), 1187--1191.  
\MR{0634306} (82k:05089).  \Zbl{476.05069}.					
\annot{} Begins an attack on the problem of characterizing by forbidden induced subgraphs the simple graphs that switch to forests.  Among them are $K_5$ and $C_n$, $n \geq 7$.  \Problem.  Find any others that may exist.  [Solved by \xref{Hage and Harju}{2004a}{Jurriaan Hage \& Tero Harju}.  Forests that switch to forests were characterized by \xref{Hage and Harju}{1998a}{Jurriaan Hage \& Tero Harju}.]
\codes{\CTG:~\CSw}%
\itemspace

\bsgitem{1982a}{}  Connected graphs switching equivalent to their iterated line graphs.  \emph{Discrete Math.}\ 41 (1982), 115--122.  
\MR{0676870} (84b:05078).  \Zbl{497.05052}.  
\codes{\CLG,~\CTG}%
\itemspace

\bsgitem{1982b}{}  Even edge colorings of a graph: II.  A lower bound for maximum even edge-coloring index.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 5 (1982), no.\ 3, 97--99.
\par
\codes{\Cbal:~\CGen}%
\itemspace

\bsgitem{1983a}{} Even edge colorings of a graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 35 (1983), 78--79.  
\MR{0723571} (85a:05034).  \Zbl{505.05032}, (\Zbl{515.05030}).
\annot{} Find the fewest colors to color the edges so that in each circle the number of edges of some color is even.  [Possibly, inspired by \S2 of \xref{Acharya and Acharya}{1983a}{B.\ Devadas Acharya \& Mukti Acharya}.]
\codes{\Cbal:~\CGen}%
\itemspace

\bsgitem{1983b}{} A characterization of consistent marked graphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 6 (1983), no.\ 12, 433--440.  
\MR{0884837} (no rev).  \Zbl{552.05052}.  							
\annot{} Converts a vertex-signed graph $(\Gamma, \mu)$ into a signed graph $\Sigma$ such that $(\Gamma, \mu)$ is consistent (as in \xref{Beineke and Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}) iff every circle in $\Sigma$ is all negative or has an even number of all-negative components.  
[See \xref{Joglekar, Shah, and Diwan}{2010a}{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan} for the definitive result on consistency.]
\codes{\CVS,~\CSG:~\Cbal}%
\itemspace

\bsgitem{1984a}{}  Some further properties of consistent marked graphs.  \emph{Indian J.\ Pure Appl.\ Math.}\ 15 (1984), 837--842.  
\MR{0757960} (86a:05101).  \Zbl{552.05053}.
\annot{} Notably: nicely characterizes consistent vertex-signed graphs in which the subgraph induced by negative vertices is connected.  [Subsumed by \xref{S.B.\ Rao}{1984a}{S.B.\ Rao}.]
\codes{\CVS:~\Cbal}%
\itemspace

\bsgitem{1984b}{}  Combinatorial aspects of a measure of rank correlation due to Kendall and its relation to social preference theory.  
In:  B.D.\ Acharya, ed., \emph{Proceedings of the National Symposium on Mathematical Modelling} (Allahabad, 1982).  M.R.I.\ Lect.\ Notes Appl.\ Math., No.\ 1.  Mehta Research Institute of Math.\ and Math.\ Physics, Allahabad, India, 1984.
\annot{} Includes an exposition of \xref{Sampathkumar and Nanjundaswamy}{1973a}{E.\ Sampathkumar \& L.\ Nanjundaswamy}.
\par
\codes{\CSG:~\CKG:~\CExp}%
\itemspace

\bsgitem{1985a}{}  \emph{Signed Graphs With Applications in Behavioural Sciences}.  M.R.I.\ Lect.\ Notes Appl.\ Math., No.\ 3.  Mehta Research Institute of Math.\ and Math.\ Physics, Allahabad, 1985.
\codes{\CSG:~\CPsS}%
\itemspace

\bsgitem{1986a}{}  An extension of Katai-Iwai procedure to derive balancing and minimum balancing sets of a social system.  \emph{Indian J.\ Pure Appl.\ Math.}\ 17 (1986), 875--882.  
\MR{0851878} (87k:92037).  \Zbl{612.92019}.						
\annot{} Expounds the procedure of \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai}.  Proposes a generalization to those $\Sigma$ that have a certain kind of circle basis.  Construct a ``dual'' graph whose vertex set is a circle basis supplemented by the sum of basic circles.  A ``dual'' vertex has sign as in $\Sigma$.  Let $T =$ set of negative ``dual'' vertices.  A $T$-join in the ``dual'', if one exists, yields a negation set for $\Sigma$.  
[A minimum $T$-join need not yield a minimum negation set.  Indeed the procedure is unlikely to yield a minimum negation set (hence the frustration index $l(\Sigma)$) for all signed graphs, since it can be performed in polynomial time while $l(\Sigma)$ is NP-complete.  \Questions.  To which signed graphs is the procedure applicable?  For which ones does a minimum $T$-join yield a minimum negation set?  Do the latter include all those that forbid an interesting subdivision or minor (\cf\ \xref{Gerards and Schrijver}{1986a}{A.M.H.\ Gerards \& A.\ Schrijver}, \xref{Gerards}{1988a}{A.M.H.\ Gerards}, \xrefd{1989a}{A.M.H.\ Gerards})?]
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2009a}{}  Role of cognitive balance in some notions of graph labelings: Influence of Frank Harary, Fritz Heider, Gustav Kirchhoff and Leonhard Euler.  \emph{Bull.\ Allahabad Math.\ Soc.}\ 24 (2009), no.\ 2, 391--413.
\MR{2597634} (no rev).  \Zbl{1221.05278}.						
\par
\codes{\CSG,~\CSD:~\CBal,~\Csw}%
\itemspace

\bsgitem{2010a}{}  Signed intersection graphs.  \emph{J.\ Discrete Math.\ Sci.\ Cryptography} 13 (2010), no.\ 6, 553--569.
\MR{2791608} (2011m:05129).  \Zbl{1217.05170}.					
\annot{} Signed hypergraph: hypergraph $H=(X,E)$ with $\sigma_H: E \to \{+,-\}$).  Canonical marking $\mu_{\sigma_H}(x) := \prod_{e \ni x} \sigma_H(e)$ ($x \in X$).  
Intersection edge sign $\sigma_\Omega(ef) := \prod_{x \in e \cap f} \mu_{\sigma_H}(x)$.  The signed intersection graph $\Omega(H,\sigma)$ is the intersection graph of $H$ with signature $\sigma_\Omega$.
Main example:  Maximal-clique hypergraph $\cK(\Xi)$ of a signed graph $\Xi$ with $X =$ \{maximal cliques of $|\Xi|\}$, signature $\sigma_\cK(Q) := \prod_{v \in Q} \mu_\sigma(Q)$ for a max clique $Q$.  Which signed graphs are $\Omega(\cK(\Xi))$?  Thm.\ 3.3:  $\Sigma$ is a maxclique signed graph iff it has an edge clique cover with the Helly property, whose members induce homogeneously signed subgraphs, an even number of which are all-negative.
\annotctd{} On orbits of the operator $\cK$:  Thm.\ 5.1:  $\cK^m(\Sigma) = \cK^n(\Sigma)$ iff $\cK^m(|\Sigma|) = \cK^n(|\Sigma|)$, $\exists\ m<n$.  
However (\S7), $m=0$ ($\Sigma$ is ``$\cK$-periodic) may hold for $|\Sigma|$ but not $\Sigma$.  \Problem\ 7.2.  Characterize $\cK$-periodic signed graphs.
\dated{28 Aug 2010}
\codes{\CSH,~\CSG:~\Clg}%
\annotctd{} \S8, ``Signed line graphs'':  Taking edges instead of max cliques defines a line graph $\Lambda_\bullet(\Sigma)$ with signature $\sigma_\bullet(ef) := \mu_\sigma(e \cap f)$ (due to \xref{M.\ Acharya [M.K.\ Gill]}{1982a}{Mukhtiar Kaur Gill}, \xref{Acharya and Acharya}{2015a}{B.\ Devadas Acharya \& Mukti Acharya}).
\dated{28 Aug 2010} 
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2010b}{}  Mathematical chemistry:  Basic issues.  In:  \emph{Graph Theory Applied to Chemistry} (Proc.\ Nat.\ Workshop, Pala, Kerala, India, 2010), Ch.\ 2.2, pp.\ 26--46.
\annot{} \S2.2.9, ``Newer vistas'':  Signed hypergraphs, signed semigraphs.
\dated{31 Aug 2010}
\codes{\CSG:~\CGen,~\CSH:~\CExp}%
\itemspace

\bsgitem{2011a}{}  On notions generalizing combinatorial graphs, with emphasis on linear symmetric dihypergraphs.  \emph{Bull.\ Allahabad Math.\ Soc.}\ 26 (2011), no.\ 2, 229--258.
\MR{2984886} (no rev).  \Zbl{1257.05054}.  
\annot{} Many generalizations of graphs and digraphs.  Mainly historical and expository.
\dated{31 Jan 2012} 
\par
\codes{\CSG,~\CSD,~\CGen:~\CExp,~\CRef}\codesctd{\CSG,~\CSD,~\CGen}%
\itemspace

\bsgitem{2012a}{}  Set-valuations of a signed digraph.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 145--167.
\Zbl{1301.05155}.  
\codes{\CSD,~\CSG}%
\itemspace

\bsgitem{2012b}{}  Minus domination in a signed graph.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 333--343.
\Zbl{1300.05118}.  
\annot{} Unlike with graphs, not every signed graph admits a minus domination function, i.e., $\mathbf{f} \in \{0,\pm1\}^V$ such that $(I+A(\Sigma))\mathbf{f} \geq \mathbf1$.  [Continued in \xref{Shreyas and Joseph}{2020a}{S.R.\ Shreyas \& M.\ Joseph}.  \Cf\ \xref{Walikar, Motammanavar, and Acharya}{2015a}{H.B.\ Walikar, Satish V.\ Motammanavar, \& B.D.\ Acharya} for signed domination.]
\dated{18 May 2018}  thru
\codes{\CSG:~\CDom:~\CLab}%
\itemspace

\bsgitem{2013a}{}  Domination and absorbance in signed graphs and digraphs I.  Foundations.  \emph{J.\ Combin.\ Math.\ Combin.\ Computing} 84 (2013), 5--20.
\MR{3076750}.  \Zbl{1274.05205}.  
\annot{} $D \subseteq V$ dominates $\Sigma$ if $D$ dominates $|\Sigma|$ and $\bigcup_{u\in D} E(u)$ is balanced.  Domination and absorbance in a signed digraph are similar, using only outgoing, resp.\ incoming, arcs at $D$.  Special attention to independent dominating and absorbing sets.
\dated{22 Apr 2023} 
\par
\codes{\CSG,~\CSD:~\CDom}%
\itemspace

								
\author{B.\ Devadas Acharya \& Mukti Acharya [M.K.\ Gill]}
\authorx{B.\ Devadas Acharya \& Mukti Acharya}
\authorspace
\bsgitem{1983a}{}  A graph theoretical model for the analysis of intergroup stability in a social system.  Manuscript, 1983.  
\annot{} The first half (most of \S1) was improved and published as \xrefd{1986a}{B.\ Devadas Acharya \& Mukti Acharya}.
\annotctd{} The second half (\S\S2--3) appears to be unpublished.  Given: a graph $\Gamma$, a vertex signing $\mu$, and a covering $\cF$ of $E(\Gamma)$ by cliques of size $\leq 3$.  Define a signed graph $S$ by $V(S) = \cF$ and $QQ' \in E(S)$ when at least half the elements of $Q$ or $Q'$ lie in $Q \cap Q'$; sign $QQ'$ negative iff there exist vertices $v \in Q \backslash Q'$, and $w \in Q' \backslash Q$ such that $\mu(v) \neq \mu(w)$.  Suppose there is no edge $QQ'$ in which $\#Q = 3$, $\#Q' = 2$, and the two members of $Q \backslash Q'$ have differing sign.  [This seems a very restrictive supposition.]  
Main result (Thm.\ 7):  $S$ is balanced.  The definitions, but not the theorem, are generalized to multiple vertex signs $\mu$, general clique covers, and clique adjacency rules that differ slightly from that of the theorem.
\par
\codes{\CGG,~\CVS,~\CSG:~\CBal}%
\itemspace

\bsgitem{1986a}{}  New algebraic models of social systems.  \emph{Indian J.\ Pure Appl.\ Math.}\ 17 (1986), 150--168.  
\MR{0830552} (87h:92087).  \Zbl{591.92029}.a
\annot{} Four criteria for balance in an arbitrary gain graph.  [Also see \xref{Harary, Lindstr\"om, and Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom}.]
\codes{\CGG:~\CBal,~\Csw}%
\itemspace

\bsgitem{2015a}{}  Dot-line signed graphs.  \emph{Ann.\ Pure Appl.\ Math.}\ 10 (2015), no.\ 1, 21--27.
\par
\codes{\CSH:~\CVS,~\CSG:~\CLG(\CGen)}%
\itemspace

\bsgitem{20xxa}{}  Consistent marked hypergraphs.  Submitted.  
\codes{\CSH(\CGen):~\CVS:~\CBal}%
\itemspace

\author{Belmannu Devadas Acharya, Mukti Acharya, \& Deepa Sinha}
\authorspace
\bsgitem{2008a}{}  Cycle-compatible signed line graphs.  \emph{Indian J.\ Math.}\ 50 (2008), no.\ 2, 407--414.
\MR{2517744} (2010h:05142).  \Zbl{1170.05032}.					
\annot{} Characterizes when $\Lambda_{BC}(\Sigma)$, the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph, with vertex signs $\sigma$ is harmonious.  
Dictionary:  ``cycle compatible'' = harmonious (the product of all edge and vertex signs on each circle is positive).
\dated{14 Oct 2009} 
\codes{\CSG,~\CVS:~\CLG:~\CBal}%
\itemspace

\bsgitem{2009a}{}  Characterization of a signed graph whose signed line graph is $S$-consistent.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ (2) 32 (2009), no.\ 3, 335--341.
\MR{2562172} (2010m:05135).  \Zbl{1176.05032}.					
\annot{} Let $\Sigma$ be a signed simple graph.  Thm.\ 2.1:  The line graph $\Lambda(|\Sigma|)$, with vertex signs $\sigma$, is consistent (as in \xref{Beineke--Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}) iff $\Sigma$ is balanced and, in $\Sigma$, a vertex of degree $\geq 4$ has only positive edges, while a trivalent vertex $v$ with negative edges has two such edges, which lie in every circle on $v$.
[\xref{Slilaty and Zaslavsky}{2015a}{Daniel C.\ Slilaty \& Thomas Zaslavsky} have a constructive approach.  \xref{Sinha and Acharya}{2016a}{Deepa Sinha \& Mukti Acharya} generalize to iterated line graphs.]
[Cor.\ 1:  A positive edge at a vertex with two negative edges is an isthmus.  Cor.\ 2:  Let $\Sigma$ be 2-connected.  $(\Lambda(|\Sigma|),\sigma)$ is consistent iff $\Sigma$ is balanced and every negative edge has endpoints of degree $\leq 2$.  \Problem.  Find a structural characterization, by means of which all such $\Sigma$ can be constructed.  \xref{Zaslavsky}{2016a}{Thomas Zaslavsky} has solutions.]  
\dated{2 Oct 2009, rev 15 Oct, 3 Nov 2013, 23 Jan 2014}
\codes{\CSG,~\CVS:~\CLG:~\CBal}%
\itemspace

\author{B.D.\ Acharya, M.K.\ Gill, \& G.A.\ Patwardhan}
\authorspace
\bsgitem{1984a}{}  Quasicospectral graphs and digraphs.
In: B.D.\ Acharya, ed., \emph{Proceedings of the National Symposium on Mathematical Modelling} (Allahabad, 1982), pp.\ 133--144.   
M.R.I.\ Lect.\ Notes Appl.\ Math., No.\ 1.  Mehta Research Institute of Math.\ and Math.\ Physics, Allahabad, 1984.  
\MR{0766920} (86c:05087).  \Zbl{556.05048}.						
\annot{} Continues \xref{M.K.\ Gill}{1981a}{Mukhtiar Kaur Gill}.  A signed graph, or digraph, is ``cycle-balanced'' if every circle, or every cycle, is positive.  Graphs, or digraphs, are ``quasicospectral'' if they have cospectral signings, ``strictly quasicospectral'' if they are quasicospectral but not cospectral, ``strongly cospectral'' if they are cospectral and have cospectral cycle-unbalanced signings.  
There exist arbitrarily large sets of strictly quasicospectral digraphs, which moreover can be assumed strongly connected, weakly but not strongly connected, etc.  There exist pairs of unbalanced, strictly quasicospectral graphs; existence of larger sets is unsolved.  
There exist arbitrarily large sets of nonisomorphic, strongly cospectral connected graphs; also, of weakly connected digraphs, which moreover can be taken to be strongly connected, unilaterally connected, etc.  Proofs, based on ideas of A.J.\ Schwenk, are sketched.
\codes{\CSD,~\CSG:~\CEig}%
\itemspace

\author{Belmannu Devadas Acharya \& Shalini Joshi}
\authorspace
\bsgitem{2003a}{}  On the complement of an ambisidigraph.  [Abstract.]
Proc.\ R.C.\ Bose Centenary Sympos.\ Discrete Math.\ Appl.\ (Kolkata, 2002).  \emph{Electronic Notes Discrete Math.}\ 15 (2003), 5.
\MR{2159023} (no rev).  \Zbl{1184.05100}.					
\annot{} The complement of a signed digraph $D$ without loops or multiple signed arcs (a loopless, simply signed digraph, or ``ambisidigraph'') is defined in the obvious way.  Observation: If $D$ or $D^c$ contains a directed cycle of length $2k+1$, then one of them contains a positive such cycle.
\codes{\CSD}%
\itemspace

\bsgitem{2004a}{}  Semibalance in signed digraphs.  In:  \emph{Proceedings of the International Conference on Recent tRends and New Directions of REsearch in Cybernetics and Systems} (Inst.\ Adv.\ Study Sci.\ and Technology, Guwahati, India, 2004).  [2004?].
\codes{\Csg:~\CSD:~\CBal}%
\itemspace

\bsgitem{2005a}{}  Mathematical modelling in social psychology--social networks.  \emph{Everyman's Science}\ 40 (2005), no.\ 2, 124--128.
\annot{} Popular exposition including ambisidigraphs (\cf\ \xrefd{2003a}{Belmannu Devadas Acharya \& Shalini Joshi}).
\dated{7 Apr 2012}
\codes{\CSD:~\CExp}%
\itemspace

\bsgitem{2011a}{}  Some reflections on discrete mathematical models in behavioral, cognitive and social sciences.  In:  Johan van Benthem, Amitabha Gupta, and Rohit Parikh, eds., \emph{Proof, Computation and Agency: Logic at the Crossroads}, Ch.\ 11, pp.\ 277--307.  Synth\`ese Library, Vol.\ 352.  Springer Netherlands, 2011.
\MR{3290119} (no rev).  \Zbl{1315.91015}.  
\codes{\CPsS:~\CSG}%
\itemspace

\author{B.D.\ Acharya, S.\ Joshi, A.R.\ Rao, \& S.B.\ Rao}	
\authorspace
\bsgitem{2003a}{}  A Ramsey theorem for strongly connected ambisidigraphs.  Manuscript, 2003.
\annot{} Sequel to \xref{Acharya and Joshi}{2003a}{Belmannu Devadas Acharya \& Shalini Joshi}.  For which loopless, simply signed digraphs $D$ do both $D$ and $D^c$ contain no positive 3-cycle?  Thm.: If strongly connected, $D$ has order $<6$.  An attempt to use this to describe all loopless, simply signed digraphs that contain no positive 3-cycle.
\par
\codes{\CSD:~\CStr}%
\itemspace

\author{Mukti Acharya [Mukhtiar Kaur Gill]}				%ACHARYA M
\authorx{Mukti Acharya}
\authorspace
\authref  See also \hyperlink{B.\ Devadas Acharya}{B.D.~Acharya}, \hyperlink{Divya Antoney}{D.~Antoney}, \hyperlink{Mukhtiar Kaur Gill}{\hyperlink{Mukhtiar Kaur Gill}{M.K.~Gill}}, \hyperlink{Rashmi Jain}{R.~Jain}, \hyperlink{Prajakta Bharat Joshi}{P.B.~Joshi}, \hyperlink{Anisha Jean Mathias}{A.J.~Mathias}, \hyperlink{Pranjali}{Pranjali}, \hyperlink{S.B.\ Rao}{S.B.~Rao}, and \hyperlink{Deepa Sinha}{D.~Sinha}. 
\authrefspace

\bsgitem{1988a}{}  Switching invariant three-path signed graphs.  
In: M.N.~Gopalan and G.A.~Patwardhan, eds., \emph{Optimization, Design of Experiments and Graph Theory} (Proc.\ Sympos.\ in Honour of Prof.\ M.N.~Vartak, 
Bombay, 1986), pp.\ 342--345.  Indian Inst.\ of Technology, Bombay, 1988.
\MR{0998809} (90b:05102).  \Zbl{744.05054}.						
\annot{} See \xref{Gill and Patwardhan}{1986a}{M.K.\ Gill \& G.A.\ Patwardhan} for the $k$-path signed graph of $\Sigma$.  The equation $\Sigma \simeq D_3(\Sigma)$ is solved.
\dated{29 Apr 2009}
\codes{\CSG,~\CSw}%
\itemspace

\bsgitem{2009a}{}  $\times$-line signed graphs.  
Int.\ Conf.\ Recent Developments Combin.\ Graph Theory (Krishnankoil, Tamil Nadu, India, 2007).  
\emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 69 (2009), 103--111.
\MR{2517311} (no rev).  \Zbl{1195.05031}.						
\annot{$\Lambda_{\times}$} $\Lambda_{\times}(\Sigma) := (\Lambda(|\Sigma|),\sigma_{\times})$ where $\sigma_{\times}(ef) := \sigma(e)\sigma(f)$.  (Contrast with line graphs of \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand}, or \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}, or M.\ Acharya in \xref{B.D.\ Acharya}{2010a}{B.\ Devadas Acharya}.)  
[The definition originated in \xref{M.K.\ Gill}{1982a}{Mukhtiar Kaur Gill}.  Publication of this article was delayed by many years.]
\dated{rev.\ 20 Dec 2010, 1 Sept 2012} 
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2010a}{}  Square-sum graphs:  Some new perspectives.  
In:  \emph{International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTGC-2010)} (Cochin, 2010) [Summaries], pp.\ 114--119.  
Dept.\ of Mathematics, Cochin Univ.\ of Science and Technology, 2010.
\annot{} P.\ 119:  Summary of $k$-square-sum signed graphs, where $k$ edges classes are square-sum with the same vertex labels.  $k=2$ is signed graphs.
\dated{30 Aug 2010}
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2012a}{}  Quasicospectrality of graphs and digraphs:  A creative review.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 241--256.
\Zbl{1301.05207}.  
\annot{} Graphs or digraphs are quasicospectral if they have cospectral signatures (signatures with the same adjacency spectrum).  Properties and examples of quasicospectral graphs and digraphs that are not cospectral.  Definitions and results from \xref{B.D.\ Acharya--Gill--Patwardhan}{1984a}{B.D.\ Acharya, M.K.\ Gill, \& G.A.\ Patwardhan} \etal, as well as new results.
\dated{4 Apr 2012} 
\par
\codes{\CSG,~\CSD:~\CEig:~\CExp}\codesctd{\CSG,~\CSD:~\CEig}%
\itemspace

\bsgitem{2017a}{}  $C$-cordial labeling in signed graphs.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 11--22.
\MR{3754786}.  \Zbl{1383.05276}.  
\par
\codes{\CVS:~\CSG,~\CSH:~\CLab}%
\itemspace

\bsgitem{2017b}{}  Total $C$-cordial labeling in signed graphs -- I.  \emph{Indian J.\ Discrete Math.}\ 3 (2017), no.\ 2, 85--96.
\MR{4317606}.  
\codes{\CVS:~\CSG:~\CLab}%
\itemspace



\author{Mukti Acharya, Rashmi Jain, \& Sangita Kansal}
\authorspace
\bsgitem{2015a}{}  Results on lict signed graphs $L_c(S)$.  \emph{J.\ Discrete Math.\ Sci.\ Cryptography} 18 (2015), no.\ 6, 727--742.
\MR{3435215} (no rev).  
\annot{} ``Lict'' = line-cutpoint, an extension of the line graph.  $V(\Sigma)$ is signed by $\mu_\sigma$, the canonical vertex signature.  $L_c(S)$ or rather $\Lambda_c(\Sigma)$ has vertex set $E\cup C$ where $C= \{\text{cutpoints of }\Sigma\}$, edge $uv$ iff $u,v$ are adjacent or incident, and $uv$ negative iff $u$ and $v$ are negative.  $\Lambda_c{:}E=\Lambda_{BC}(\Sigma)$, the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph.  
Characterized: (1) The signed $K_n$, $C_n$, and $K_{r,s}$ that are $\Lambda_{BC}(\Sigma)$ or $\Lambda_c(\Sigma)$.  (2) $\Sigma$ such that $\Lambda_{BC}(\Sigma)$ or $\Lambda_c(\Sigma)$ is balanced.  (3)  $\Sigma$ that are switching equivalent to $\Lambda_{BC}(\Sigma)$ or $\Lambda_c(\Sigma)$ or their negatives.
\dated{8 Jan 2016} 
\codes{\CSG:~\CLG(\CGen):~\CBal,~\CKG}%
\itemspace

\bsgitem{2016a}{}  On $\bullet$-lict signed graphs $L_{\bullet_c}(S)$ and $\bullet$-line signed graphs $L_\bullet(S)$.  \emph{Trans.\ Combin.}\ 5 (2016), no.\ 1, 37--48.
\MR{3462889}.  \Zbl{1463.05234}.  
\codes{}%
\itemspace

\bsgitem{2017a}{}  Vertex equitable labeling of signed graphs.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 461--468.
\MR{3754836}.  \Zbl{1383.05139}.  
\codes{\CSG:~\CLab}%
\itemspace

\author{Mukti Acharya \& Joseph Varghese Kureethara}
\authorspace
\bsgitem{2021a}{}  Parity labeling in signed graphs.  \emph{J.\ Prime Res.\ Math.}\ 17 (2021), no.\ 2, 1--7.
\MR{4307053}.  
\arXiv{2012.07737}.					
\annot{} Parity labeling:  $f: V \leftrightarrow [n]$ such that $\sigma(uv) = (-1)^{f(u)+f(v)}.$  [I.e., $E^- =$ cut with nearly equal sides.]  \emph{rna} number $:= \min |E^-|$ over all parity signatures.  Minor results.  \Cf\ \xref{Acharya--Kureethara--Zaslavsky}{2021a}{Mukti Acharya, Joseph Varghese Kureethara, \& Thomas Zaslavsky}, \xref{Sehrawat--Bhattacharjya}{20xxb}{Deepak Sehrawat \& Bikash Bhattacharjya}, and esp.\ \xref{Kang, Chen, and Jin}{2022a}{Yingli Kang, Xiaoyue Chen, \& Ligang Jin}.
\dated{7 Oct 2019} 
\codes{\CLab:~\CSG}%
\itemspace

\author{Mukti Acharya, Joseph Varghese Kureethara, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2021a}{}  Characterizations of some parity signed graphs.  \emph{Australasian J.\ Combin.}\ 81 (2021), no.\ 1, 89--100.
\MR{4312564}.  \Zbl{1483.05071}.  \arXiv{2006.03584}.				
\annot{} Continues \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara}.  Characterizations of parity signed graphs and in more detail parity signed stars, bistars, cycles, paths and complete bipartite graphs.  \emph{rna} numbers for some parity signed graphs.  
More examples in \xref{Ranjith--Kureethara}{2020a}{Athira P Ranjith \& Joseph Varghese Kureethara}, \xref{Reshma--Gayathri--Rajendra}{2021a}{Reshma R, Gayathri H, \& Supriya Rajendra}.
\dated{13 Dec 2020} 
\codes{\CLab:~\CSG}%
\itemspace

\author{Mukti Acharya, Pranjali, Atul Gaur, \& Amit Kumar}
\authorspace
\bsgitem{2017a}{}  Line signed graph of a signed total graph.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 389--397.
\MR{3754828}.  \Zbl{1383.05140}.  
\codes{\CSG:~\CLG}%
\itemspace

\author{Mukti Acharya \& Pranjali Sharma}
\authorspace
\bsgitem{2016a}{}  Balanced signed total graphs of commutative rings.  \emph{Graphs \& Combin.}\ 32 (2016), no.\ 4, 1585--1597.
\MR{3514985}.  \Zbl{1342.05060}.  
\codes{\CSG:~\CAlgeb:~\CBal}%
\itemspace

\author{Mukti Acharya \& Tarkeshwar Singh}
\authorspace
\bsgitem{2002a}{}  Characterization of sigraphs whose negations are switching equivalent to their iterated line sigraphs.  Manuscript, 2002(?).
\annot{} The signed simple graphs $\Sigma$ (which necessarily are signed circles) such that $-\Sigma$ is switching isomorphic to any of its iterated \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graphs.
\dated{20 July 2009} 
\codes{\CSG:~\CSw,~\CLG}%
\itemspace

\bsgitem{2003a}{}  Graceful signed graphs: III, \ The case of signed cycles in which the negative sections form a maximum matching.  
\emph{Graph Theory Notes N.Y.}\ 45 (2003), 11--15.
\MR{2040207} (no rev).  
\annot{} See \xrefd{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  Here the graph is a circle and the second color class is a maximum matching.
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2003b}{}  Skolem graceful signed graphs.  
Proc.\ R.C.\ Bose Centenary Sympos.\ Discrete Math.\ Appl.\ (Kolkata, 2002).  \emph{Electronic Notes Discrete Math.}\ 15 (2003), 10--11.
\MR{2159025} (no rev).  \Zbl{1184.05108}.					
\annot{} Announcement of \xrefd{2010a}{Mukti Acharya \& Tarkeshwar Singh}.
\codes{\CSGc:~\CLab:~\CExp}%
\itemspace

\bsgitem{2003c}{}  Construction of certain infinite families of graceful sigraphs from a given graceful sigraph.  Manuscript, 2003.
\annot{} Let $\vee$ denote the join of graphs or (defined in \xref{B.D.\ Acharya}{1980c}{B.\ Devadas Acharya}) signed graphs.  Thms.:  If $\Sigma$ is gracefully numbered, so are $\Sigma \cupdot\, K_t^c$ and $\big(\Sigma \cupdot\; K_{\#E-\#V+1}^c\big) \vee K_t^c$.  All $(K_2\vee K_r^c,\sigma)$ are gracefully numbered.  
\dated{20 July 2009} 
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2004a}{}  Graceful signed graphs.  \emph{Czechoslovak Math.\ J.}\ 54(129) (2004), no.\ 2, 291--302.
\MR{2059251} (2005a:05193).  \Zbl{1080.05529}.					
\annot{} [Generalizing the definition in the article:  Given: a graph with $r$-colored edges; integers $k, d > 0$.  Required: a \emph{$(k,d)$-graceful labelling}, i.e., an injection $\lambda: V \to \{0,1,\ldots,k+(\#E-1)d\}$ so that, if $f(vw) := |\lambda(v)-\lambda(w)|$, then $f$ restricted to each color class is injective with range $k, k+d, \ldots$.]  
\annotctd{} The article concerns the case $r=2$ with ``results of our preliminary investigation''.   
\Conjecture.  Every 2-colored circle of length $\geq3$ is $(k,d)$-graceful.
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2004b}{}  Graceful signed graphs V.  The case of union of signed cycles of the length three with one vertex in common.  \emph{Int.\ J.\ Management Systems} 20 (2004), no.\ 3, 245--254.  
\annot{} $\Sigma$ is a signed windmill with $k>1$ blades.  Only rim edges may be negative.
Thm.:  $\Sigma$ is graceful $\implies$ $k\equiv0 \mod 4$ and $\#E^-$ is even, or $k\equiv1 \mod 4$, or $k\equiv2 \mod 4$ and $\#E^-$ is odd.
Thm.:  If $k\equiv0,1 \mod 4$ and all rim edges are negative, then $\Sigma$ is graceful.
Thm.:  If $k\equiv2 \mod 4$ and all rim edges but one are negative, then $\Sigma$ is graceful.
Also see \xref{Singh}{2009a}{Tarkeshwar Singh}.
\dated{21 July 2010} 
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2004c}{}  A characterization of signed graphs whose negation is switching equivalent to its iterated line sigraphs.  In: R.J.\ Wilson, R.\ Balakrishnan and G.\ Sethuraman, eds., \emph{Proceedings of the Conference of Graph Theory and Applications} (CGTA-2001), pp.\ 15--24.  Narosa Publishing House, New Delhi, 2004.
\codes{\CSG:~\CSw,~\CLG}%
\itemspace

\bsgitem{2005a}{}  Graceful signed graphs: II.  The case of signed cycles with connected negative sections.  
\emph{Czechoslovak Math.\ J.}\ 55(130) (2005), no.\ 1, 25--40.
\MR{2121654} (2005m:05192).  \Zbl{1081.05097}.					
\annot{} Proof of the conjecture of \xrefd{2004a}{Mukti Acharya \& Tarkeshwar Singh} for a circle of length $\not\equiv1\ (\mod4)$ where the negative edge set is connected.
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2009a}{}  Skolem graceful signed stars.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 69 (2009), 113--124.
\MR{2517312} (2010e:05257).  \Zbl{1195.05065}.					
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2010a}{}  Skolem graceful signed graphs.  \emph{Utilitas Math.}\ 82 (2010), 97--109.
\MR{2663369} (2011h:05218).  \Zbl{1232.05198}.					
\annot{} From \xref{Singh}{2003a}{Tarkeshwar Singh}, Ch.\ III.  See \xrefd{2003b}{Mukti Acharya \& Tarkeshwar Singh}, \xref{Singh}{2008a}{Tarkeshwar Singh}.  ``Skolem gracefulness'' is the $(0,1)$-gracefulness of \xrefd{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  Thm.:  A signed $k$-edge matching is Skolem graceful iff $k \equiv 0 \pmod 4$ and $\#E^-$ is even, or $k \equiv 2 \pmod 4$ and $\#E^-$ is odd, or $k \equiv 1 \pmod 4$.  Curiously complementary to the theorem of \xref{Singh}{2008ba}{Tarkeshwar Singh}.
\dated{20 July 2009}
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2013a}{}  Embedding of signed graphs in graceful signed graphs.  \emph{Ars Combin.}\ 108 (2013), 421--426.
\MR{3112764}.  \Zbl{1313.05162}.
\annot{} See \xrefd{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  Every signed graph whose vertices have distinct nonnegative integral labels is an induced subgraph of a signed graph with $(1,1)$-graceful labels.
\codes{\CSGc:~\CLab}%
\itemspace

\author{Mukti Acharya \& Deepa Sinha}
\authorspace
\bsgitem{2002a}{}  A characterization of signed graphs that are switching equivalent to their jump signed graphs.  
\emph{Graph Theory Notes N.Y.}\ 43 (2002), 7--8.
\MR{1960487} (no rev).  
\par
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2003a}{}  A characterization of sigraphs whose line sigraphs and jump sigraphs are switching equivalent.  
\emph{Graph Theory Notes N.Y.}\ 44 (2003), 30--34.
\MR{2002894}.  
\par
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2003b}{}  A characterization of line sigraphs.  Proc.\ R.C.\ Bose Centenary Sympos.\ Discrete Math.\ Appl.\ (Kolkata, 2002).  \emph{Electronic Notes Discrete Math.}\ 15 (2003), 12.
\MR{2159026} (no rev).  
\annot{} Abstract of \xrefd{2005a}{Mukti Acharya \& Deepa Sinha}.
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2005a}{}  Characterizations of line sigraphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 28 (2005), no.\ 1-2, 31--34.
\MR{2127289} (no rev).  
\annot{} Thm.:  A signed simple graph $\Sigma$ is the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph of a signed graph iff the underlying graph is a line graph and $\Sigma$ is ``sign compatible'' (\xref{Sinha}{2005a}{Deepa Sinha}).
\dated{27 Apr 2009, 12 Oct 2010}
\par
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2006a}{}  Common-edge sigraphs.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 3 (2006), no.\ 2, 115--130.
\MR{2285459} (2007k:05083).  \Zbl{1119.05053}.					
\annot{} The common-edge signed graph $C_E(\Sigma)$ is the second line graph $\Lambda^2(|\Sigma|)$ with signs $\sigma_{C_E}\{ef,fg\} = \sigma(f)$.  
Characterized in whole or part:  When this is balanced (rarely), or isomorphic to $\Sigma$ (rarely), or switching isomorphic to the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph $\Lambda_{BC}(\Sigma)$ (rarely), or switching equivalent to $\Lambda_{BC}^2(\Sigma)$.  There are notions of consistency and compatibility of $C_E(\Sigma)$ with respect to a vertex signature of $\Sigma$, that seem ill defined.	
\codes{\CSG:~\CLG:~\CGen}%
\itemspace

\bsgitem{2013a}{}  Characterization of signed line digraphs.  \emph{Discrete Appl.\ Math.}\ 161 (2013), no.\ 9, 1170--1172.
\MR{3030606}.  \Zbl{1277.05078}.  
\codes{\CSD:~\CLG}%
\itemspace

\author{Sibel Adal\i}							%ADALI
\authorx{Sibel Adali}
\authorspace
\authref  See \hyperlink{Yi Qian}{Y.~Qian}.
\itemspace

\authorx{Gbemisola Adejumo}						%ADEJUMO
\author{Gbemisola Adejumo, P.\ Robert Duimering, \& Zhehui Zhong}
\authorspace
\bsgitem{2008a}{}  A balance theory approach to group problem solving.  \emph{Social Networks} 30 (2008), 83--99.
\codes{\CPsS,~\CSG:~\CFr}%
\itemspace

\authorx{Bibhas Adhikari}						%ADHIKARI
\authorspace
\authref  See also \hyperlink{Ranveer Singh}{R.~Singh}.
\authrefspace

\author{Bibhas Adhikari, Satyabrata Adhikari, \& Subhashish Banerjee}
\authorspace
\bsgitem{2012a}{}  Graph representation of quantum states.  Manuscript, 2012.  
\arXiv{1205.2747}.  
\annot{} A somewhat confusing attempt to model quantum states by a graph with weighted edges and vertices.  There are several systems; in the most general, the edge weights may be complex, the vertex weights real.  The adjacency matrix is Hermitian; for complex units that implies the edge weights are gains, but not so in general.
\dated{13 Jan 2015} 
\par
\codes{\Cpar:~\CAdj}\codesctd{\Cgg(\CGen):~\CAdj}%
\itemspace

\author{Bibhas Adhikari, Amrik Singh, \& Sandeep Kumar Yadav}
\authorspace
\bsgitem{20xxa}{}  Corona product of signed graphs and its application to modeling signed networks.  \emph{Discrete Math.\ Algor.\ Appl.}\ (in press).
\arXiv{1908.10018}.	
\annot{} Defines a corona product.
\codes{\CSG,~\CVS}%
\itemspace

\author{Satyabrata Adhikari}						%ADHIKARI
\authorspace
\authref  See \hyperlink{Bibhas Adhikari}{B.~Adhikari}.
\itemspace

\authorx{Chandrashekar Adiga}						%ADIGA
\author{Chandrashekar Adiga, E.\ Sampathkumar, \& M.A.\ Sriraj}
\authorspace
\bsgitem{2014a}{}  Color energy of a unitary Cayley graph.  \emph{Discuss.\ Math.\ Graph Theory} 34 (2014), 707--721.
\MR{3268686}.  \Zbl{1303.05056}.  
\annot{} Energy of $L$-matrix (\cf\ \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj}) of a colored unitary Cayley graph and a colored gcd-graph.
\dated{14 Oct 2014} 
\par
\codes{\Csgw:~\CAdj:~\CEig}%
\itemspace

\author{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}
\authorspace
\bsgitem{2013a}{}  Color energy of a graph.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 16 (2013), no.\ 3, 335--351.
\MR{3100089}.  \Zbl{1306.05140}.  
\annot{} Definitions at \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj}.  
``Color matrix'' := $A(\Sigma)$; ``color energy'' := energy of $A(\Sigma)$.  Elementary results on characteristic polynomial, eigenvalues of $A(\Sigma)$.  Thm.\ 2.4:  Energy $\leq$ $\sqrt{2n\#E(\Sigma)}$.  ``Complement'' formed by complementing $\Sigma^+$ in $(K_n{:}\pi)^c$.  Examples:  $K_n$, $K_{n,m}$, $C_n$, CP$(n)$, etc., and complements.
[No mention of signed graphs.]  
[2.2, 2.4 for all signed (di)graphs are in \xref{Bhat}{2017a}{Mushtaq A.\ Bhat}.]  
Sequels: \xref{Adiga--Sampathkumar--Sriraj}{2014a}{Chandrashekar Adiga, E.\ Sampathkumar, \& M.A.\ Sriraj}, \xref{Sampathkumar--Pushpa\-latha--Sri\-raj}{2016a}{E.\ Sampathkumar, L.\ Pushpalatha, \& M.A.\ Sriraj}, \etal]  
[For a strange ``signed-graphic'' generalization: \xref{Joshi--Joseph--Acharya}{20xxa}{Prajakta Bharat Joshi, Mayamma Joseph, \& Mukti Acharya}.]
\dated{21 Dec 2018} 
\par
\codes{\Csgw:~\CAdj:~\CEig}%
\itemspace

\author{Chandrashekar Adiga, Shrikanth A.S., \& Shivakumar Swamy C.S.}
\authorspace
\bsgitem{2012a}{}  A note on 1-edge balance index set.  \emph{Int.\ J.\ Math.\ Combin.}\ 2012 (2012), no.\ 3, 113--117.
\Zbl{1276.05102}.  
\annot{} Like \xref{Adiga, Subbaraya, Shrikanth, and Sriraj}{2011a}{Chandrashekar Adiga, C.K.\ Subbaraya, A.S.\ Shrikanth, \& M.A.\ Sriraj} for wheels and certain graphs of Mycielski.
\dated{29 Dec 2015} 
\codes{\Csgw:~\Cvsw:~\CInvar}%
\itemspace

\author{Chandrashekar Adiga, C.K.\ Subbaraya, A.S.\ Shrikanth, \& M.A.\ Sriraj}
\authorspace
\bsgitem{2011a}{}  On 1-edge balance index set of some graphs.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 14 (2011), no.\ 3, 319--331.
\MR{3183856} (no rev).  \Zbl{1226.05209}.  
\annot{} Let $I(\sigma) := \sum_v |d^\pm(v)|$ ($d^\pm=$ net degree).  
The titular index set is $\{ I(\sigma) : \#E^+(\sigma) = \lfloor \frac12\#E\rfloor \text{ or } \lceil \frac12\#E \rceil \}.$  
The set is determined for two kinds of graph.
\dated{29 Dec 2015} 
\annotctd{} [Simplified:  For $S \subseteq E$, $I(S) := \sum_v |d_\Gamma(v)-2d_S(v)|$.  The set is $\{ I(S) : |\#E-2\#S|\leq1 \}$.]
\dated{19 May 2019} 
\codes{\Csgw:~\Cvsw:~\CInvar}%
\itemspace

\bsgitem{2013a}{}  On vertex balance index set of some graphs.  \emph{Bull.\ Iranian Math.\ Soc.}\ 39 (2013), no.\ 4, 627--634.
\MR{3108880}.  \Zbl{1301.05283}.  
\annot{} Given $\zeta: V \to \signs$, set $\sigma(uv) := \zeta(u)\zeta(v)$ and $J(\zeta) := \big| \#E^+(\sigma)-\#E^-(\sigma) \big|.$  
The titular set is $\{ J(\zeta) : \#\zeta\inv(0) = \lfloor \frac12\#V\rfloor \text{ or } \lceil \frac12\#V \rceil \}.$  
The set is determined for four kinds of graph including $K_n$, $K_{r,s}$.  
\dated{29 Dec 2015} 
\annotctd{} [Simplified:  For $X \subseteq V$, $J(X) := |\#E - 2\#E(X,X^c)|$.  The set is $\{ J(X) : |\#V-2\#X|\leq1 \}$.]
\dated{19 May 2019} 
\codes{\Cvsw:~\Csgw:~\CInvar}%
\itemspace

\authorx{L.\ Adler}							%ADLER L
\author{L.\ Adler \& S.\ Cosares}
\authorspace
\bsgitem{1991a}{}  A strongly polynomial algorithm for a special class of linear programs.  \emph{Operations Res.}\ 39 (1991), 955--960.  
\MR{1139965} (92k:90042).  \Zbl{749.90048}.
\annot{} The class is that of the transshipment problem with gains.  Along the way, a time bound on the uncapacitated, demands-only flows-with-gains problem.
\codes{\CGN:~\CInc(\CDual),~\CAlgor}%
\itemspace

\author{Florian Adriaens}						%ADRIAENS
\author{Florian Adriaens \& Simon Apers}
\authorspace
\bsgitem{20xxa}{}  Testing properties of signed graphs.  Submitted.  \emph{}\ 
\arXiv{2102.07587}.  
\par
\codes{\CSG:~\CAlgor:~\CBal,~\CClu}%
\itemspace

\author{S.N.\ Afriat}							%AFRIAT
\authorspace
\bsgitem{1963a}{} The system of inequalities $a_{rs} > X_r - X_s$.  \emph{Proc.\ Cambridge Philos.\ Soc.}\ 59 (1963), 125--133.  
\MR{0141674} (25 \#5071).  \Zbl{118.14901} (118, p.\ 149a).
\annot{} \Cf\ \xref{Roy}{1959a}{Bernard Roy}.
	
\codes{\CGG:~\COG,~\CSw,~\Cbal}%
\itemspace

\bsgitem{1974a}{} On sum-symmetric matrices.  \emph{Linear Algebra Appl.}\ 8 (1974), 129--140.  
\MR{0332838} (48 \#11163).  \Zbl{281.15017}.
\codes{\CGG:~\CSw,~\Cbal}%
\itemspace

\author{Amit Agarwal}						%AGARWAL
\authorspace
\authref  See \xref{Harary, Lim, \etal{}}{2004a}{Frank Harary, Meng-Hiot Lim, Amit Agarwal, \& Donald C.\ Wunsch}.
\itemspace

\authorx{A.A.\ Ageev}						%AGEEV
\author{A.A.\ Ageev, A.V.\ Kostochka, \& Z.\ Szigeti}
\authorspace
\bsgitem{1995a}{}  A characterization of Seymour graphs.  In: Egon Balas and Jens Clausen, eds., \emph{Integer Programming and Combinatorial Optimization} (4th Int.\ IPCO Conf., Copenhagen, 1995), pp.\ 364--372.  \LNCS 920.  Springer, Berlin, 1995.   
\MR{1367995} (96h:05157).  
\annot{} A Seymour graph satisfies with equality a general inequality between $T$-join size and $T$-cut packing.  Thm.:  A graph is not a Seymour graph iff it has a conservative $\pm 1$-weighting such that there are two circles with total weight $0$ whose union is an antibalanced subdivision of $-K_n$ or $-Pr_3$ (the triangular prism).
\codes{\CSGw:~\CStr,~\CBal,~\CPar}%
\itemspace

\bsgitem{1997a}{}  A characterization of Seymour graphs.  \emph{J.\ Graph Theory} 24 (1997), 357--364.
\MR{1437297} (97m:05217).  \Zbl{970.24507}.
\annot{} Virtually identical to \xrefd{1995a}{A.A.\ Ageev, A.V.\ Kostochka, \& Z.\ Szigeti}.
\codes{\CSGw:~\CStr,~\CBal,~\CPar}%
\itemspace

\author{Charu Aggarwal}						%AGGARWAL C
\authorspace
\authref  See \hyperlink{Jiliang Tang}{J.-L.~Tang}.
\itemspace

\author{J.K.\ Aggarwal}						%AGGARWAL JK
\authorspace
\authref   See \hyperlink{M.\ Malek-Zavarei}{M.~Malek-Zavarei}.
\itemspace

\authorx{Kalin Agrawal}						%AGRAWAL K
\author{Kalin Agrawal \& William H.\ Batchelder}
\authorspace
\bsgitem{2012a}{}  Cultural Consensus Theory:  Aggregating signed graphs under a balance constraint.  In:  S.J.\ Yang, A.M.\ Greenberg, and M.\ Endsley, eds., \emph{Social Computing, Behavioral-Cultural Modeling and Prediction} (Proc.\ 5th Int.\ Conf., SBP 2012, College Park, Md.), pp.\ 53--60.  \LNCS 7227.  Springer, Berlin, 2012.
\codes{\CSG:~\CKG,~\CPsS,~\CRand}%
\itemspace

\authorx{Priyanka Agrawal}						%AGRAWAL P
\author{Priyanka Agrawal, Vikas K.\ Garg, \& Ramasuri Narayanam}
\authorspace
\bsgitem{2013a}{}  Link label prediction in signed social networks.  In:  \emph{Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence} (IJCAI '13), pp.\ 2591--2597.  AAAI Press, 2013.
\codes{\CSG:~\CPred,~\CPsS}%
\itemspace

\author{Boris Aguilar}						%AGUILAR
\authorspace
\authref  See \hyperlink{Alan Veliz-Cuba}{A.~Veliz-Cuba}.
\itemspace

\author{F.\ Aguilera-Granja}						%AGUILERA-GRANJA
\authorspace
\authref  See \hyperlink{M.C.\ Salas-Solis}{M.C.~Salas-Sol{\'\i}s}.
\itemspace

\authorx{Ron Aharoni}						%AHARONI
\author{Ron Aharoni, Rachel Manber, \& Bronislaw Wajnryb}
\authorspace
\bsgitem{1990a}{}  Special parity of perfect matchings in bipartite graphs.  \emph{Discrete Math.}\ 79 (1990), 221--228.  
\MR{1044222} (91b:05140).  \Zbl{744.05036}. 
\annot{} When do all perfect matchings in a signed bipartite graph have the same sign product?  Solved.
\codes{\Csg:~\Cbal,~\CAlgor}\codesctd{\Cqm:~\CSol}%
\itemspace

\author{R.\ Aharoni, R.\ Meshulam, \& B.\ Wajnryb}
\authorspace
\bsgitem{1995a}{} Group weighted matchings in bipartite graphs.  \emph{J.\ Algebraic Combin.}\ 4 (1995), 165--171.  
\MR{1323746} (96a:05111).  \Zbl{950.25380}.					
\annot{} Given an edge weighting $w: E \to \fK$ where $\fK$ is a finite abelian group.  Main topic: perfect matchings $M$ such that $\sum_{e \in M} w(e) = 0$ [I'll call them $0$-weight matchings].  (Also, in \S2, $= c$ where $c$ is a constant.)  Generalizes and extends \xref{Aharoni, Manber, and Wajnryb}{1990a}{Ron Aharoni, Rachel Manber, \& Bronislaw Wajnryb}.  [Continued by \xref{Kahn and Meshulam}{1998a}{Jeff Kahn \& Roy Meshulam}.]
\codes{\CGGw}%
\annotctd{} Prop.\ 4.1 concerns vertex-disjoint circles whose total sign product is $+$ in certain signed digraphs.
\codes{\CSD:~\CCircles}%
\itemspace

\author{Amnon Aharony}						%AHARONY
\authorspace
\bsgitem{1978a}{}  Low-temperature phase diagram and critical properties of a dilute spin glass.  \emph{J.\ Phys.\ C}\ 11 (1978), L457--L463.
\annot{} Physics of a random signed subgraph of $\Gamma$: $p, q, r =$ probabilities of $+$, $-$, or no edge.  $r=0$ is a randomly signed $\Gamma$.  $p=0$ is a random subgraph $-\Gamma_1$.  Edges may have weights but the signs are most significant (pp.\ L461--2).  Bipartite graphs (``simple systems, with two sublattices'') give easier results; e.g., switching exchanges $p$ and $q$, and transforms all-negative to all-positive.  Analysis by the replica method: replicate the graph randomly $n$ times.  For temperature $T\to0$:  The case $p=q$ has special properties.  The limit $r \to 0$ gives all-positive (ferromagnetic) behavior because ``only [constant states $\zeta: V \to \signnos$] contribute to the partition function.''  $T>0$:  Special cases for equal weights, similarly to \xref{Houtappel}{1950b}{R.M.F.\ Houtappel}, \xref{Newell}{1950a}{G.F.\ Newell}.  The replica method's limitations include failure at $T\to0$ when the signed subgraph is unbalanced (``frustrated'') (p.\ L463).
[An interesting study.  \Problem.  Interpret the replica method and results in terms of random signed graphs.]
\dated{21 Jun 2012} 
\codes{\CPhys,~\CSG,~\CWG:~\CRand,~\CFr,~\Csw}%
\itemspace

\authorx{Uzma Ahmad}						%AHMAD
\author{Uzma Ahmad, Saieed Akbari, Saira Hameed, Mohammad Ali Nematollahi, \& Faiza Saeed}
\authorspace
\bsgitem{2022a}{}  Addendum to ``Spectral characterizations of signed cycles''.  \emph{Linear Algebra Appl.}\ 651 (2022), 83--89.
\MR{4445166}.  \Zbl{1493.05141}.  
\annot{} Corrects oversights about negative even circles in \xref{Akbari, Belardo, Dodongeh, and Nematollahi}{2018a}{Saieed Akbari, Francesco Belardo, Ebrahim Dodongeh, \& Mohammad Ali Nematollahi}.
\dated{26 Dec 2022} 
\par
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\authorx{Saba Ahmadi}						%AHMADI S
\author{Saba Ahmadi, Samir Khuller, \& Barna Saha}
\authorspace
\bsgitem{2019a}{}  Min-max correlation clustering via multicut.  In:  Andrea Lodi, ed., \emph{Integer Programming and Combinatorial Optimization} (20th Int.\ Conf., IPCO 2019, Ann Arbor, Mich.), pp. 13--26.  \LNCS 11480.  Springer, Cham, 2019. 
\MR{3950874}.  \Zbl{1436.90114}.  \arXiv{907.00117}.  
\annot{} Cluster $\Sigma$ to minimize negative edges within each cluster.
\dated{17 Sept 2022} 
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\authorx{Saeed Ahmadizadeh}						%AHMADIZADEH
\author{Saeed Ahmadizadeh, Iman Shames, Samuel Martin, \& Dragan Ne\v{s}i\'c}
\authorspace
\bsgitem{2017a}{}  On eigenvalues of Laplacian matrix for a class of directed signed graphs.  \emph{Linear Algebra Appl.}\ 523 (2017), 281--306.  
\MR{3624677}.  \Zbl{1369.05132}.  \arXiv{1705.04406}.				
Corrigendum.  \emph{Linear Algebra Appl.}\ 530 (2017), 541--557.
\MR{3672976} (no rev).  \Zbl{1391.05155}.  
\codes{\CSD:~\CLap:~\CEig}%
\itemspace

\author{Luis von Ahn}						%AHN
\authorspace
\bsgitem{2008a}{}  Science of the Web: 15-396.  Networks II: Structural Balance.  Course slides.  URL \url{http://www.scienceoftheweb.org}.
Dept.\ of Computer Science, Carnegie Mellon University.
\annot{} Triangle (``triad'') balance and balance.
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\authorx{Ravindra K.\ Ahuja}						%AHUJA
\author{Ravindra K.\ Ahuja, Thomas L.\ Magnanti, \& James B.\ Orlin}
\authorspace
\bsgitem{1993a}{}  \emph{Network Flows: Theory, Algorithms, and Applications}.  Prentice Hall, Englewood Cliffs, N.J., 1993.  
\MR{1205775} (94e:90035).  
\annot{} \S12.6: ``Nonbipartite cardinality matching problem''.  Nicely expounds theory of blossoms and flowers (\xref{Edmonds}{1965a}{Jack Edmonds}, etc.).  Historical notes and references at end of chapter.
\codes{\Cpar:~\Cori,~\CAlgor:~\CExp,~\CRef}%
\annotctd{} \S5.5: ``Detecting negative cycles'' (i.e., sum $<0$); \S12.7, subsection ``Shortest paths in directed networks''.  Weighted arcs with negative weights allowed.  Techniques for detecting negative cycles and, if none exist, finding a shortest path.
\codes{\COG:~\CAlgor:~\CExp}%
\annotctd{} Ch.\ 16: ``Generalized flows''.  \S15.5: ``Good augmented forests and linear programming bases'', Thm.\ 15.8, makes clear the connection between flows with gains and the frame matroid of the underlying gain graph.  Some terminology:  ``breakeven cycle'' = balanced circle; ``good augmented forest'' = basis of the frame matroid, assuming the gain graph is connected and unbalanced. 
\codes{\CGN:~\CMatrd(\CBases),~\CAlgor:~\CExp,~\CRef}%
\itemspace

\author{Martin Aigner}						%AIGNER
\authorspace
\bsgitem{1979a}{}  \emph{Combinatorial Theory}.  Grundl.\ math.\ Wiss., Vol.\ 234.  Springer-Verlag, Berlin, 1979.  
Reprint:  Classics in Mathematics.  Springer-Verlag, Berlin, 1997.
\MR{0542445} (80h:05002).  \Zbl{415.05001}, \Zbl{858.05001} (reprint).
\annot{} In \S~VII.1, pp.\ 333--334 and Exerc.\ 13--15 treat the Dowling lattices of $\GF(q)^{\times}$ and higher-weight analogs.
\codes{\CGG,~\CGG(\CGen):~\CMatrd:~\CInvar,~\CStr}%
\itemspace

\bsgitem{1982a}{}  (as M.\ A\u{\i}gner)  \emph{Kombinatornaya teoriya}.  ``Mir'', Moscow, 1982.  
\MR{0694072} (84b:05002).  
\annot{} Russian translation of \xref{Aigner}{1979a}{Martin Aigner} by V.V.\ Ermakov and V.N.\ Lyamin.  Ed.\ and preface by G.P.\ Gavrilov.
\par
\codes{\CGG,~\CGG(\CGen):~\CMatrd:~\CInvar,~\CStr}%
\itemspace

\authorx{Nir Ailon}							%AILON
\author{Nir Ailon, Moses Charikar, \& Alantha Newman}
\authorspace
\bsgitem{2005a}{}  Aggregating inconsistent information: ranking and clustering.  In: \emph{STOC'05: Proceedings of the 37th Annual ACM Symposium on the Theory of Computing} (Boston, 2005), pp.\ 684--693.  Assoc.\ for Computing Machinery, New York, 2005.
\MR{2181673}.  \Zbl{1192.90252}.						
\annot{} Conference version of \xrefd{2008a}{Nir Ailon, Moses Charikar, \& Alantha Newman}.
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\bsgitem{2008a}{}  Aggregating inconsistent information: ranking and clustering.  \emph{J.\ ACM} 55 (2008), no.\ 5, Art.\ 23, 27 pp.
\MR{2456548} (2009k:68280).  
\par
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\author{Saieed Akbari}						%AKBARI
\authorspace
\authref  See also \hyperlink{Uzma Ahmad}{U.~Ahmad}.
\authrefspace

\author{Saieed Akbari, Francesco Belardo, Ebrahim Dodongeh, \& Mohammad Ali Nematollahi}	
\authorspace
\bsgitem{2018a}{}  Spectral characterizations of signed cycles.  \emph{Linear Algebra Appl.}\ 553 (2018), 307--327. 
\MR{3809382}.  \Zbl{1391.05126}.  
\annot{} $(C_n,\sigma)$ is spectrally unique for $A(\Sigma)$ iff $n$ is odd or $n=4$; finds most cospectral signed graphs otherwise.  It is spectrally nonunique for $\Lap(\Sigma)$ iff even and balanced; finds all cospectral signed graphs otherwise.
Corrected for negative even circles in \xref{Ahmad, Akbari, \etal{\!}}{2022a}{Uzma Ahmad, Saieed Akbari, Saira Hameed, Mohammad Ali Nematollahi, \& Faiza Saeed}.
\dated{9 May 2018, rev 26 Dec 2022}
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Saieed Akbari, Francesco Belardo, Farideh Heydari, Mohammad Maghasedi, \& Mona Souri}
\authorspace
\bsgitem{2019a}{}  On the largest eigenvalue of signed unicyclic graphs.  \emph{Linear Algebra Appl.}\ 581 (2019), 145--162.
\MR{3982012}.  \Zbl{1420.05070}.  
\annot{} \S2, ``Signed graphs perturbations'':  For any $\Sigma$, effect on $\lambda_1 = \max \lambda_i$ of adding an edge.  $\lambda_1$ and $\lambda_n$ for $\sigma\equiv+1,-1$ vs.\ arbitrary $\sigma$.  \S3, ``Unbalanced unicyclic graphs with extremal index'':  Those with max and min $\lambda_1$.
\dated{23 Jan 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Akbari, S.\ Bolouki, P.\ Hatami, \& M.\ Siami}
\bsgitem{2009a}{}  On the signed edge domination number of graphs.  \emph{Discrete Math.} 309 (2009), no.\ 3, 587--594.
\MR{2499011} (2010e:05213).  Zbl 1186.05089.  \arXiv{1008.3217}.			
\annot{} See \xref{B.-G.\ Xu}{2001a}{Baogen Xu}).  Assume $s_e>0,\ \forall e$.  Then $s \geq -n^2/16$, and $s=-(1+o(1)) n^2/54$ is attainable.  [What is interesting is that $s<0$ is possible.]  Improved in \xref{Cherkashin and Prozorov}{2022a}{Danila Cherkashin \& Pavel Prozorov}.
\dated{15 Aug 2022} 
\codes{\CSGw}
\itemspace

\author{S.\ Akbari, S.M.\ Cioab\u{a}, S.\ Goudarzi, A.\ Niaparast, \& A.\ Tajdini}
\authorspace
\bsgitem{2021a}{}  On a question of Haemers regarding vectors in the nullspace of Seidel matrices.  \emph{Linear Algebra Appl.}\ 615 (2021), 194--206.
\MR{4200822}.  \Zbl{1459.05154}.  \arXiv{2011.06435}.  
\annot{} Nonsingular $A(K_n,\sigma)$ need not have a $\pm1$-nullvector.
\dated{6 Feb 2021} 
\codes{\CSG:~\CKG:~\CAdj}%
\itemspace

\author{S.\ Akbari, A.\ Daemi, O.\ Hatami, A.\ Javanmard, \& A.\ Mehrabian}
\authorspace
\bsgitem{2015a}{}  Nowhere-zero unoriented flows in hamiltonian graphs.  \emph{Ars Combin.}\ 120 (2015), 51--63. 
\MR{3363263}.  \Zbl{1363.05104}.  
\annot{} Every signed Hamiltonian graph without a coloop has a nowhere-zero 12-flow:  an improved result towards \xref{Bouchet's}{1983a}{Andre Bouchet} conjecture.  The proofs are for unoriented flows on a graph (i.e., flows on an all-negative signed graph, which are equivalent to signed-graph flows).  Better results if there is a negative Hamilton circle $C$.  Thm.\ 3.2:  An 8-flow if $\Sigma\setm C$ is connected.  Thm.\ 3.3:  A 6-flow if $\Sigma\setm C$ is unbalanced.
\dated{5 Feb 2010} 
\codes{\CSG:~\CFlows,~\Cori}%
\itemspace

\author{S.\ Akbari, M.\ Dalirrooyfard, K.\ Ehsani, \& R.\ Sherkati}
\authorspace
\bsgitem{2016a}{}  A note on signed $k$-matching in graphs.  \emph{Australasian J.\ Combin.}\ 64 (2016), no.\ 2, 341--346.
\MR{3442494}.  \Zbl{1333.05233}.  
\annot{} Follows \xref{C.-P.\ Wang}{2013a}{Changping Wang}.  Thm.\ 2:  $\beta_S^k \geq n - k - c(\Gamma)$.  Thm.\ 3:  If $\Gamma$ is connected and not Eulerian, $\beta_S^k = \max$ \# odd trails in a minimal trail decomposition.
\dated{17 Dec 2020} 
\codes{\Csg}%
\itemspace

\author{Saieed Akbari, Soudabeh Dalvandi, Farideh Heydari, \& Mohammad Maghasedi}
\authorspace
\bsgitem{2019a}{}  On the eigenvalues of signed complete graphs.  \emph{Linear Multilinear Algebra} 67 (2019), no.\ 3, 433--441.
\MR{3909000}.  \Zbl{1407.05119}.  
\annot{} Thms.:  Multiplicity of eigenvalue $\pm1 \geq$ number of homogeneously positive (negative) vertices.  Spectrum of regular $\Gamma$ is related to that of $K_\Gamma$.
\dated{17 Dec 2020} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  Signed complete graphs with maximum index.  \emph{Discuss.\ Math.\ Graph Theory} 40 (2020), no.\ 2, 393--403.
\MR{4060991}.  \Zbl{1433.05138}.  
\annot{} Thm.\ 8:  For $\Sigma^- =$ tree of $k$ edges, $\lambda_1$ is max iff $\Sigma^-=K_{1,k}$.  Thm.\ 10 relates eigenvalues of $H \subseteq K_n$ to those of $K_H$.  Thm.\ 16:  For nonspanning $H \subseteq K_n$ and most eigenvalues $\lambda$ of $H$, $-1-2\lambda$ is an eigenvalue of $K_n(-H)$.  [\Cf\ \xref{Ebrahim Ghorbani \& Arezoo Majidi}{2021a}{Ebrahim Ghorbani \& Arezoo Majidi}.]
\dated{17 Dec 2020} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020b}{}  On the multiplicity of $-1$ and $1$ in signed complete graphs.  \emph{Utilitas Math.}\ 116 (2020), 21--32.
\MR{4243163}.  \Zbl{1469.05069}.  
\annot{} The multiplicities of $\pm1$ in $\Spec A(K_n,\sigma)$.  Also, $\Spec A$ if $(K_n,\sigma)^- \cong K_{p,q,r}$.
\dated{26 Dec 2022} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Akbari, M.\ Einollahzadeh, M.M.\ Karkhaneei, \& M.A.\ Nematollahi}
\authorspace
\bsgitem{2020a}{}  Proof of a conjecture on the Seidel energy of graphs.  \emph{European J.\ Combin.}\ 86 (2020), art.\ 103078.
\MR{4047519}.  \Zbl{1437.05130}.  \arXiv{1901.06692}.  
\annot{} ``Seidel'' means signed complete graphs.  Thm.\ 2:  Proof of \xref{Haemers'}{2012a}{Willem H.\ Haemers} conjecture.  Thm.\ 1:  Lower bound on $p$-power energy.
\dated{10 Dec 2020} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Akbari, K.\ Etemadi, P.\ Ezzati, \& M.\ Ghadiri}
\authorspace
\bsgitem{2015a}{}  Even and odd cycles passing a given edge or a vertex.  Manuscript, 2015.
\arXiv{1512.02443}.  
\codes{\Csg:~\CPar:~\CCircles}%
\itemspace

\author{S.\ Akbari, F.A.M.\ Fran\c{c}a, E.\ Ghasemian, M.\ Javarsineh, \& L.S.\ de Lima}
\authorspace
\bsgitem{2021a}{}  The main eigenvalues of signed graphs.  \emph{Linear Algebra Appl.}\ 614 (2021), 270--280.
\MR{4209003}.  \Zbl{1459.05155}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Akbari, A.\ Ghafari, K.\ Kazemian, \& M.\ Nahvi}
\authorspace
\bsgitem{2020a}{}  Some criteria for a signed graph to have full rank.  \emph{Discrete Math.}\ 343 (2020), no.\ 8, art.\ 111910.
\MR{4081468}.  \Zbl{1441.05097}.  \arXiv{1708.07118}.				
\annot{} Given simple $\Gamma$, there is $\sigma$ such that $\rk A(\Gamma,\sigma) = n$ iff $\Gamma$ has a $\{1,2\}$-factor.
\dated{10 Dec 2020} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{S.\ Akbari, A.\ Ghafari, M.\ Nahvi, \& M.A.\ Nematollahi}
\authorspace
\bsgitem{2020a}{}  Mixed paths and cycles determined by their spectrum.  \emph{Linear Algebra Appl.}\ 586 (2020), 325--346.
\MR{4027760}.  \Zbl{1429.05117}.  \arXiv{1806.03634}.  
\annot{} The graphs are $\{\pm1,\pm\mathrm{i}\}$-gain graphs.
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Saieed Akbari, Ebrahim Ghorbani, Jack [Jacobus] H.\ Koolen, \& Mohammad Reza Oboudi}
\authorspace
\bsgitem{2010a}{}  A relation between the Laplacian and signless Laplacian eigenvalues of a graph.  \emph{J.\ Algebraic Combin.}\ 32 (2010), no.\ 3, 459--464.
\MR{2721061} (2011i:05125).  \Zbl{1230.05196}.  
\annot{} The sign-corrected coefficients of the characteristic polynomial of $\Lap(-\Gamma)$ dominate those of $\Lap(+\Gamma)$.  
[\Problem\ 1.  Prove they dominate those of $\Lap(\Gamma,\sigma)$ for any $\sigma$.  \Problem\ 2.  Generalize to any pair of signatures of $\Gamma$.]
\dated{22 Nov 2010} 
\codes{\CPar:~\CEig,~\CInc}%
\itemspace

\bsgitem{2010b}{}  On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs.  \emph{Electronic J.\ Combin.}\ 17 (2010), art.\ R115, 8 pp.
\MR{2679569} (2011j:05189).  \Zbl{1218.05086}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{S.\ Akbari, E.\ Ghorbani, \& M.R.\ Oboudi}
\authorspace
\bsgitem{2009a}{}  A conjecture on square roots of Laplacian and signless Laplacian eigenvalues of graphs.  Manuscript.
\arXiv{0905.2118}.  
\annot{} \Conjecture.  The sum $s$ of singular values is larger for $\Eta(-\Gamma)$ than for $\Eta(+\Gamma)$.  Dictionary:  ``incidence matrix'' = the unoriented incidence matrix $\Eta(-\Gamma)$;  ``directed incidence matrix'' = oriented incidence matrix $\Eta(+\Gamma)$.  
[\Problem.  Prove that $\max_\sigma s(\Gamma,\sigma) = s(-\Gamma)$, achieved uniquely for $[-\Gamma]$.]
\dated{8 Oct 2010, rev 10 Dec 2020} 
\codes{\Csg:~\CPar:~\CInc:~\CEig}%
\itemspace

\author{Saieed Akbari, Willem H.\ Haemers, Hamid Reza Maimani, \& Leila Parsaei Majd}
\authorspace
\bsgitem{2018a}{}  Signed graphs cospectral with the path.  \emph{Linear Algebra Appl.}\ 553 (2018), 104--116. 
\MR{3809370}.  \Zbl{1391.05156}.  \arXiv{1709.09853}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Akbari, H.R.\ Maimani, \& L.\ Parsaei Majd}
\authorspace
\bsgitem{2018a}{}  On the spectrum of some signed complete and complete bipartite graphs.  \emph{Filomat} 32 (2018), no.\ 17, 5817--5826.
\MR{3899319}.  
\annot{} Spectra of $K_n(-M)$ and $K_{p,q}(-M)$ for a matching $M$.  A family of $(K_n,\sigma)$ with symmetric spectrum.  [\Cf\ \xref{Etsuo Segawa \& Yusuke Yoshie}{20xxa}{Etsuo Segawa \& Yusuke Yoshie}.]
\dated{17 Dec 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Jin Akiyama}						%AKIYAMA
\author{J.\ Akiyama, D.\ Avis, V.\ Chv\'atal, \& H.\ Era}
\authorx{J.\ Akiyama, D.\ Avis, V.\ Chvatal, \& H.\ Era}
\authorspace
\bsgitem{1981a}{\dag\dag}  Balancing signed graphs.  \emph{Discrete Appl.\ Math.}\ 3 (1981), 227--233.  
\\ \MR{0675687} (83k:05059).  \Zbl{468.05066}.					
\annot{} Bounds for $D(\Gamma)$, the largest frustration index $l(\Gamma, \sigma)$ over all signings of a fixed graph $\Gamma$ (not necessarily simple) of order $n$ and size $m = \#E$.  Main Thm.: $\frac 12 m - \sqrt{mn} \leq D(\Gamma) \leq \frac 12 m$.  Thm.\ 4: $D(K_{t,t}) \leq \frac 12 t^2 - c_0 t^{3/2}$, where $c_0$ can be taken $= \pi/480$.  Probabilistic methods are used.  Thus, Thm.\ 2: Given $\Gamma$, Prob$(l(\Gamma, \sigma) > \frac 12 m - \sqrt{mn}) \geq 1 - (\frac 2e)^n$.  
Moreover, let $n_{\textb}(\Sigma)$ be the largest order of a balanced subgraph of $\Sigma$.  Thm.\ 5: Prob$(n_{\textb}(K_n, \sigma) \geq k) \leq \binom{n}{k}/2^{\binom{k}{2}}$.  (The problem of evaluating $n - n_{\textb}$ was raised by \xref{Harary}{1959b}{Frank Harary}.)  Finally, Thm.\ 1:  If $\Sigma$ has vertex-disjoint balanced induced subgraphs with $m'$ edges, then $l(\Sigma) \leq \frac 12(m-m')$.  
[See \xref{Poljak and Turz\'{\i}k}{1982a}{Svatopluk Poljak \& Daniel Turzik} for an upper bound on $D(\Gamma)$, \xref{Sol\'e and Zaslavsky}{1994a}{Thomas Zaslavsky} for lower and (bipartite) upper bounds; \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}, \xref{Gordon and Witsenhausen}{1972a}{Y.\ Gordon \& H.S.\ Witsenhausen} for $D(K_{t,t})$; \xref{Harary, Lindstr\"om, and Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom} for a result similar to Thm.\ 1.]
\par
\codes{\CSG:~\CFr,~\CRand}%
\itemspace

\authorx{Tatsuya Akutsu}						%AKUTSU
\author{Tatsuya Akutsu, Sven Kosub, Avraham A.\ Melkman, \& Takeyuki Tamura}
\authorspace
\bsgitem{2012a}{}  Finding a periodic attractor of a Boolean network.  \emph{IEEE/ACM Trans.\ Comput.\ Biol.\ Bioinformatics} 9 (2012), no.\ 5, 1420--1421.
\codes{\CSG:~\CDyn,~\CBiol}%
\itemspace

\author{Tatsuya Akutsu, Avraham A.\ Melkman, \& Takeyuki Tamura}
\authorspace
\bsgitem{2012a}{}  Singleton and 2-periodic attractors of sign-definite Boolean networks.  \emph{Inform.\ Processing Lett.}\  112 (2012), 35--38.
\\ \MR{2895503} (2012k:68098).  \Zbl{1233.68176}.  
\codes{\CSD:~\CDyn}%
\itemspace

\authorx{Ghadeer Alabandi}						%ALABANDI
\author{Ghadeer Alabandi, Jelena Te\v{s}i\'c, Lucas Rusnak, \& Martin Burtscher}
\authorx{Ghadeer Alabandi, Jelena Tesic, Lucas Rusnak, \& Martin Burtscher}
\authorspace
\bsgitem{2021a}{}  Discovering and balancing fundamental cycles in large signed graphs.  In:  \emph{SC '21: Proc.\ Int.\ Conf.\ for High Performance Computing, Networking, Storage and Analysis} (St.\ Louis, 2021), art.\ 68, 14 pp. 
\codes{\CSG:~\CBal:~\CAlgor}%
\itemspace

\author{Ansam I.\ Al-Aqtash}						%AL-AQTASH
\authorspace
\bsgitem{2014a}{}  \emph{The Minimum Semidefinite Rank of Signed Graphs}.  Doctoral dissertation, Central Michigan University, 2014.
\annot{} 
\Cf\ \xref{Matar, Mitchell, and Narayan}{20xxa}{Nancy Matar, Lon H.\ Mitchell, \& Sivaram K.\ Narayan}.
\codes{\CSG:~\CAdj:~\CQM}%
\itemspace

\authorx{M.J.\ Alava}							%ALAVA
\author{M.J.\ Alava, P.M.\ Duxbury, C.F.\ Moukarzel, \& H.\ Rieger}
\authorspace
\bsgitem{2001a}{}  Exact combinatorial algorithms:  Ground states of disordered systems.  In:  C.\ Domb and J.L.\ Lebowitz, eds., \emph{Phase Transitions and Critical Phenomena}, Vol.\ 18.  Academic Press, San Diego, 2001.
\MR{2014388} (2004k:82040).  
\annot{} \S7.1, ``Random Ising magnets'', (iv), ``Frustrated magnets and spin glasses'', introduces \S7.4, ``Ising spin glasses and Euclidean matching''.  
\S7.4.1, ``Introduction and overview'':  Frustration index, in terms of Hamiltonian $H(s):=-\sum J_{ij}s_is_j$ where, mainly, $J_{ij}=\pm1$ randomly (random signed graphs).  Frustrated (negative) plaquettes (girth circles) in a lattice.  
\S7.4.2, ``Mapping to optimization problems'':  (i) ``Mapping to a matching problem'':  Planar solution by dual matching as in \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai} [not cited], \xref{Bieche, Maynard, Rammal, and Uhry}{1980a}{I.\ Bieche, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}, \xref{Barahona}{1982b}{Francisco Barahona}, \etal\  (ii) ``Mapping to a cut problem'':  Equivalence to max cut.
\codes{\CPhys:~\CSG:~\CFr:~\CRand:~\CExp,~\CRef}%
\annotctd{} \S7.4.3, ``Ground-state calculation in two dimensions'':  Behavior of ground state (fewest frustrated edges) as function of negative-edge density.  Remarks on external magnetic field, cubic grid graphs.
\dated{29 Aug 2012} 
\codes{\CPhys:~\CSG:~\CFr,~\CState(\Cfr):~\CExp,~\CRef}%
\itemspace

\authorx{Abdullah Alazemi}						%ALAZEMI
\author{Abdullah Alazemi, Milica An{\dj}eli\'c, Francesco Belardo, Maurizio Brunetti, \& Carlos M.\ da Fonseca}
\authorx{Abdullah Alazemi, Milica Andelic, Francesco Belardo, Maurizio Brunetti, \& Carlos M.\ da Fonseca}
\authorspace
\bsgitem{2019a}{}  Line and subdivision graphs determined by $\bbT_4$-gain graphs.  \emph{Mathematics} 7 (2019), no.\ 10, art.\ 926, 12 pp.
\annot{} The gain group is $\{\pm1,\pm\mathrm{i}\}$.		
\codes{\CGG:~\CLG,~\CLG(\CGen),~\CEig,~\CInc}%
\itemspace

\author{\c{S}ahin Albayrak}						%ALBAYRAK
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Istv\'an Albert}						%ALBERT I
\authorspace
\authref  See \hyperlink{Assieh Saadatpour}{A.~Saadatpour}.
\itemspace

\author{J.\ James Albert}						%ALBERT JJ
\authorspace
\authref  See \hyperlink{Santhi M}{Santhi.~M}.
\itemspace

\author{R\'eka Albert}						%ALBERT R
\authorx{Reka Albert}
\authorspace
\authref  See \hyperlink{Assieh Saadatpour}{A.~Saadatpour}.
\itemspace

\author{Siavash Alemzadeh}						%ALEMZADEH
\authorspace
\authref  See \hyperlink{Mathias Hudoba de Badyn}{M.H.~de~Badyn}.
\itemspace

\author{John C.\ Alessio}						%ALESSIO
\authorspace
\bsgitem{1990a}{}  A synthesis and formalization of Heiderian balance and social exchange theory.  \emph{Social Forces} 68 (1990), no.\ 4, 1267--1285.
\annot{} Signed digraphs and graphs, with weights, used to describe a novel kind of ``balance'', different from normal signed-graph balance, based on exchange between persons.
\dated{24 Jan 2016} 
\par
\codes{\CPsS:~\CSD,~\CSG,~\CWG:~\CBal(\CGen)}%
\itemspace

\authorx{S.\ Alexander}						%ALEXANDER S
\author{S.\ Alexander \& P.\ Pincus}
\authorspace
\bsgitem{1980a}{} Phase transitions of some fully frustrated models.  \emph{J.\ Phys.\ A:  Math.\ Gen.}\ 13 (1980), no.\ 1, 263--273.  
\annot{} Certain all-negative signed graphs where every edge is in a triangle: $d=2$-dimensional triangular lattice and $d\geq3$-dimensional face-centered cubic lattice.  Phase phenomena depend on the parity of $d$.  Odd $d$ implies interesting infinities of switchings with minimum $\#E^-$.
\dated{12 Aug 2012} 
\codes{\CPar:~\CPhys}%
\itemspace

\author{Leonidas G.\ Alexopoulos}					%ALEXOPOULOS
\authorspace
\authref  See \hyperlink{Ioannis N.\ Melas}{I.N.~Melas}.
\itemspace

\authorx{Artiom Alhazov}						%ALHAZOV
\author{Artiom Alhazov, Ion Petre, \& Vladimir Rogojin}
\authorspace
\bsgitem{2009a}{}  The parallel complexity of signed graphs: Decidability results and an improved algorithm.  
\emph{Theor.\ Comput.\ Sci.}\ 410 (2009), no.\ 24--25, 2308--2315.
\MR{2522435} (2011a:68045).  \Zbl{1167.68022}.					
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Luiz Emilio Allem}						%ALLEM
\authorspace
\authref  See \hyperlink{Rafael Esteves Mansano}{R.E.~Mansano}.
\itemspace

\author{Noga Alon}							%ALON
\authorspace
\bsgitem{1996a}{}  Bipartite subgraphs.  \emph{Combinatorica} 16 (1996), no.\ 3, 301--311.
\MR{1417340} (97h:05081).  \Zbl{860.05043}.					
\annot{} Lower bound on the largest bipartite subgraph of a simple graph with $m$ edges.  [I.e., upper bound on $l(-\Gamma)$.  \Problem.  Generalize to $l(\Sigma)$.]
\dated{8 Mar 2011, 19 May 2012} 
\codes{\Csg:~\Cpar:~\CFr}%
\itemspace

\author{Noga Alon \& Yoshimi Egawa}
\authorspace
\bsgitem{1985a}{}  Even edge colorings of a graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 38 (1985), no.\ 1, 93--94. 
\MR{0782628} (86f:05059).  \Zbl{556.05026}.					
\annot{} Proves and improves a conjecture of \xref{B.D.\ Acharya}{1983a}{B.\ Devadas Acharya}.  Thm.:  The minimum number of colors for an ``even edge coloring'' = minimum number of colors so each color class is bipartite = $\lceil \log_2 \chi(\Gamma) \rceil$.  [\xref{Zaslavsky}{1987b}{Thomas Zaslavsky} generalizes the latter to $\Sigma$.]
\codes{\Cpar:~\Cbal:~\CGen}%
\itemspace

\author{Noga Alon, Gregory Gutin, Eun Jung Kim, Stefan Szeider, \& Anders Yeo}
\authorspace
\bsgitem{2010a}{}  Solving MAX-$r$-SAT above a tight lower bound.  In:  Moses Charikar, ed., \emph{Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA 2010, Austin, Tex.), pp.\ 511--517.  
Soc.\ for Industrial and Appl.\ Math., Philadelphia, and Assoc.\ for Computing Machinery, New York, 2010.
\MR{2809695} (2012h:68266).  
\annot{} Extended abstract of \xrefd{2011a}{Noga Alon, Gregory Gutin, Eun Jung Kim, Stefan Szeider, \& Anders Yeo}.
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2011a}{}  Solving MAX-$r$-SAT above a tight lower bound.  \emph{Algorithmica} 61 (2011), no.\ 3, 638--655.
\MR{2824999} (2012g:68101).  \Zbl{1242.68118}.  \arXiv{0907.4573}.			
\par
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Noga Alon, Konstantin Makarychev, Yury Makarychev, \& Assaf Naor}
\authorspace
\bsgitem{2005a}{}  Quadratic forms on graphs.  In:  \emph{STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing} (Baltimore, 2005), pp.\ 486--493.  ACM Press, New York, 2005.
\MR{2181652} (2006g:05207).  \Zbl{1192.05168}.  
\annot{} Extended abstract of \xrefd{2006a}{Noga Alon, Konstantin Makarychev, Yury Makarychev, \& Assaf Naor}.
\itemspace

\bsgitem{2006a}{}  Quadratic forms on graphs.  \emph{Invent.\ Math.}\ 163 (2006), no.\ 3, 499--522.
\MR{2207233} (2008a:05156).  \Zbl{1082.05051}.  
\itemspace

\author{N.\ Alon \& T.H.\ Marshall}
\authorspace
\bsgitem{1998a}{}  Homomorphisms of edge-colored graphs and Coxeter groups.  \emph{J.\ Algebraic Combin.}\ 8 (1998), no.\ 1, 5--13.
\MR{1635549} (99i:05074).  \Zbl{911.05034}.  
\annot{} (E.g., sign-colored graphs.)  Thm.:  For edge-colored graphs $\exists$ universal planar target.  Bounds on target.  [Relevant for signed-graph homomorphisms.]
\dated{28 Dec 2019} 
\codes{\Csgc(\CGen):~\CHom}%
\itemspace

\authorx{Stefanu Elias Aloysius}					% ALOYSIUS
\author{Stefanu Elias Aloysius, ed.}
\authorspace
\bsgitem{2011a}{}  \emph{Dowling Geometry}.  AlphaScript Publishing, 2011.
\annot{} ``\dots\ primarily consists of articles available from Wikipedia or other free sources online.''  This seems to mean copying Wikipedia.  \Cf\ Chris Rand, ``The odd tale of Alphascript Publishing and Betascript Publishing'', URL \url{http://www.chrisrand.com/blog/2010/02/odd-tale-alphascript-publishing-betascript-publishing/}
\dated{22 Jun 2018} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Yu.\ A.\ Al'pin}						%ALPIN
\authorspace
\bsgitem{2014a}{\dag}  Harary's theorem on signed graphs and reversibility of Markov chains.  \emph{J.\ Math.\ Sci.}\ 199 (2014), no.\ 4, 375--380.  Trans.\ from Russian in \emph{Zap.\ Nauchn.\ Semin.\ POMI} 419 (2013), 5--15.
\MR{3477903}.  \Zbl{1343.60104}.  
\annot{} \S2, ``Harary's theorem for digraphs over groups'':  Thm.\ 3.  A gain digraph $(D,\phi,\fG)$, where $D$ is strongly connected, is cycle balanced (every cycle has gain $1$) $\Leftrightarrow$ every closed walk has gain $1$ $\Leftrightarrow$ for each $u,v$ all $uv$-paths have the same gain $\Leftrightarrow$ $\exists$ potential for $\phi$.  
This makes explicit properties implicit in previous work, e.g., \xref{Belitski\u\i\ and Lyubich}{1984a}{G.R.\ Belitskii \& Yu.I.\ Lyubich}.  
[This actually generalizes \xref{Harary--Norman--Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright}, Thm.\ 13.11 for signed digraphs, not \xref{Harary}{1953a}{Frank Harary} for undirected graphs.]
Thm.\ 4.  Two potentials $\theta, \theta'$ for $\phi$ differ by $\theta' = h \theta$ for some $h \in \fG$.
Application to Markov chains.
Dictionary:  ``path'' = directed walk, ``weight'' = gain, ``graph over a group'' = gain digraph.  
\annotctd{} [The important aspect:  Strong connection suffices for gain-graph properties; the arcs need not be reversible as in a gain graph.]
\dated{22 Feb 2021} 
\codes{\CGG:~\CBal,~\CAppl}%
\itemspace

\author{Claudio Altafini}						%ALTAFINI
\authorspace
\authref  See also  \hyperlink{Nuria Ballber Torres}{N.~Ballber~Torres}, \hyperlink{Giuseppe Facchetti}{G.~Facchetti}, \hyperlink{Angela Fontan}{A.~Fontan}, \hyperlink{Giovanni Iacono}{G.~Iacono}, \hyperlink{Guodong Shi}{G.D.~Shi}, and \hyperlink{Nicola Soranzo}{N.~Soranzo}.
\authrefspace

\bsgitem{2012a}{}  Dynamics of opinion forming in structurally balanced social networks.  \emph{PLoS ONE} 7 (2012), no.\ 6, art.\ 38135, 8 pp.\ + 6 supplements.
URL \url{https://doi.org/10.1371/journal.pone.0038135}
\codes{\CSD,~\Csg:~\CBal,~\CDyn,~\CPsS}%
\itemspace

\bsgitem{2012b}{}  Achieving consensus on networks with antagonistic interactions.  In:  \emph{51st IEEE Conference on Decision and Control} (CDC 2012, Maui, Hawaii), pp.\ 1966--1971.  IEEE, 2012.
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{2013a}{}  Consensus problems on networks with antagonistic interactions.  \emph{IEEE Trans.\ Automatic Control} 58 (2013), no.\ 4, 935--946.
\MR{3038795}.  \Zbl{1369.93433}.  
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{2013b}{}  Stability analysis of diagonally equipotent matrices.  \emph{Automatica J.\ IFAC} 49 (2013), no.\ 9, 2780--2785.
\MR{3084465}.  \Zbl{1364.93318}.  
\codes{\CSD:~\CQM:~\CSta}%
\itemspace

\author{Claudio Altafini \& Gabriele Lini}
\authorspace
\bsgitem{2015a}{}  Predictable dynamics of opinion forming for networks with antagonistic interactions.  \emph{IEEE Trans.\ Automatic Control} 60 (2015), no.\ 2, 342--357.
\MR{3310162}.  \Zbl{1360.91114}.  
\codes{\CSG:~\CBal,~\CDyn}%
\itemspace

\author{Dora Altbir}							%ALTBIR
\authorspace
\authref  See  \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\author{Randolf Altmeyer}						%ALTMEYER
\authorspace
\authref  See \hyperlink{Derek Feng}{D.~Feng}.
\itemspace

\author{[Susan S.\ D'Amato]}						%AMATO
\authorspace
\authref  See \hyperlink{Susan S.\ D'Amato}{S.S.\ D'Amato} (under `D').
\itemspace

\authorx{C.\ Amoruso}						%AMORUSO
\author{C.\ Amoruso \& A.K.\ Hartmann}
\authorspace
\bsgitem{2004a}{}  Domain-wall energies and magnetization of the two-dimensional random-bond Ising model.  \emph{Phys.\ Rev.\ B} 70 (2004), art.\ 134425, 7 pp.
\arXiv{cond-mat/0401464}.  
\codes{\CPhys,~\CSG:~\CFr:~\CState:~\CAlgor}%
\itemspace

\authorx{Meirav Amram}						%AMRAM
\author{Meirav Amram, Robert Shwartz, \& Mina Teicher}
\authorspace
\bsgitem{2010a}{}  Coxeter covers of the classical Coxeter groups.  \emph{Int.\ J.\ Algebra Comput.}\ 20 (2010), no.\ 8, 1041--1062.
\MR{2747415} (2012c:20104).  \Zbl{1237.20031}.  \arXiv{0803.3010}.			
\annot{} The structure of a quotient of a generalized Coxeter group depends on the presence of loops in the associated signed graph.
\dated{17 Dec 2011} 
\codes{\CSG}%
\itemspace

\authorx{Javeria Amreen}						%AMREEN
\author{Javeria Amreen \& Sudev Naduvath}
\authorspace
\bsgitem{20xxa}{}  Order sum signed graph of a group.  Submitted.  
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{Xinhui An}							%AN XH
\authorspace
\authref  See \hyperlink{Jianfeng Wang}{J.F.~Wang}.
\itemspace

\author{Milica An{\dj}eli\'c}						%ANDELIC
\authorx{Milica Andelic}
\authorspace
\authref  See also \hyperlink{Abdullah Alazemi}{A.~Alazemi} and \hyperlink{Slobodan K.\ Simic}{S.K.~Simi\'c}.
\authrefspace

\author{Milica An{\dj}eli\'c, Carlos M.\ da Fonseca, Slobodan K.\ Simi\'c, \& Dejan V.\ To\v{s}i\'c}
\authorx{Milica Andelic, Carlos M.\ da Fonseca, Slobodan K.\ Simic, \& Dejan V.\ Tosic}
\authorspace
\bsgitem{2012a}{}  Connected graphs of fixed order and size with maximal $Q$-index: Some spectral bounds.  \emph{Discrete Appl.\ Math.}\ 160 (2012), no.\ 4-5, 448--459.
\MR{2876327} (2012k:05219).  \Zbl{1239.05115}.  
\annot{} Bounds on $\lambda_1(\Lap(-\Gamma))$ where $\Gamma$ is a nested split graph.  (\Cf\ \xref{Cvetkovi\'c, Rowlinson, and Simi\'c}{2007b}{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}, which shows nested split graphs maximize $\lambda_1(\Lap(-\Gamma))$.)
\dated{2 Feb 2012} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\bsgitem{2012b}{}  Some further bounds for the $Q$-index of nested split graphs.  \emph{J.\ Math.\ Sci.\ (N.Y.)}\ 182 (2012), no.\ 2, 193--199.  Trans.\ from \emph{Sovrem.\ Mat.\ Prilozh.}\ 71 (2011).
\MR{3141339}.  \Zbl{1254.05095}.  
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Milica An{\dj}eli\'c, Tamara Koledin, \& Zoran Stani\'c}
\authorx{Milica Andelic, Tamara Koledin, \& Zoran Stanic}
\authorspace
\bsgitem{2019a}{}  A note on the eigenvalue free intervals of some classes of signed threshold graphs  \emph{Special Matrices} 7 (2019), 218--225.
\MR{4044596}.  \Zbl{1431.05098}.  
\annot{} 
[\Cf\ \xref{Xiong and Hou}{2022a}{Zhuang Xiong \& Yaoping Hou}, \xref{Mandal and Mehatari}{20xxa}{Santanu Mandal \& Ranjit Mehatari}.]
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  On regular signed graphs with three eigenvalues.  \emph{Discuss.\ Math.\ Graph Theory} 40 (2020), no.\ 2, 405--416.
\MR{4060992}.  \Zbl{1433.05139}.  
\annot{} Constructions for signed regular graphs with exactly three eigenvalues.  Determines all of order at most 10.
\dated{29 Apr 2022} 
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Inequalities for Laplacian eigenvalues of signed graphs with given frustration number.  \emph{Symmetry} 2021 (2021), no.\ 13, art.\ 1902, 8 pp.
\codes{\CSG:~\CFr,~\CLap:~\CEig}%
\itemspace

\bsgitem{20xxa}{}  Signed graphs whose all Laplacian eigenvalues are main.  \emph{Linear Multilinear Algebra} (to appear).
\annot{} That is, up to switching equivalence.  Order $\leq7$ is sufficient but $8$ is not.  Two problems:  (1)  Find all signed graphs with the property up to switching equivalence.  (2) Find all switchings with this property; solved for threshold graphs and for some cographs.
\dated{7 Aug 2022} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Milica An{\dj}eli\'c \& Slobodan K.\ Simi\'c}
\authorx{Milica Andelic \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2010a}{}  Some notes on the threshold graphs.  \emph{Discrete Math.}\ 310 (2010), no.\ 17-18, 2241--2248. 
\MR{2659175} (2011h:05145).  \Zbl{1220.05035}.					
\codes{\Csg:~\Cpar:~\CLap}%
\itemspace

\authorx{Lars D{\o}vling Andersen}					%ANDERSEN
\authorx{Lars Dovling Andersen}
\author{Lars D{\o}vling Andersen \& Douglas D.\ Grant}
\authorx{Lars Dovling Andersen \& Douglas D.\ Grant}
\authorspace
\bsgitem{1981a}{}  Homopolar circuits in polar graphs.  \emph{Czechoslovak Math.\ J.}\ 31 (1981), no.\ 2, 218--228.
\MR{0611077} (82e:05086).  \Zbl{0472.05038}.  
\annot{} \Cf\ \xref{Zelinka}{1976b}{Bohdan Zelinka}.  
If there are no coherent circles, no loops, and no parallel edges with the same orientation, then $\#E \leq 4n-4$ (equality is characterized) and $\delta \leq 6$.  Sufficient conditions for an antidirected Hamiltonian circle.
Dictionary:  ``polar graph'' = switching class of bidirected graphs, ``homopolar circuit'' = antidirected circle.  
[Later work, only on digraphs: e.g., \cf\ \xref{Diwan, Frye, Plantholt, and Tipnis}{2011a}{Ajit A.\ Diwan, Josh B.\ Frye, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}.]
\dated{27 Jul 2013} 
\codes{\Cgg:~\CCircles:~\CStr}\codesctd{\Csg:~\COri:~\CCircles}%
\itemspace

\authorx{Laura Anderson}						%ANDERSON L
\author{Laura Anderson, Ting Su, \& Thomas Zaslavsky}
\authorspace
\bsgitem{20xxa}{}  Matroids of gain signed graphs.  \emph{Discrete Comput.\ Geom.}\ (to appear).
\arXiv{2206.00237}.  
\annot{\dag\dag} Combines frame matroid $\bfF(\Sigma)$ of signed graph with lift matroid $\bfL(\Phi)$ of abelian gain graph to form extended matroid $\bfM_\infty(\Gamma,\sigma,\phi)$ of graph with gains upon signed edges.  Basic properties of gain signed graphs and the matroid.  Examples:  Dimension of edge polytope (\cf\ \xref{Ohsugi and Hibi}{1998a}{Hidefumi Ohsugi \& Takayuki Hibi}), symmetric edge polytope (\cf\ \xref{Matsui \etal}{2011a}{Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, \& Takayuki Hibi}), and bidirected generalization.
\dated{1, 4 Jun 2022} 
\par
\codes{\CGG,~\CSG:~\CMatrd,~\CSw,~\CBal}%
\itemspace

\author{P.W.\ Anderson}						%ANDERSON PW
\authorspace
\authref  See \hyperlink{S.F.\ Edwards}{S.F.~Edwards}.
\itemspace

\author{Ascensi\'on Andina-D\'{\i}az}					%ANDINA-DIAZ
\authorx{Ascension Andina-Diaz}
\authorspace
\authref  See \hyperlink{Antonio Parravano}{A.~Parravano}.
\itemspace

\authorx{Kazutoshi Ando}						%ANDO
\author{Kazutoshi Ando \& Satoru Fujishige}
\authorspace
\bsgitem{1994a}{}   $\sqcup, \sqcap$-closed families and signed posets.  Report no.\ 93813, Forschungsinstitut f\"ur Diskrete Math., Universit\"at Bonn, 1994.
\codes{\Csg:~\COri}%
\itemspace

\bsgitem{1996a}{}  On structures of bisubmodular polyhedra.  \emph{Math.\ Programming} 74 (1996), 293--317.  
\MR{1407690} (97g:90102).  \Zbl{855.68107}.					
\codes{\Csg:~\COri}%
\itemspace

\author{Kazutoshi Ando, Satoru Fujishige, \& Takeshi Naitoh}
\authorspace
\bsgitem{1997a}{}  Balanced bisubmodular systems and bidirected flows.  \emph{J.\ Operations Res.\ Soc.\ Japan} 40 (1997), 437--447.
\MR{1476834} (98k:05073).  \Zbl{970.61830}.  					
\annot{} A balanced bisubmodular system corresponds to a bidirected graph that is balanced.  The ``flows'' are arbitrary capacity-constrained functions, not satisfying conservation at a vertex.
\codes{\Csg:~\COri,~\CBal}
\itemspace

\author{Kazutoshi Ando, Satoru Fujishige, \& Toshio Nemoto}
\authorspace
\bsgitem{1996a}{}  Decomposition of a bidirected graph into strongly connected components and its signed poset structure.  \emph{Discrete Appl.\ Math.}\ 68 (1996), 237--248.  
\MR{1398276} (97c:05096).  \Zbl{960.53208}.					
\codes{\Csg:~\COri}%
\itemspace

\bsgitem{1996b}{}  The minimum-weight ideal problem for signed posets.  \emph{J.\ Operations Res.\ Soc.\ Japan} 39 (1996), 558--565.  
\MR{1435185} (98j:90084).  \Zbl{874.90188}.  
\codes{\Csg:~\COri}
\itemspace

\author{Thomas Andreae} 						%ANDREAE
\authorspace
\bsgitem{1978a}{}  Matroidal families of finite connected nonhomeomorphic graphs exist.  \emph{J.\ Graph Theory} 2 (1978), 149--153.  
\MR{0493394} (80a:05160).  \Zbl{401.05070}.
\annot{} Partially anticipates the ``count'' matroids of graphs (see \xref{Whiteley}{1996a}{Walter Whiteley}).
\codes{\CBic,~\CECyc:~\CGen}%
\itemspace

\authorx{Eleonora Andreotti}						%ANDREOTTI
\author{Eleonora Andreotti \& Raffaella Mulas}
\authorspace
\bsgitem{20xxa}{}  Spectra of signless normalized Laplace operators for hypergraphs.  Submitted.  \emph{}\ 
\arXiv{2005.14484}.  
\codes{\Csh:~\CPar:~\CLap:~\CEig}%
\itemspace

\authorx{David Angeli}						%ANGELI
\author{D.\ Angeli, M.\ Banaji, \& C.\ Pantea}
\authorspace
\bsgitem{2014a}{}  Combinatorial approaches to hopf bifurcations in systems of interacting elements.  \emph{Commun.\ Math.\ Sci.}\ 12 (2014), no.\ 6, 1101--1133.
\MR{3194372}.  \Zbl{1315.15007}.  \arXiv{1301.7076}.				
\codes{\CSD:~\CQM,~\CDyn}%
\itemspace

\author{David Angeli, Patrick De Leenheer, \& Eduardo Sontag}
\authorspace
\bsgitem{2010a}{}  Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates.  \emph{J.\ Math.\ Biol.}\ 61 (2010), no.\ 4, 581--616.
\MR{2672536} (2011d:92054).  \Zbl{1204.92038}.  
\annot{} Dictionary:  ``J-graph'' = a signed graph of a Jacobian matrix.  
``Species-reaction graph'' (``SR-graph'') = signed bipartite graph $(V_S,V_R,E,\sigma) =: \Sigma$; ``reaction graph'' (``R-graph'') = $-\Sigma^2{:}V_R$; ``species graph'' (``S-graph'') = $-\Sigma^2{:}V_S$ [where $\Sigma^2$ is the distance-2 signed graph: $V(\Sigma^2) := V_R \cup V_S$, $T_iT_j \in E^\epsilon(\Sigma^2) \iff \exists\ \text{path}\ T_iU_kT_j \text{ with } \sigma(T_iU_kT_j) = \epsilon$].  
Dictionary:  ``Simple loop'' $\approx$ circle; ``positive-loop property'' = balance.  In a signed bipartite graph, ``e-loop, o-loop'' = circle with $(-1)^{\#C/2}\sigma(C) = +$ or $-$.  Prop.\ 4.5:  $\Sigma^2{:}V_R$ is antibalanced iff all circles in $\Sigma$ are e-loops and max deg$(\Sigma{:}V_S) \leq 2$.  
Thm.\ 1 (oversimplified):  A certain differential system is monotone iff $\Sigma^2{:}V_R$ is antibalanced (the R-graph is balanced).
\dated{19 Feb 2010} 
\codes{\CSG:~\CBal,~\Csw,~\CGeom,~\CChem}%
\annotctd{} [A bipartite multiplicative gain graph $\Phi := (V_S,V_R,E,\phi)$ may be defined by $\phi(S_iR_j) := $(a value from the stoichiometry matrix $\Gamma$).  Circle $C$ is ``unitary'' if $(-1)^{\#C/2}\phi(C) = +1$.]  $\Phi$ is implicated in the proof of geometrical Lemma 6.1.
\dated{19 Feb 2010} 
\codes{\Cgg:~\CBal,~\CGeom}%
\itemspace

\author{David Angeli \& Eduardo Sontag}
\authorspace
\bsgitem{2003a}{}  A note on multistability and monotone I/O systems.  In:  \emph{Decision and Control, 2003} (42nd IEEE Conf., Maui, Hawaii, 2003), Vol.\ 1, pp.\ 67--72.
\annot{} \S IV, ``Graphical conditions for strong monotonicity'': Mentions signed (di)graph balance and monotonicity.
\dated{1 Jan 2012} 
\par
\codes{\CSD,~\CSG:~\CBal,~\CChem:~\CExp}%
\itemspace

\bsgitem{2004a}{}  Interconnections of monotone systems with steady-state characteristics.  In: Marcio S.\ de Queiroz, Michael Malisoff, and Peter Wolenski, eds., \emph{Optimal Control, Stabilization and Nonsmooth Analysis}, pp.\ 135--154.  Lect.\ Notes in Control Inform.\ Sci., Vol.\ 301.  Springer, Berlin, 2004.
\MR{2079681} (2005f:93132).  \Zbl{1259.93092}.  
\par
\codes{\CSD,~\CSG:~\CBal,~\CChem:~\CExp}%
\itemspace

\bsgitem{2004b}{}  Multi-stability in monotone input/output systems.  \emph{Systems Control Lett.}\ 51 (2004), 185--202.
\MR{2037252} (2004k:93128).  \Zbl{1157.93509}.  
\par
\codes{\CSD,~\CSG:~\CBal,~\CChem}%
\itemspace

\bsgitem{2008a}{}  Oscillations in I/O monotone systems under negative feedback.  \emph{IEEE Trans.\ Automatic Control} 53 (2008), Special Issue on Systems Biology, 166--176. 
\MR{2492561} (2009k:92005), \MR{2605139} (no rev).  
\annot{} \S II, p.\ 167, mentions signed (di)graph balance and monotonicity.
\dated{1 Jan 2012} 
\codes{\CSD,~\CSG:~\CBal,~\CChem:~\CExp}%
\itemspace

\bsgitem{2012a}{}  Remarks on the invalidation of biological models using monotone systems theory.  In:  \emph{51st IEEE Conference on Decision and Control} (Maui, Hawaii, 2012), pp.\ 2989--2994.
\codes{\CBiol,~\CSD}%
\itemspace

\authorx{J.C.\ Angles d'Auriac}						%ANGLES D'AURIAC
\author{J.C.\ Angles d'Auriac \& R.\ Maynard}
\authorspace
\bsgitem{1984a}{}  On the random antiphase state of the $\pm J$ spin glass model in two dimensions.  \emph{Solid State Commun.}\ 49 (1984), no.\ 8, 785--790.
\annot{} Signed square lattice graph: frustration index and ground states (minimum $\#E^-$ of switched $\Sigma$) via matching [\cf\ \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai}, \xref{Barahona}{1981a}{Francisco Barahona}, \xrefd{1982a}{Francisco Barahona}].  Observed: natural clusters with relatively fixed spins (vertex signs) if the density of negative edges is in $(0.1,0.2)$.
\dated{18 Aug 2012} 
\codes{\CPhys,~\CSG:~\CFr,~\CState(\Cfr):~\CAlgor}%
\itemspace

\authorx{Marcin Anholcer}						%ANHOLCER
\author{Marcin Anholcer, Bart{\l}omiej Bosek, \& Jaros{\l}aw Grytczuk}
\authorx{Marcin Anholcer, Bartlomiej Bosek, \& Jaroslaw Grytczuk}
\authorspace
\bsgitem{2019a}{}  Weight choosability of oriented hypergraphs.  \emph{Ars Math.\ Contemp.}\ 16 (2019), 111--117.
\MR{3904720}.  \Zbl{1416.05243}.  
\annot{} The ``orientation'' maps $I \to \bbC^\times$, where $I$ is the set of vertex-hyperedge incidences.
	
\codes{\CSH(\CGen):~\Cgg}%
\itemspace

\authorx{Achu Aniyan}						%ANIYAN
\author{Achu Aniyan \& Sudev Naduvath}
\authorspace
\bsgitem{2020a}{}  Induced signed graphs of some classes of graphs.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 23 (2020), no.\ 2, 283--291.
\MR{4099090}.  
\annot{} Induced signed graph $\Sigma_f$ of $\Gamma$: given $f: V \to \bbZ$, $\sigma(uv) := (-1)^{f(u)-f(v)}$.  
\S2: $f(u) = d(u)$, degree.  Thm.\ 2.6:  $\Sigma_d$ is balanced.  Thm. 2.7:  $-\Sigma_d$ is balanced iff $\Gamma$ is bipartite.  \S3: $f(u) = \epsilon(u)$, eccentricity.  Similar results.  [Elementary.  Balance of all $\Sigma_f$ by \xref{Harary}{1953a}{Frank Harary}.  $-\Sigma_f$ property because $\Sigma_f$ is balanced.]
\S3: $f(u) =$ eccentricity.  Similar results.
Dictionary:  ``switched signed graph'' = negation $-\Sigma$, not a switching of $\Sigma$.
\dated{26 Sep, 13 Oct 2020} 
\codes{\CSG:~\CLab:~\CBal}%
\itemspace

\bsgitem{2020b}{}  On degree product induced signed graphs of graphs.  In:  Samayan Narayanamoorthy, ed., \emph{Advances in Applicable Mathematics} (ICAAM2020, Coimbatore, India), art.\ 030018, 5 pp.  AIP Conf.\ Proc., Vol.\ 2261.  AIP Publishing, 2020.
\annot{} \Cf\ \xrefd{2020a}{Achu Aniyan \& Sudev Naduvath}.  $\sigma(uv) := (-1)^{d(u)d(v)}$.  Simple results.  [Thms.\ 2, 4, 5, 7 are wrong.]
[In general, define $\sigma(uv) := (-1)^{f(u)f(v)}$.  Then $E^- = E{:}V_\text{odd}$ where $V_\text{odd} := \{v: f(v) \text{ odd}\}.$] 
\dated{17 Oct 2020} 
\codes{\CSG:~\CBal,~\CFr}%
\itemspace

\author{Christopher E.\ Anson}						%ANSON
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\authorx{T.\ Antal}							%ANTAL
\author{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner}
\authorspace
\bsgitem{2005a}{\dag\dag}  Dynamics of social balance on networks.  \emph{Phys.\ Rev.\ E} 72 (2005), art.\ 036121, 10 pp.
\MR{2179924} (2006e:91124).  
\annot{} Models for the evolution of a signed $K_n$ towards balance, with conclusions about the probable long-term behavior.  A ``state'' of the graph is a signature.  The unit of time $t$ is $\#E = \binom{n}{2}$ steps of the process.  The density of edges is $\rho := \#E^+/\binom{n}{2}$.  The number of triangles with $k$ negative edges (type $k$) is $N_k$; their density is $n_k := N_k / \binom{n}{3}$.  The average density of type $k$ triangles on a positive edge is $n_k^+ = (3-k)N_k/(n-2)\#E^+ = (3-k)n_k/(3n_0+2n_1+n_2).$  Similarly, $n_k^- = k n_k/(3n_0+2n_1+n_2).$  
\annotctd{} ``Local triad dynamics'':  At each step a random triangle $T$ is chosen.   If it is all negative, a random edge in $T$ is chosen and negated.  If it has one negative edge, a random edge in $T$ is chosen and negated with probability $p$ if it is negative and $1-p$ if positive.  If it is balanced there is no change.  The process is repeated \emph{ad infinitum}.  
Finite [i.e., fixed] $n$:  For $p > 1/2$ the graph reaches all-positivity (``paradise'') in time $C \log t$ and for $p = 1/2$ in time $C/\sqrt{2t}$.  For $p < 1/2$ the graph reaches a balanced state which is not all positive, in super-exponential time.  (Time is in the units described.)
``Infinite'' $n$ [i.e., $n \to \infty$]:  For $p < 1/2$ the density of negative edges approaches the stationary value $(1+\surd(3(1-2p))\inv$.  For $p > 1/2$ the network approaches all-positivity.  Thus, at $p = 1/2$ there is a phase transition.
Differential equations arise in the densities, with coefficients $\pi^+, \pi^-$ where $\pi^\epsilon :=$ the probability that, in one step, the sign change is from $\epsilon$ to $-\epsilon$; thus $\pi^+ = (1-p)n_1$ and $\pi^- = p n_1 + n_3$.
A stationary state has $\pi^+ = \pi^-$.  For infinite $n$ the stationary states are in \S~III.B and temporal evolution of $\rho=\rho(t)$ is treated in \S~III.C.  Finite $n$ is in \S~III.D.
\annotctd{} ``Constrained triad dynamics'':  An edge is chosen randomly and is negated with probability $1$ if the number of positive triangles increases, $0$ if the number decreases, and $1/2$ if the number remains the same.  This corresponds to an Ising model with Hamiltonian $-\sum \sigma_i\sigma_j\sigma_k$, summed over all edge triples that form a triangle.  This model approaches balance in time $C \log t$ with high probability if $n$ is large.  
The other alternatives are to reach an unbalanced absorbing state, where every edge is more positive than negative triangles (a ``jammed state''), or a trajectory where every edge is in equally many triangles of each sign (a ``blinker'').  Blinkers were not observed in the simulations.  The probability of a jammed state decreases quickly as $n \to \infty$.  
The ``final'' state, if balanced, has Harary bipartition $V = V_1 \cupdot V_2$.  For $\rho(0) \lesssim .4$, $\#V_1/\#V_2 approx 1$.  As $\rho(0) \to \beta \approx .65$, $\#V_1/\#V_2 \to \infty$, i.e., one set becomes dominant.  When $\rho(0) > \beta$, $V_1=V$ and all edges are positive.  (\S~IV.B.)
A jammed state can occur only when $n=9$ or $n\geq11$ (\S~IV.C), e.g., certain $3$-cluster states as in \xref{Davis}{1967a}{James A.\ Davis}.  
The number of jammed signatures $> 3^n \gg 2^{n-1} =$ number of balanced ones, notwithstanding that the probable long-term state is balanced (\S~IV.C).
[See \xref{Marvel, Strogatz, and Kleinberg}{2009a}{Seth A.\ Marvel, Steven H.\ Strogatz, \& Jon M.\ Kleinberg}, \xref{Abell and Ludwig}{2009a}{Peter Abell \& Mark Ludwig}, \xref{Kujawski, Ludwig, and Abell}{2010a}{Bernard Kujawski, Mark Ludwig, \& Peter Abell}, \xref{Deng and Abell}{2010a}{Hongzhong Deng \& Peter Abell}.]
\annotctd{} Proposed research:  Allow type 3 triangles (i.e., clustering).  Allow incomplete graphs.
\annotctd{} Dictionary: ``network'' = complete graph.
\dated{27 Apr 2009}
\par
\codes{\CSG:~\CKG:~\CDyn:~\CBal}%
\itemspace

\bsgitem{2006a}{}  Social balance on networks: the dynamics of friendship and enmity.  \emph{Physica D} 224 (2006), no.\ 1-2, 130--136.
\MR{2301516} (2007k:91210).  \Zbl{1130.91041}.					
\annot{} Similar to \xrefd{2005a}{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner}, with some details omitted and some additional results.
\dated{27 Apr 2009} 
\codes{\CSG:~\CKG:~\CDyn:~\CBal}%
\itemspace

\authorx{St.\ Antohe}							%ANTOHE
\author{St.\ Antohe \& E.\ Olaru}
\authorspace
\bsgitem{1981a}{} Singned graphs homomorphism \sic.  [Signed graph homomorphisms.]  \emph{Bul.\ Univ.\ Galati Fasc.\ II Mat.\ Fiz.\ Mec.\ Teoret.}\ 4 (1981), 35--43. 
\MR{0668767} (83m:05057).  
\annot{} A ``congruence'' is an equivalence relation $R$ on $V(\Sigma)$ such that no negative edge is within an equivalence class.  The quotient $\Sigma/R$ has the obvious simple underlying graph and signs $\bar{\sigma}(\bar{x}\bar{y}) = \sigma(xy)$ [which is ambiguous].  A signed-graph homomorphism is a function $f: V_1 \to V_2$ that is a sign-preserving homomorphism of underlying graphs.  [This is inconsistent, since the sign of edge $f(x)f(y)$ can be ill defined.  The defect might perhaps be remedied by allowing multiple edges with different signs or by passing entirely to multigraphs.]  
The canonical map $\Sigma \to \Sigma/R$ is such a homomorphism.  Composition of homomorphisms is well defined and associative; hence one has a category ${\text{\rm Graph}}^{\text{\rm sign}}$.  The categorial product is $\prod_{i \in I} \Sigma_i :=$ Cartesian product of the $|\Sigma_i|$ with the component-wise signature $\sigma((\ldots, u_i, \dots)(\ldots, v_i, \dots)) := \sigma_i(u_iv_i)$.  Some further elementary properties of signed-graph homomorphisms and congruences are proved.  
[The paper is hard to interpret due to mathematical ambiguity and grammatical and typographical errors.]
\codes{\CSGc:~\CHom}%
\itemspace

\authorx{Divya Antoney}						%ANTONEY
\author{Divya Antoney, Tabitha Agnes Mangam, \& Mukti Acharya}
\authorspace
\bsgitem{20xxa}{}  Signed graphs from proper coloring of graphs.  Submitted.  \emph{}\ 
\annot{} Parity signs (\cf\ \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara}) on $\Gamma$ derived from minimal proper colorings.
\dated{12 Mar 2021} 
\codes{\CLab:~\CSG}%
\itemspace

\bsgitem{20xxa}{}  S-cordial and total S-cordial labeling in signed graphs.  Submitted.  \emph{}\ 
\annot{} When does $\Lambda_{BC}(\Sigma)$ have nearly equal numbers of $+$ and $-$ vertices, or edges, or both together?
\dated{1 Oct 2021} 
\codes{\CSG,~\CVS:~\CLG}%
\itemspace

\authorx{M.\ Venkata Anusha}						%ANUSHA
\author{M.\ Venkata Anusha, R.\ Lakshmi, M.\ Siva Parvathi, \& G.S.\ Shanmuga Priya}
\authorspace
\bsgitem{2021a}{}  Seidel energy and Seidel matrix energy of Euler totient Cayley graph.  \emph{Int.\ J.\ Analytical Exper.\ Modal Anal.}\ 13 (2021), no.\ 7, 309--320.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Katsuaki Aoki}						%AOKI
\authorspace
\authref  See \hyperlink{Masao Iri}{M.\ Iri}.
\itemspace

\authorx{Mustapha Aouchiche}						%AOUCHICHE
\author{Mustapha Aouchiche \& Pierre Hansen}
\authorspace
\bsgitem{2010a}{}  A survey of automated conjectures in spectral graph theory.  \emph{Linear Algebra Appl.}\ 432 (2010), 2293--2322.
\MR{2599861} (2011b:05139).  \Zbl{1218.05087}.  
\annot{} Computer-generated conjectures.  \S4, ``Signless Laplacian'':  Several computer-generated conjectures about eigenvalues of $\Lap(-\Gamma)$; some are proved (mainly in \xref{Cvetkovi\'c, Rowlinson, and Simi\'c}{2007b}{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}) or disproved; some are difficult.  
[\Question.  How many generalize to all $\Sigma$, with or without proofs?]
\dated{22 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2013a}{}  A survey of Nordhaus--Gaddum type relations.  \emph{Discrete Appl.\ Math.}\ 161 (2013), no.\ 4-5, 466--546.
\MR{3015299}.  \Zbl{1259.05083}.  
\annot{} \S6, ``Spectral invariants'':  \S6.3, ``The eigenvalues of the signless Laplacian matrix'':  Nordhaus--Gaddum-type relations imply theorems from \xref{Gutman, Kiani, Mirzakhah, and Zhou}{2009a}{Ivan Gutman, Dariush Kiani, Maryam Mirzakhah, \& Bo Zhou} about the eigenvalues, singular values, incidence energy of $\Lap(-\Gamma)$.  Conjecture 6.19, generated by a computer---\cf\ \xrefd{2010a}{Mustapha Aouchiche \& Pierre Hansen}: $\lambda_1(\Lap(-\Gamma)) + \lambda_1(\Lap(-\Gamma^c)) \leq 3n-4$; $\lambda_1(\Lap(-\Gamma)) \cdot \lambda_1(\Lap(-\Gamma^c)) \leq 2n(n-2)$; $=$ iff $\Gamma$ is a star.
\dated{22 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2013b}{}  Two Laplacians for the distance matrix of a graph.  \emph{Linear Algebra Appl.}\ 439 (2013),  23--33.
\MR{3045220}.  \Zbl{1282.05086}.  
\annot{} The ``distance signless Laplacian'' is $\cD^L(-\Gamma) := D+\cD(\Gamma)$, where $D=$ ,diagonal degree matrix, $\cD=$ distance matrix.  Contrasts to the ``distance Laplacian'' $\cD^L(\Gamma) := D-\cD(\Gamma)$, in analogy to the Laplacian matrix $\Lap(\Gamma)=D-A$ vs.\ signless Laplacian $\Lap(-\Gamma)=D+A$.)  [\Question.  Is there a signed-graphic distance matrix $\cD(\Sigma)$ generalizing $\cD(\Gamma)$ and $-\cD(\Gamma)$, analogously to $A(\Sigma)$?  E.g., is distance algebraically additive?]
\dated{20 Mar 2016} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\bsgitem{2016a}{}  On the distance signless Laplacian of a graph.  \emph{Linear Multilinear Algebra} 64 (2016), no.\ 6, 1113--1123. 
\MR{3479404}.  \Zbl{1381.05015}.  
\annot{3479404} Further development of \xrefd{2013b}{Mustapha Aouchiche \& Pierre Hansen}.
\dated{20 Mar 2016} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Mustapha Aouchiche, Pierre Hansen, \& Claire Lucas}
\authorspace
\bsgitem{2011a}{}  On the extremal values of the second largest $Q$-eigenvalue.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10,  2591--2606.
\MR{2811141} (2012h:05184).  \Zbl{1222.05146}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Simon Apers}						%APERS
\authorspace
\authref  See \hyperlink{Florian Adriaens}{F.~Adriaens}.
\itemspace

\author{Gautam Appa}						%APPA
\authorspace
\authref  See also \hyperlink{Leonidas S.\ Pitsoulis}{L.S.~Pitsoulis}.
\authrefspace

\author{Gautam Appa \& Bal\'azs Kotnyek}
\authorx{Gautam Appa \& Balazs Kotnyek}
\authorspace
\bsgitem{2004a}{}  Rational and integral $k$-regular matrices.  \emph{Discrete Math.}\ 275 (2004), 1--15.
\MR{2026273} (2004m:05005).  \Zbl{1043.15011}.					
\annot{} $2$-regular matrices include binet matrices \xrefd{2006a}{Gautam Appa \& Balazs Kotnyek}.  A key property of $k$-regular matrices is that solutions of integral equations are $1/k$-integral.
\par
\codes{\Csg:~\CInc:~\COri}%
\itemspace

\bsgitem{2006a}{}  A bidirected generalization of network matrices.  \emph{Networks} 47 (2006), no.\ 4, 185--198.
\MR{2229861} (2008a:05157).  \Zbl{1097.05025}.					
\annot{} Binet matrices are the network matrices of bidirected (or signed) graphs.  Basic theory of binet matrices, generalizing that of network matrices, notably half-integrality theorems.  [For a slight simplification see \xref{Bolker and Zaslavsky}{2006a}{Ethan D.\ Bolker \& Thomas Zaslavsky}.]
\codes{\Csg:~\CInc:~\COri}%
\itemspace

\author{Gautam Appa, Bal\'azs Kotnyek, Konstantinos Papalamprou, \& Leonidas Pitsoulis}
\authorx{Gautam Appa, Balazs Kotnyek, Konstantinos Papalamprou, \& Leonidas Pitsoulis}
\authorspace
\bsgitem{2007a}{}  Optimization with binet matrices.  \emph{Operations Res.\ Lett.}\ 35 (2007), 345--352.
\MR{2320128} (2008a:90052).  \Zbl{1169.90407}.					
\codes{\COri:~\CInc(\CGen),~\Cmatrd}%
\itemspace

\author{Julio Aracena}						%ARACENA
\authorspace
\authref  See also \hyperlink{Felipe Benitez}{F.~Benitez}, \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, \hyperlink{Jacques Demongeot}{J.~Demongeot}, and \hyperlink{Marco Montalva}{M.~Montalva}.
\authrefspace

\bsgitem{2008a}{}  Maximum number of fixed points in regulatory Boolean networks.  \emph{Bull.\ Math.\ Biol.}\ 70 (2008), no.\ 5, 1398--1409.
\MR{2421503} (2009d:05088).  \Zbl{1144.92323}.					
\annot{} A regulatory Boolean network $N$ is built on a signed digraph $D$.  
Thm.\ 6:  If all (directed) cycles are positive then $N$ has at least 2 fixed points.  
Thm.\ 9:  $N$ has at most $2^p$ fixed points, where $p :=$ minimum number of vertices that cover all positive cycles [unusually, not negative cycles!], and this is best possible.
\dated{9 July 2009} 
\par
\codes{\CSD:~\CDyn:~\CFr(\CGen),~\CBiol}%
\itemspace

\author{J.\ Aracena, S.\ Ben Lamine, M.A.\ Mermet, O.\ Cohen, \& J.\ Demongeot}
\authorspace
\bsgitem{2003a}{}  Mathematical modeling in genetic networks:  Relationships between the genetic expression and both chromosomic breakage and positive circuits.  \emph{IEEE Trans.\ Systems, Man, Cybernetics B} 33 (2003), no.\ 5, 825--834.
\par
\codes{\CSD,~\CBiol:~\CDyn:~\CFr(\CGen)}%
\itemspace

\author{Julio Aracena, Jacques Demongeot, \& Eric Goles}
\authorspace
\bsgitem{2004a}{}  Positive and negative circuits in discrete neural networks.  \emph{IEEE Trans.\ Neural Networks} 15 (2004), no.\ 1, 77--83. 
\annot{} Existence and upper bound on the number of fixed points of a ``discrete neural network'' $\cN$, which consists of a real $n \times n$ matrix $W$, the associated signed digraph $D$ of order $n$, and a real vector $b$.  A state is $x \in \{-1,+1\}^n$.  A transition is $x \mapsto f(x) := \sgn_+(Wx-b)$ where $\sgn_+(t) := \sgn(t)$ except $\sgn_+(0) := +1$.  Assume: $D$ is connected; no component of $f$ is constant, hence a cycle exists.  
Lemma 1:  A cycle is positive iff it has a satisfied state.  
Thm.\ 1:  If all cycles are positive, $f$ has a fixed point.  
Thm.\ 2:  If all cycles are negative, $f$ has no fixed point.  
Thm.\ 3:  \#\{fixed points\} $\leq 2^p$ where $p := \min$(size of vertex cover of positive cycles), and this is sharp.
Dictionary:  ``positive feedback vertex set'' = vertex cover of positive cycles = vertex set that covers all positive cycles; ``circuit'' = (directed) cycle; {\bf 1} = $\sgn_+$.
\dated{20 July 2009} 
\par
\codes{\CSD:~\CDyn:~\CBal,~\CFr(\CGen)}%
\itemspace

\author{J.\ Aracena, E.\ Fanchon, M.\ Montalva, \& M.\ Noual}
\authorspace
\bsgitem{2011a}{}  Combinatorics on update digraphs in Boolean networks.  \emph{Discrete Math.}\ 159 (2011), no.\ 6, 401--409.
\MR{2765431} (2012b:05271).  \Zbl{1209.05103}.  
\annot{} Dictionary:  ``labeled digraph'' = signed digraph.
\codes{\CSD:~\CDyn}%
\itemspace

\author{J.\ Aracena, E.\ Goles, A.\ Moreira, \& L.\ Salinas}
\authorspace
\bsgitem{2009a}{}  On the robustness of update schedules in Boolean networks.  \emph{BioSystems} 97 (2009), 1--8.
\annot{} A ``labelled digraph'' is essentially a signed digraph.
	
\par
\codes{\CSD:~\CDyn}\codesctd{\Csd:~\CDyn,~\CBiol}%
\itemspace

\author{Julio Aracena, Mauricio Gonz\'alez, Alejandro Zu\~niga, Marco A.\ Mendez, \& Ver\'onica Cambiazo}
\authorx{Julio Aracena, Mauricio Gonzalez, Alejandro Zuniga, Marco A.\ Mendez, \& Veronica Cambiazo}
\authorspace
\bsgitem{2006a}{}  Regulatory network for cell shape changes during \emph{Drosophila} ventral furrow formation.  \emph{J.\ Theoretical Biol.}\  239 (2006), 49--62.
\MR{2224512} (no rev).  
\annot{} Figs.\ 2, 3 show particular proposed genetic regulatory networks based on signed digraphs.  \S3.2 describes how the mathematical model of \xref{Aracena, Ben Lamine, \etal{}}{2003a}{J.\ Aracena, S.\ Ben Lamine, M.A.\ Mermet, O.\ Cohen, \& J.\ Demongeot} and \xref{Aracena, Demongeot, and Goles}{2004a}{Julio Aracena, Jacques Demongeot, \& Eric Goles} applies to the situation of this paper.
\dated{20 July 2009} 
\par
\codes{\CSD:~\CDyn:~\CAppl(\CBiol)}%
\itemspace

\author{Julio Aracena, Adrien Richard, \& Lilian Salinas}
\authorspace
\bsgitem{2017a}{}  Fixed points in conjunctive networks and maximal independent sets in graph contractions.  \emph{J.\ Computer System Sci.}\ 88 (2017), 145--163.
\MR{3659350}.  \Zbl{1371.68204}.  \arXiv{1507.06141}.				
\codes{\CSD:~\CDyn}%
\itemspace

\authorx{Juli\'an Ar\'aoz}						%ARAOZ
\authorx{Julian Araoz}
\authorspace

\author{Juli\'an Ar\'aoz, William H.\ Cunningham, Jack Edmonds, \& Jan Green-Kr\'otki}
\authorx{Julian Araoz, William H.\ Cunningham, Jack Edmonds, \& Jan Green-Krotki}
\authorspace
\bsgitem{1983a}{}  Reductions to $1$-matching polyhedra.  Proc.\ Sympos.\ on the Matching Problem:  Theory, Algorithms, and Applications (Gaithersburg, Md., 1981).  \emph{Networks} 13 (1983), 455--473.  
\MR{0723693} (85d:90059).  \Zbl{525.90068}.					
\annot{} ``Minimum-cost capacitated $b$-matching problem in a bidirected graph $\Beta$'': minimize $\sum_e c_ex_e$ subject to $0 \leq x \leq u \in \{0, 1, \ldots, \infty\}^E$ and $\incid(\Beta)x = b \in \bbZ^V$.  Proves, by reduction to the ordinary perfect matching problem, \xref{Edmonds and Johnson's}{1970a}{Jack Edmonds \& Ellis L.\ Johnson} description of the convex hull of feasible solutions.  
Dictionary:  ``lobe'' = half edge.
\par
\codes{\Csg:~\COri:~\CInc,~\CAlgor,~\CGeom}%
\itemspace

\authorx{Marina Arav}						%ARAV
\author{Marina Arav, F.\ Scott Dahlgren, \& Hein van der Holst}
\authorspace
\bsgitem{2021a}{}  Signed graphs with stable maximum nullity at most two.  \emph{Linear Algebra Appl.}\ 620 (2021), 124--146.
\MR{4227776}.  
\annot{} \Cf\ \xref{Arav, Hall, Li, and Hein van der Holst}{2021a}{Marina Arav, Frank J.\ Hall, Zhongshan Li, \& Hein van der Holst}.
\codes{\CSG:~\CAdj(\CGen)}%
\itemspace

\author{Marina Arav, Frank J.\ Hall, Zhongshan Li, \& Hein van der Holst}
\authorspace
\bsgitem{2013a}{}  The inertia set of a signed graph.  \emph{Linear Algebra Appl.}\ 439 (2013), no.\ 5, 1506--1529.
\MR{3067819}.  \Zbl{1282.05058}.  \arXiv{1208.5285}.				
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Two-connected signed graphs with maximum nullity at most two.  \emph{Linear Algebra Appl.}\ 611 (2021), 82--93.
\MR{4185137}.  \Zbl{1461.05102}.  \arXiv{1407.2525}.  
\annot{} Studies $\max\nul(A)$ for symmetric matrix $A$ with sign pattern $-A(\Sigma)$, where $|\Sigma|$ is simple and 2-connected.  Equivalently, $A$ also has Strong Arnold Property.  
Thm.\ 15:  $\max\nul(A) \leq 2$ iff $\Sigma$ has no $+K_4$, $-K_4$, $\pm K_3$, or $K_{2,3}(-e)$ minor.
Dictionary:  ``even, odd'' = positive, negative; ``re-signing'' = switching.
\dated{6 Feb 2021} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{Marina Arav, Hein van der Holst, \& John Sinkovic}
\authorspace
\bsgitem{2015a}{}  On the inertia set of a signed graph with loops.  \emph{Linear Algebra Appl.}\ 471 (2015), 169--183.
\MR{3314332}.  \Zbl{1307.05091}.  \arXiv{1402.4326}.				
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{2016a}{}  On the inertia set of a signed tree with loops.  \emph{Linear Algebra Appl.}\ 510 (2016), 361--372.
\MR{3551638}.  \Zbl{1352.05081}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2016b}{\dag}  Signed graphs whose signed Colin de Verdi\`ere parameter is two.  \emph{J.\ Combin.\ Theory Ser.\ B} 116 (2016), 440--455.
\MR{3425251}.  \Zbl{1327.05142}.  \arXiv{1209.4628}.				
\codes{\CSG:~\CAdj,~\CStr}%
\itemspace

\author{Dan Archdeacon}						%ARCHDEACON*
\authorspace
\bsgitem{1992a}{}  The medial graph and voltage-current duality.  \emph{Discrete Math.}\ 104 (1992), no.\ 2,  111--141.
\MR{1172842} (93i:05051).  \Zbl{757.05045}.					
\annot{} The medial graph of $\Gamma \subset S$, a graph embedded in a surface, is a 4-regular graph $M \subset S$ that encodes $\Gamma$ and its surface dual.  Gains (``voltages'') on $\Gamma$ transfer to gains (``voltages'') on $M$.  Graphs have both gains and signs; signs are determined by face orientations.  
[\Question.  Does this suggest a gain-graphic, surface-embedding theory of 4-regular gain graphs?  It gives such a theory under certain conditions: faces are 2-colorable and black face boundaries must have balanced gains.]
Dictionary:  ``straight, twisted edge'' = positive, negative edge.
\dated{16 Jan 2012, rev 20 Nov 2016}
\codes{\CTop:~\CGG,~\CSG,~\CDual}%
\itemspace

\bsgitem{1995a}{}  Problems in topological graph theory.  Manuscript, 1995.  
URL (2/1998) \url{http://www.emba.uvm.edu/~archdeac/papers/papers.html}
\annot{} A compilation from various sources and contributors, updated every so often.  ``The genus sequence of a signed graph'', p.\ 10:  A conjecture due to  \v{S}ir\'a\v{n} (?) on the demigenus range (here called ``spectrum'' [though unrelated to matrices]) for orientation embedding of $\Sigma$, namely, that the answer to Question 1 under \xref{\v{S}ir\'a\v{n}}{1991b}{Jozef Siran} is affirmative.
[The term ``parity embedding'' is mistakenly used for orientation embedding of any signed graph; parity embedding is of an unsigned graph.]
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{1996a}{}  Topological graph theory: a survey.  
Surveys in Graph Theory (Proc., San Francisco, 1995).  \emph{Congressus Numer.}\ 115 (1996), 5--54.  
Updated version at URL (2/1998) \url{http://www.emba.uvm.edu/~archdeac/papers/papers.html}\\
\MR{1411236} (98g:05044).  \Zbl{897.05026}.
\annot{} \S2.5 describes orientation embedding (called ``signed embedding'' [although there are other kinds of signed embedding]) and switching (called ``sequence of local switches of sense'') of signed graphs with rotation systems.  \S5.5, ``Signed embeddings'', briefly mentions \xref{\v{S}ir\'a\v{n}}{1991b}{Jozef Siran}, \xref{\v{S}ir\'a\v{n} and \v{S}koviera}{1991a}{Jozef Siran \& Martin Skoviera}, and \xref{Zaslavsky}{1993a}{Thomas Zaslavsky}, \xrefd{1996a}{Thomas Zaslavsky}.
\codes{\CSG:~\CTop:~\CExp}%
\itemspace

\bsgitem{2005a}{}  Variations on a theme of Kuratowski.  \emph{Discrete Math.}\ 302 (2005), 22--31.
\MR{2179233} (2006g:05055).  \Zbl{1076.05027}.					
\annot{} Mentions [and conflates] the theorems of \xref{Zaslavsky}{1993a}{Thomas Zaslavsky}.
\dated{20 Jun 2011} 
\codes{\CTop:~\CSG:~\CExp}%
\itemspace

\author{Dan Archdeacon \& Marisa Debowsky}
\authorspace
\bsgitem{2005a}{}  A characterization of projective-planar signed graphs.  \emph{Discrete Math.}\ 290 (2005), no.\ 2--3, 109--332.
\MR{2123383} (2005j:05041).  \Zbl{1060.05039}.					
\annot{} Similar to \xref{Archdeacon--\v{S}ir\'a\v{n}}{1998a}{Dan Archdeacon \& Jozef Siran}, but for the projective plane.
\par
\codes{\CSG,~\CSw:~\CTop}%
\itemspace

\author{Dan Archdeacon, Joan Hutchinson, Atsuhiro Nakamoto, Seiya Negam [Seiya Negami], \& Katsuhiro Ota}
\authorspace
\bsgitem{2001a}{}  Chromatic numbers of quadrangulations on closed surfaces.  \emph{J.\ Graph Theory} 37 (2001), no.\ 2, 100--114.
\MR{1829924} (2002j:05044).  \Zbl{0979.05034}.  
\annot{} 	
[\Cf\ \xref{Nakamoto, Negami, and Ota}{2002a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}, \xrefd{2004a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}.]
\codes{\Csg:~\CTop:~\Csw}%
\itemspace

\author{Dan Archdeacon \& Jozef \v{S}ir\'a\v{n}}
\authorx{Dan Archdeacon \& Jozef Siran}
\authorspace
\bsgitem{1998a}{}  Characterizing planarity using theta graphs.  \emph{J.\ Graph Theory} 27 (1998), 17--20.  
\MR{1487782} (98j:05055).  \Zbl{887.05016}.					
\annot{} A ``claw'' consists of a vertex and three incident half edges.  Let $C$ be the set of claws in $\Gamma$ and $T$ the set of theta subgraphs.  Fix a rotation of each claw.  Call $t \in T$ an ``edge'' with endpoints $c, c'$ if $t$ contains $c$ and $c'$; sign it $+$ or $-$ according as $t$ can or cannot be embedded in the plane so the rotations of its trivalent vertices equal the ones chosen for $c$ and $c'$.  This defines, independently (up to switching) of the choice of rotations, the ``signed triple graph'' $T^{\pm}(\Gamma)$.  Theorem: $\Gamma$ is planar iff $T^{\pm}(\Gamma)$ is balanced.
\codes{\CSG,~\CSw:~\CTop}%
\itemspace

\author{Federico Ardila}						%ARDILA
\authorspace
\bsgitem{2002a}{}  The Tutte polynomial of a hyperplane arrangement (extended abstract).  \emph{}\ 
\annot{} Extended abstract of \xrefd{2007a}{Federico Ardila}.
\dated{4 Oct 2014} 
\par
\codes{\Cgg:~\Cbal:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2003a}{}  \emph{Enumerative and Algebraic Aspects of Matroids and Hyperplane Arrangements}.  Ph.D.\ thesis, Massachusetts Institute of Technology, Cambridge, Mass., 2003.
MR{2717078} (no rev).  
\annot{} Ch.\ 2 is published as \xrefd{2007a}{Federico Ardila}.
\dated{4 Oct 2014} 
\par
\codes{\Cgg:~\Cbal:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2007a}{}  Computing the Tutte polynomial of a hyperplane arragement [\sic].  \emph{Pacific J.\ Math.}\ 230 (2007), no.\ 2, 1--26.
\MR{2318445} (2008g:52034).  \Zbl{1152.52011}.  \arXiv{math/0409211}.			
\annot{} Applies the ``finite field method'' of \xref{Athansiadis}{1996a}{Christos A.\ Athanasiadis} (in \S5 this is actually the modular gains method for computing chromatic polynomials of integral gain graphs), cleverly extended, to compute Tutte polynomials (in the equivalent form of coboundary polynomials) of various gain graphs (in the equivalent form of affine hyperplane arrangements).  
\annotctd{} Thms.\ 4.2, 4.3:  Coboundary polynomial of complete signed graph $\pm K_n\full$ (``$\cB_n$ arrangement'') and complete signed link graph $\pm K_n$ (``$\cD_n$ arrangement'') in terms of generating functions.  
Thm.\ 4.4:  Balanced coboundary polynomial of a contrabalanced multigraph $(\Gamma,\eset)$ (coboundary polynomial of ``$\cA_n^\#$ arrangement'' [equivalently, the bicircular lift matroid $\bfL(\Gamma,\eset)$]).  [Also computed, slightly generalized, in \xref{Zaslavsky}{1995b}{Thomas Zaslavsky}, Ex.\ 3.4, $q^\textb_{(\Gamma,\eset)}$.]
Thm.\ 4.5:  Coboundary polynomial of $-K_n$ (``threshold arrangement'' $\cT_n = \cH[-K_n]$).  
\annotctd{} \S5, ``Deformations of the braid arrangement'':  Balanced subgraphs of $A$-expansions $A\cdot K_n$, where $A$ is a finite set of integers, are employed to compute the coboundary polynomials of integral deformations $\cH[A\cdot K_n]$ of the complete-graph (```braid'') arrangement $\cH[K_n]$.  Prop.\ 5.8 treats the Catalan arrangement $\cH[\{0,\pm1\}\cdot K_n]$ and its subarrangements.  Thm.\ 5.14 treats the Catalan arrangement.  Prop.\ 5.9 and Thm.\ 5.11 treat the Lineal arrangement $\cH[\{1\}\cdot K_n$.  Thm.\ 5.12 treats the Shi arrangement $\cH[\{0,1\}\cdot K_n]$.  Thm.\ 5.13 treats the semiorder arrangement $\cH[\{\pm1\}\cdot K_n]$.  
\annotctd{} Dictionary:  ``finite field method'' usually = modular gains method (\xref{Zaslavsky}{2002a}{Thomas Zaslavsky}, \S11.4 after (11.3); or see \xref{Berthom\'e \etal\!}{2009a}{Pascal Berthome, Raul Cordovil, David Forge, Veronique Ventos, \& Thomas Zaslavsky}, Lemma 6.3); ``type'' = gain of edge; ``planted graded graph with height function'' = balanced integral gain graph with potential function; ``plant\-ed graded $A$-graph'' = balanced subgraph of $A\cdot K_n$; $\cE_n$ = $\cH[A\cdot K_n]$; ``threshold arrangement'' = all-negative complete graph arrangement.  
\dated{4 Oct 2014} 
\codes{\Csg,~\Cgg:~\Cbal:~\CGeom,~\CInvar}%
\itemspace

\author{Federico Ardila, Federico Castillo, \& Michael Henley}
\authorspace
\bsgitem{2014a}{}  The arithmetic Tutte polynomials of the classical root systems.  In:  \emph{26th International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC 2014, Chicago), pp.\ 851--862.  Discrete Math.\ Theor.\ Computer Sci.\ Proc., AT.  Assoc.\ Discrete Math.\ Theor.\ Computer Sci., Nancy, 2014.
\MR{3466427}.  
\annot{} Extended abstract of \xrefd{2015a}{Federico Ardila, Federico Castillo, \& Michael Henley}.
\par
\codes{\CSG:~\CMatrd(\CGen):~\CInvar,~\CEnum}\codesctd{\CSG:~\CGeom:~\CExp}%
\itemspace

\bsgitem{2015a}{}  The arithmetic Tutte polynomials of the classical root systems.  \emph{Int.\ Math.\ Res.\ Notices} 2015 (2015), no.\ 12, 3830--3877.
\MR{3356740}.  \Zbl{1317.05091}.  \arXiv{1305.6621}.				
\annot{} These invariants of signed graphs are obtained by a substitution in generating functions of the ordinary Tutte polynomials.  G.f.\ computed for $A_{n-1}, B_n, C_n, D_n$ with respect to integer, root, and weight lattices.  
\S4.1.2, ``Enumeration of signed graphs'':  Counted by several parameters, e.g., number of balanced components 
\S4.2,  ``From classical root systems to signed graphs'':  Expository.  $A_{n-1}, B_n, C_n, D_n \leftrightarrow +K_n, \pm K_n', \pm K_n', \pm K_n$, ${}'$ denoting half edges [but $C_n$ should $\leftrightarrow \pm K_n^\circ$, with negative loops].  
\S4.3, ``Computing the Tutte polynomials by signed graph enumeration''.   
\S5,  ``Arithmetic characteristic polynomials'':  Explicit, for some cases.  
Dictionary:  ``loop'' = half edge (Lemma 4.9 depends on that).  [Lemma 4.9: special case of \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Lemma 8A.2.]  
\annotctd{} [\Question.  How to compute arithmetic Tutte polynomials and characteristic polynomials for other signed graphs?  Preferably, directly.  How does the lattice framework fit in?]
\dated{26 May 2018} 
\par
\codes{\CSG:~\CMatrd(\CGen):~\CInvar,~\CEnum}\codesctd{\CSG:~\CGeom:~\CExp}%
\itemspace

\author{Federico Ardila \& Alexander Postnikov}
\authorspace
\bsgitem{2010a}{}  Combinatorics and geometry of power ideals.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 362 (2010), no.\ 8, 4357--4384.
\MR{2608410} (2011g:05322).  \Zbl{1226.05019}.  \arXiv{0809.2143}.			
\annot{} The polynomials $\widetilde{Z}_\cA$ (\xref{Sokal's}{2005a}{Alan D.\ Sokal} ``multivariate Tutte polynomial''), slightly normalized, and $S_\cA$ are specializations of \xref{Traldi's}{1989a}{Lorenzo Traldi} ``weight\-ed dichromatic polynomial'', hence of \xref{Zaslavsky's}{1992b}{Thomas Zaslavsky} ``para\-metrized'' dichromatic and corank-nullity polynomials.
\dated{17 Oct 2017} 
\par
\codes{\CSGw(\CGen):~\CInvar,~\CGeom}%
\itemspace

\bsgitem{2015a}{}  Correction to ``Combinatorics and geometry of power ideals'':  two counterexamples for power ideals of hyperplane arrangements.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 367 (2015), no.\ 5, 3759--3762. 
\MR{3314823}.  \Zbl{1315.05009}.  \arXiv{1211.1368}.				
\codes{\CGeom}%
\itemspace

\author{Samin Aref}							%AREF
\authorspace
\bsgitem{2019a}{}  \emph{Signed Network Structural Analysis and Applications with a Focus on Balance Theory}.  Doctoral thesis, Univ.\ of Auckland, 2019. 
\arXiv{1901.06845}.  
\par
\codes{\CSG:~\CBal}%
\itemspace

\author{Samin Aref, Andrew J.\ Mason, \& Mark C.\ Wilson}
\authorspace
\bsgitem{2018a}{}  Computing the line index of balance using integer programming optimisation.  In:  Boris Goldengorin, ed., \emph{Optimization Problems in Graph Theory: In Honor of Gregory Z.\ Gutin's 60th Birthday}, pp.\ 65--84.  Springer Optim.\ Appl., Vol.\ 139.  Springer, Cham, 2018.
\MR{3838926}.  \Zbl{1416.05128}.  \arXiv{1710.09876}.				
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2020a}{}  A modeling and computational study of the frustration index in signed networks.  \emph{Networks} 75 (2020), no.\ 1, 95--110.
\MR{4045043}.  
\arXiv{1611.09030}.								
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{Samin Aref \& Zachary Neal}
\authorspace
\bsgitem{20xxa}{}  Legislative effectiveness hangs in the balance: Studying balance and polarization through partitioning signed networks.   Submitted.  \emph{}\ 
\arXiv{1906.01696}.								
\annot{} Two-step algorithm for the NP-complete problem of frustration index $l(\Sigma)$: first find upper and lower bounds by fast methods (by examples and LP relaxation), then use that to control the amount of computation.  It also finds an optimal bipartition.
\dated{15 Jul 2019} 
\codes{\CSG:~\CFr:~\CAlgor,~\CPsS}%
\itemspace

\author{Samin Aref \& Mark C.\ Wilson}
\authorspace
\bsgitem{2018a}{}  Measuring partial balance in signed networks.  \emph{J.\ Complex Networks} 6 (2018), no.\ 4, 566--595.
\MR{3842390}.  \Zbl{1460.90039}.  \arXiv{1509.04037}.				
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2019a}{}  Balance and frustration in signed networks.  \emph{J.\ Complex Networks} 7 (2019), no.\ 2,163--189.
\MR{3941371}.  
\arXiv{1712.04628}.								
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{Alex Arenas}							%ARENAS
\authorspace
\authref  See \hyperlink{Sergio Gomez}{S.~G\'omez}.
\itemspace

\authorx{Srinivasa R.\ Arikati}						%ARIKATI
\author{Srinivasa R.\ Arikati \& Uri N.\ Peled}
\authorspace
\bsgitem{1993a}{}  A linear algorithm for the group path problem on chordal graphs.  \emph{Discrete Appl.\ Math.}\ 44 (1993), 185--190.  
\MR{1227703} (94h:68084).  \Zbl{779.68067}.
\annot{} Given a graph with edges weighted from a group.  The weight of a path is the product of its edge weights in order (not inverted, as with gains).  Problem: to determined whether between two given vertices there is a chordless path of given weight.  This is NP-complete in general but for chordal graphs there is a fast algorithm (linear in $(\#E+\#V)\cdot(\text{group order})$).  
[\Question.  What if the edges have gains rather than weights?]
\codes{\CWG:~\Cpar(\CGen):~\CAlgor}%
\itemspace

\bsgitem{1996a}{}  A polynomial algorithm for the parity path problem on perfectly orientable graphs.  \emph{Discrete Appl.\ Math.}\ 65 (1996), 5--20.  
\MR{1380065} (96m:05120).  \Zbl{854.68069}.
\annot{} Problem: Does a given graph contain an induced path of specified parity between two prescribed vertices?  A polynomial-time algorithm for certain graphs.  (\Cf\ \xref{Bienstock}{1991a}{Dan Bienstock}.)  
[\Problem.  Generalize to paths of specified sign in a signed graph.]
\codes{\Cpar:~\CAlgor}\codesctd{\CRef}%
\itemspace

\authorx{Nejat Ar{\i}n{\i}k}						%ARINIK
\authorx{Nejat Arinik}	
\authorspace

\author{Nejat Ar{\i}n{\i}k, Rosa Figueiredo, \& Vincent Labatut}
\authorx{Nejat Arinik, Rosa Figueiredo, \& Vincent Labatut}
\authorspace
\bsgitem{2017a}{}  Signed graph analysis for the interpretation of voting behavior.  In:  \emph{International Conference on Knowledge Technologies and Data-driven Business (i-KNOW)} (Graz, Austria, 2017). 
\arXiv{1712.10157}.  \HAL{hal-01583133}.  
\codes{\CSG}%
\itemspace

\bsgitem{2020a}{}  Multiple partitioning of multiplex signed networks: Application to European Parliament votes.  \emph{Social Networks} 60 (2020), 83--102.
\arXiv{1811.10337}.  \HAL{hal-02082574}.	 
\codes{\CSG:~\CClu:~\CAlgor,~\CAppl}%
\itemspace

\bsgitem{2020b}{}  Multiplicity and diversity: Analyzing the optimal solution space of the correlation clustering problem on complete signed graphs.  \emph{J.\ Complex Networks} 8 (2020), no.\ 6, art.\ cnaa025.
\arXiv{2011.05196}.  \HAL{hal-02994011}.						
\par
\codes{\CSG:~\CKG:~\CFr}%
\itemspace

\bsgitem{20xxb}{}  2-Edge connected balanced subgraphs for correlation clustering problem.  Submitted.  \emph{}\ 
\HAL{hal-02428305}.						
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\authorx{Esther M.\ Arkin}						%ARKIN
\author{Esther M.\ Arkin \& Christos H.\ Papadimitriou}
\authorspace
\bsgitem{1985a}{}  On negative cycles in mixed graphs.  \emph{Operations Res.\ Lett.}\ 4 (1985), 113--116.  
\MR{0821170} (87h:68061).  \Zbl{585.05017}.					
\annot{} Reinterpreted:  Treat a mixed graph as a bidirected graph $\Beta$ where a walk can only enter a vertex along an in-arrow (thus introverted edges are irrelevant).  Given integral gains in $\bbZ^+$, finding negative cycles is NP-complete, though it is P if $\Beta$ is all positive or all negative.  Also, ``it is NP-complete to find shortest cycles \ldots, \emph{even when the graph is known not to have negative cycles.}''
\dated{18 Sept 2018} 
\codes{\Cori:~\COG:~\CCycles}%
\itemspace

\bsgitem{1986a}{}  On the complexity of circulations.  \emph{J.\ Algorithms} 7 (1986), 134--145.
\MR{0834086} (88a:68033).  \Zbl{603.68039}.					
\annot{} As in \xrefd{1985a}{Esther M.\ Arkin \& Christos H.\ Papadimitriou}, finding a feasible circulation is NP-complete for a general bidirected $\Beta$ but is P for all-negative (and all-positive) $\Beta$.
\dated{18 Sept 2018} 
\codes{\Cori:~\COG:~\CFlows}%
\itemspace

\author{Drew Armstrong}						%ARMSTRONG
\authorspace
\bsgitem{2011a}{}  Hyperplane arrangements and diagonal harmonics.  In:  \emph{23rd International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC 2011, Reykjavik, 2011), pp.\ 39--50.  Discrete Math.\ Theor.\ Comput.\ Sci.\ Proc., AO.  Assoc.\ Discrete Math.\ Theor.\ Comput.\ Sci., Nancy, 2011.
\MR{2820696}.  \Zbl{1355.52014}.	  
\annot{} Introduces the Ish arrangement; \cf\ \xref{Armstrong and Rhoades}{2012b}{Drew Armstrong \& Brendon Rhoades}.  Extended abstract of \xrefd{2013a}{Drew Armstrong}.
\dated{14 Mar 2013} 
\par
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2013a}{}  Hyperplane arrangements and diagonal harmonics.  \emph{J.\ Combin.}\ 4 (2013), no.\ 2, 157--190.
\MR{3096132}.  \Zbl{6222541}.  \arXiv{1005.1949}.					
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\author{Drew Armstrong \& Brendon Rhoades}
\authorspace
\bsgitem{2012a}{}  The Shi arrangement and the Ish arrangement.  In:  \emph{23rd International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC 2011, Reykjavik, 2011), pp.\ 51--62.  Discrete Math.\ Theor.\ Comput.\ Sci.\ Proc., AO.  Assoc.\ Discrete Math.\ Theor.\ Comput.\ Sci., Nancy, 2011.
\Zbl{1355.52015}.  
\annot{} Extended abstract of \xrefd{2012b}{Drew Armstrong \& Brendon Rhoades}.
\dated{14 Mar 2013} 
\par
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2012b}{}  The Shi arrangement and the Ish arrangement.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 364 (2012), no.\ 3, 1509--1528.
\MR{2869184}.  \Zbl{1238.05271}.  \arXiv{1009.1655}.					
\annot{} The integral gain graphs $0K_n \cup 1\vec\Gamma$ (the Shi gain graph, if $\Gamma=K_n$) and $0K_n \cup \{ie_{1j}: i \in [j-1]\}$ (the Ish gain graph, if $\Gamma=K_n$) have the same chromatic polynomial (but not Tutte polynomial) and the same numbers of acyclic orientations with specified properties.  [$\vec\Gamma$ has $V=[n]$ and edges oriented upwards.]  Proofs are [in effect] by counting proper integral colorations modulo a large prime.  The results and proofs are cast in terms of the Shi and Ish hyperplane arrangements; \cf\ \xref{Athanasiadis}{1996a}{Christos A.\ Athanasiadis}, whose method is used, called the ``finite field method''.  
[\Cf\ \xref{Leven, Rhoades, and Wilson}{2014a}{Emily Leven, Brendon Rhoades, \& Andrew Timothy Wilson}.]
\dated{14 Mar 2013} 
\par
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\author{S.\ Arockiaraj}						%AROCKIARAJ
\authorspace
\authref  See \hyperlink{P.\ Jeyalakshmi}{P.~Jeyalakshmi}.
\itemspace

\authorx{A.\ Aromsawa}						%AROMSAWA
\author{A.\ Aromsawa and J.\ Poulter}
\authorspace
\bsgitem{2007a}{}  Domain wall entropy of the bimodal two-dimensional Ising spin glass.  \emph{Phys.\ Rev.\ B} 76 (2007), art.\ 064427, 5 pp.
\codes{\CSG:~\Cfr,~\CPhys}%
\itemspace

\author{S.\ Arumugam}						%ARUMUGAM
\authorspace
\authref  See \hyperlink{V.\ Vasanthi}{V.~Vasanthi}.
\itemspace

\author{Ashraf P K}							%ASHRAF PK
\authorx{P.K.\ Ashraf}
\authorspace
\authref  See also \hyperlink{K.A.\ Germina}{K.A.~Germina} and \hyperlink{Naduvath K.\ Sudev}{N.K.~Sudev}.
\authrefspace

\author{P.K.\ Ashraf \& K.A.\ Germina}
\authorspace
\bsgitem{2014a}{}  Neighbourhood balanced domination in signed graphs.  \emph{Int.\ J.\ Math.\ Sci.\ Engineering Appl.}\ 8 (2014), no.\ 3, 193--203.
\codes{\CSG:~\CDom}%
\itemspace

\bsgitem{2015a}{}  On minimal dominating sets for signed graphs.  \emph{Adv.\ Appl.\ Discrete Math.}\ 15 (2015), no.\ 2, 101--112.
\MR{3381741}.  \Zbl{1328.05138}.  
\annot{} Definition (from \xref{B.D.\ Acharya}{2013a}{B.\ Devadas Acharya}):  $D \subseteq V(\Sigma)$ is ``dominating'' if for some vertex signature $\mu$, $u\notin D$ implies $\exists\ v \in N(u)\cap D$, and $\mu(v) = \sigma(uv)\mu(u), \forall\ v \in N(u)\cap D$.  Characterizes minimal such sets $D$ (dominating in $|\Sigma|$ and three side conditions), and more.
\dated{24 Mar 2017} 
\codes{\CSG:~\CDom:~\CVS,~\CInvar}%
\itemspace

\bsgitem{2016a}{}  Double domination in signed graphs.  \emph{Cogent Math.}\ 3 (2016), art.\ 1186135, 9 pp.
\MR{3625344}.  
\annot{} Double dominating set $D$ of $\Sigma$: double dominating set for $|\Sigma|$ such that the cut $E(D,D^c)$ is balanced.  $\gamma_{\times2}(\Sigma) := \min|D|$ can be 2 or $n$.  
\codes{\CSG:~\CDom}%
\itemspace

\author{P.K.\ Ashraf, K.A.\ Germina, \& N.K.\ Sudev}
\authorspace
\bsgitem{2018a}{}  A study on set-valuations of signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2018 (2018), no.\ 1, 34--40.
\arXiv{1610.00698}.  
\annot{}  Set-valuation: $f: V \to \cP(X)$ (injective), $X$ a set, $f^\oplus(vw) := f(v) \oplus f(w)$; define $\Sigma$ by $\sigma(vw) := (-1)^{|f(vw|}$.  Long proofs, results trivial or false except Thms.\ 2.9--10:  $\Sigma$ has a set-valuation iff balanced. [Quick proof: switch by $\zeta(v)=(-1)^{|f(v)|}$.]
\dated{3 Mar 2019} 
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{2019a}{}  On certain associated graphs of set-valued signed graphs.  \emph{Malaya J.\ Mat.}\ 7 (2019), no.\ 1, 113--117.
\MR{3938245}.  
\HAL{hal-02263292}.	  
\annot{} Continues \xrefd{2018a}{P.K.\ Ashraf, K.A.\ Germina, \& N.K.\ Sudev}.  Long proofs of trivial balance properties.
\dated{3 Mar 2019} 
\codes{\CSG:~\CBal}%
\itemspace

\author{Ali Reza Ashrafi}						%ASHRAFI
\authorspace
\authref  See \hyperlink{Zahra Yarahmadi}{Z.~Yarahmadi}.
\itemspace

\authorx{Jalal Askari}							%ASKARI
\author{Jalal Askari, Ali Iranmanesh, \& Kinkar Ch Das}
\authorspace
\bsgitem{2016a}{}  Seidel-Estrada index.  \emph{J.\ Inequalities Appl.}\ 2016 (2016), art.\ 120, 9 pp.
\MR{3486633}.  \Zbl{1335.05106}.  
\annot{} Seidel--Estrada index $:= \sum e^\lambda$ over all eigenvalues of $A(K_\Gamma)$.  (Estrada index applied to signed complete graphs.)  Bounds, and comparison with Seidel energy $\sum |\lambda|$.
\dated{9 Nov 2020} 
\par
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Fatihcan M.\ Atay}						%ATAY
\authorspace
\bsgitem{2012a}{}  On delay-induced stability in diffusively coupled discrete-time systems.  \emph{Afrika Mat.}\ 23 (2012), no.\ 1, 109--119.
\MR{2897768}.  \Zbl{1253.93078}.  
\par
\codes{\CSG,~\CSD:~\CWG:~\CDyn}%
\itemspace

\author{Fatihcan M. Atay \& Bobo Hua}
\authorspace
\bsgitem{2016a}{}  On the symmetry of the Laplacian spectra of signed graphs.  \emph{Linear Algebra Appl.}\ 495 (2016), 24--37.
\MR{3462985}.  \Zbl{1331.05134}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Fatihcan M.\ Atay \& Shiping Liu}
\authorspace
\bsgitem{2020a}{}  Cheeger constants, structural balance, and spectral clustering analysis for signed graphs.  \emph{Discrete Math.}\ 343 (2020), no.\ 1, art.\ 111616, 26 pp.
\MR{4039398}.  \Zbl{1429.05120}.  \arXiv{1411.3530}.				
\codes{\CSG:~\CBal,~\CFr:~\CInvar,~\CEig,~\CAlgor}%
\itemspace

\author{Fatihcan M.\ Atay \& Hande Tun\c{c}el}							%
\authorspace
\bsgitem{2014a}{}  On the spectrum of the normalized Laplacian for signed graphs: Interlacing, contraction, and replication.  \emph{Linear Algebra Appl.}\ 442 (2014), 165--177.
\MR{3134361}.  \Zbl{1282.05088}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Christos A.\ Athanasiadis}					%ATHANASIADIS
\authorspace
\bsgitem{1996a}{\dag}  Characteristic polynomials of subspace arrangements and finite fields.  \emph{Adv.\ Math.}\ 122 (1996), 193--233.  
\MR{1409420} (97k:52012).  \Zbl{872.52006}.					
\annot{} Treats the canonical lift representations (as affine hyperplane arrangements) of various gain graphs and signed gain graphs with additive gain group $\bbZ^+$.  The article is largely a series of (sometimes brilliant) calculations of chromatic polynomials (\emph{mutatis mutandis}, the characteristic polynomials of the representing arrangements) modulo a large integer $q$ using gain graph coloring, though disguised as applications of Crapo--Rota's Critical Theorem.  The fundamental principle is that, if $q$ is larger than the largest gain of a circle, then $\bbZ^+$ can be replaced as gain group by $\bbZ_q^+$ without changing the chromatic polynomial (a consequence of \xref{Zaslavsky}{1995b}{Thomas Zaslavsky}, Thm.\ 4.2)---and the analog for signed gain graphs, whose theory needs to be developed.  A non-graphical result of the general method is a unified proof (Thm.\ 2.4) of the theorem of \xref{Blass and Sagan}{1998a}{Andreas Blass \& Bruce Sagan}.
\annotctd{} \S3, ``The Shi arrangements'': these represent $\Latb \{0,1\}\vec K_n$ and signed-graph analogs.  \S4: ``The Linial arrangement'': it represents $\Latb \{1\}\vec K_n$.
\S5, ``Other interesting hyperplane arrangements'', treats: the arrangement representing $\Latb AK_n$ where $A = \{-m, \ldots, m-1, m\}$ [which is the semilattice of $m$-composed partitions; see \xref{Zaslavsky}{2002a}{Thomas Zaslavsky}, Ex.\ 10.5, also \xref{Edelman--Reiner}{1996a}{Paul H.\ Edelman \& Victor Reiner}], and several generalizations, including to arbitrary sign-symmetric gain sets $L$ and to Weyl analogs; also, an antibalanced analog of the $A_n$ Shi arrangement (Thm.\ 5.4); and more.  Most impressive result:  Thm.\ 5.2:  Let $A$ be a finite set of integers such that $0 \notin A = -A$ and let $A^0 = A \cup \{0\}$.  For $\Phi = A^0K_n$ and large integral $\lambda$, $\chi^*_\Phi(\lambda)/\lambda$ is the coefficient of $x^{\lambda-n}$ in $(1-x)\inv - f_A(x)/x$ where $f_A$ is the ordinary generating function for $A$.  From this $\chi^*_{AK_n}(\lambda)/\lambda$ is derived.
\annotctd{} [The signed affinographic arrangements represent a kind of signed gain graph whose exact nature has not yet been penetrated by gain graph theory.]
\codes{\Csg,~\Cgg:~\CGeom,~\CMatrd,~\CInvar}%
\itemspace

\bsgitem{1996b}{}  \emph{Algebraic Combinatorics of Graph Spectra, Subspace Arrangements and Tutte Polynomials}.  Doctoral dissertation, Massachusetts Institute of Technology, 1996.
\MR{2716640} (no rev).  
\codes{\CSG:~\CGeom,~\CInvar,~\CCol:~\CExp}%
\itemspace

\bsgitem{1997a}{}  A class of labeled posets and the Shi arrangement of hyperplanes.  \emph{J.\ Combin.\ Theory Ser.\ A} 80 (1997), 158--162.  
\MR{1472110} (98d:05008).  \Zbl{970.66662}.  
\annot{} The arrangement represents $\Latb \{0,1\}\vec K_n$.
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar}%
\itemspace

\bsgitem{1998a}{}  On free deformations of the braid arrangement.  \emph{European J.\ Combin.}\ 19 (1998), 7--18.
\MR{1600259} (99d:52008).  \Zbl{898.52008}.  
\annot{} The arrangements considered are the subarrangements of the projectivized Shi arrangements of type $A_{n-1}$ that contain $A_{n-1}$.  Thms.\ 4.1 and 4.2 characterize those that are free or supersolvable.  The extended Shi arrangements, representing $\bfL_0([1-a,a]\vec K_n)$ where $a \geq 1$, and a mild generalization, are of use in the proof.
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar}%
\itemspace

\bsgitem{1998b}{}  On noncrossing and nonnesting partitions for classical reflection groups.  \emph{Electronic J.\ Combin.}\ 5 (1998), Research Paper R42, 16 pp.
\MR{1644234} (99i:05204).  \Zbl{898.05004}.  
\annot{} \S5, ``Nonnesting partitions of fixed type'', has calculations like those in \xrefd{1996a}{Christos A.\ Athanasiadis} for affinographic arrangements representing additional types of integral gain graph [of a kind that is not yet fully understood].
\par
\codes{\Cgg:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\bsgitem{1999a}{}  Extended Linial hyperplane arrangements for root systems and a conjecture of Postnikov and Stanley.  \emph{J.\ Algebraic Combin.}\ 10 (1999), 207--225.
\MR{1723184} (2000i:52039).  \Zbl{0948.52012}.  \arXiv{math/9705223}.			
\annot{} Brief description as at \xrefd{1998b}{Christos A.\ Athanasiadis}.  Solves a conjecture of \xref{Postnikov and Stanley}{2000a}{Alexander Postnikov \& Richard P.\ Stanley}.
\dated{27 May 2018} 
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{1999b}{}  Piles of cubes, monotone path polytopes, and hyperplane arrangements.  \emph{Discrete Comput.\ Geom.}\ 21 (1999), no.\ 1, 117--130.
\MR{1661295} (99j:52015).  \Zbl{979.52002}.					
\annot{} The proof of Proposition 4.2 is essentially gain-graphic.
\par
\codes{\Cgg:~\Cmatrd:~\CGeom:~\CInvar}%
\itemspace

\bsgitem{2000a}{}  Deformations of Coxeter hyperplane arrangements and their characteristic polynomials.  
In:  Michael Falk and Hiroaki Terao, eds., \emph{Arrangements---Tokyo, 1998}, pp.\ 1--26.  Adv.\ Studies Pure Math., 27.  Kinokuniya, for the Mathematical Soc.\ of Japan, Tokyo, 2000.
\MR{1796891} (2001i:52035).  \Zbl{976.32016}.					
\par
\codes{\Cgg:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\bsgitem{2004a}{}  Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes.  \emph{Bull.\ London Math.\ Soc.}\ 36 (2004), 294--302.
\MR{2038717} (2005b:52055).  \Zbl{1068.20038}.					
\codes{\Cgg:~\CGeom:~\CGen:~\CInvar}%
\itemspace

\bsgitem{2004b}{}  On a refinement of the generalized Catalan numbers for Weyl groups.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 357 (2004), no.\ 1, 179--196.
\MR{2098091} (2005h:20091).  \Zbl{1079.20057}.					
\codes{\Cgg:~\CGeom:~\CGen:~\CInvar}%
\itemspace

\author{Christos A.\ Athanasiadis \& Svante Linusson}
\authorspace
\bsgitem{1999a}{}  A simple bijection for the regions of the Shi arrangement of hyperplanes.  \emph{Discrete Math.}\ 204 (1999), 27--39.
\MR{1691861} (2000f:52031).  \Zbl{959.52019}.					
\codes{\Cgg:~\CGeom}%
\itemspace

\author{David Avis}							%AVIS
\authorspace
\authref  See \hyperlink{Jin Akiyama}{J.~Akiyama}.
\itemspace

\authorx{Remi C.\ Avohou}						%AVOHOU
\author{Remi C.\ Avohou, Joseph Ben Geloun, \& Etera R.\ Livine}
\authorspace
\bsgitem{2014a}{}  On terminal forms for topological polynomials for ribbon graphs: The $N$-petal flower.  \emph{European J.\ Combin.}\ 36 (2014), 348--366.
\MR{3131901}.  \Zbl{1284.05067}.  \arXiv{1212.5961}.				
\codes{\Csg:~\CTop:~\CInvar}%
\itemspace

\authorx{F.\ Ayoobi}							%AYOOBI
\author{F.\ Ayoobi, G.R.\ Omidi, \& B.\ Tayfeh-Rezaie}
\authorspace
\bsgitem{2011a}{}  A note on graphs whose signless Laplacian has three distinct eigenvalues.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 6, 701--706.
\MR{2801363} (2012i:05158).  \Zbl{1223.05169}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ghodratollah Azadi}						%AZADI
\authorspace
\authref  See \hyperlink{Vahid Ghorbani}{V.~Ghorbani}.
\itemspace

\author{Habib Azanchiler}						%AZANCHILER
\authorspace
\authref  See \hyperlink{Vahid Ghorbani}{V.~Ghorbani}.
\itemspace


\label{LETTER B} \hypertarget{LETTER B}{}	% LETTER  B


\authorx{L\'aszl\'o Babai}						%BABAI
\authorx{Laszlo Babai}
\author{L.\ Babai \& P.J.\ Cameron}
\authorspace
\bsgitem{2000a}{}  Automorphisms and enumeration of switching classes of tournaments.  \emph{Electronic J.\ Combin.}\ 7 (2000), Research Paper R38, 25 pp.
\MR{1773295} (2001h:\-05048).  \Zbl{956.05050}.					
\annot{} Tournaments are treated as nowhere-zero $\GF(3)^+$-gain graphs based on $K_n$; ``switching'' is by negation in $\GF(3)^+$.  
[\xref{Cheng--Wells}{1986a}{Ying Cheng \& Albert L.\ Wells, Jr.} treats all digraphs as $K_n$ with $\GF(3)^+$-gains.  $\GF(3)^+$-gain switching differs from Babai--Cameron's switching.]
\dated{rev.\ 7 Jan 2016, 4 Nov 2017}
\codes{\Cgg:~\CKG:~\CSw,~\CAut,~\CEnum}%
\itemspace

\author{Maxim A.\ Babenko}						%BABENKO
\authorspace
\bsgitem{2006a}{}  Acyclic bidirected and skew-symmetric graphs: algorithms and structure.  
In:  Dima Grigoriev, John Harrison and Edward A.\ Hirsch, eds., \emph{Computer Science---Theory and Applications} (Proc.\ 1st Int.\ Symp.\ Computer Sci.\ in Russia, CSR 2006, St.\ Petersburg, 2006), pp.\ 23--34.  \LNCS 3967.  Springer, Berlin, 2006.
\MR{2260979} (2007f:05165).  \Zbl{1185.05133}.					
\annot{} ``Skew-symmetric graph'' = double covering digraph of a bidirected $-\Gamma$.  ``Weak acyclicity'':  No positive dicycle.  ``Strong acyclicity'':  No positive closed diwalk.  Algorithm to test for weak acyclicity.  Construction of weakly acyclic graphs from strongly acyclic ones.
\dated{9 Sept 2010} 
\par
\codes{\Csg:~\COri:~\CStr,~\CCov,~\CAlgor}%
\itemspace

\bsgitem{2006b}{}  On flows in simple bidirected and skew-symmetric networks.  (In Russian.)  
\emph{Problemy Peredachi Informatsii} 42 (2006), no.\ 4, 104--120.  
English trans.\ \emph{Probl.\ Inf.\ Transm.}\ 42 (2006), no.\ 4, 356--370.
\MR{2278815} (2008i:90013).  
\annot{} $O(mn^{2/3})$ algorithm for integral max flow, improving on \xref{Gabow}{1983a}{H.N.\ Gabow}, showing that max flow takes no longer on a bidirected graph than on a digraph.  The time bound follows from an upper bound on the max flow value.  Also, an acyclic flow of value $v$ is zero on all but $O(n v^{1/2})$ arcs.  The technique involves transferring the flow to the double covering digraph.
\dated{9 Sept 2010} 
\codes{\Csg:~\COri:~\CFlows,~\CAlgor,~\CCov}%
\itemspace

\bsgitem{2007a}{}  On an application of the structural theory of acyclic skew-symmetric digraphs.  (In Russian.)  \emph{Vestnik Moskov.\ Univ.\ Ser.\ I Mat.\ Mekh.}\ (2007), no.\ 2, 65--66, 80.  English trans.\ \emph{Moscow Univ.\ Math.\ Bull.}\ 62 (2007), no.\ 2, 85--86.
\MR{2357046} (2008i:05146).  \Zbl{1164.05056}.
\annot{} The double covering graph of suitably oriented $-\Gamma$ [matching edges are introverted; nonmatching edges are extraverted] yields a proof that, if $\Gamma$ has a unique perfect matching $M$, then $M$ contains an isthmus.
\dated{9 Sept 2010}
\codes{\Cpar:~\COri}%
\itemspace

\author{Maxim A.\ Babenko \& Alexander V.\ Karzanov}
\authorspace
\bsgitem{2007a}{}  Free multiflows in bidirected and skew-symmetric graphs.  \emph{Discrete Appl.\ Math.}\ 155 (2007), 1715--1730. 
\MR{2348356} (2008j:90102).  \Zbl{1152.90574}.					
\annot{} Optimization of integral odd-vertex flows on a bidirected graph, without or with capacities.
\dated{9 Sept 2010} 
\codes{\Csg:~\COri:~\CFlows:~\CAlgor}%
\itemspace

\bsgitem{2009a}{}  Minimum mean cycle problem in bidirected and skew-symmetric graphs.  \emph{Discrete Optimization} 6 (2009), no.\ 1, 92--97.  
\MR{2483322} (2010a:05105).  \Zbl{1161.05327}.					
\annot{} Minimizing the average weight in a cycle, or a closed trail, of an edge-weighted bidirected graph, in time $O(n^2\min\{n^2,m\log n\})$.
\dated{9 Sept 2010} 
\codes{\Csg:~\COri:~\CAlgor}%
\itemspace

\authorx{Jayapal Baskar Babujee}					%BABUJEE
\author{Jayapal Baskar Babujee \& Shobana Loganathan}
\authorspace
\bsgitem{2011a}{}  On signed product cordial labeling.  \emph{Appl.\ Math.\ (Irvine)} 2 (2011), 1525--1530.
\annot{} Does $\Gamma$ admit a balanced edge signature $\sigma = (+)^\zeta$ ($\zeta=$ switching function), such that $\#\zeta\inv(+) \approx \#\zeta\inv(-)$ and $\#E^+ \approx \#E^-$?  ($\approx$ means difference $\leq 1$.)  If so, $\zeta$ is a ``signed product cordial labeling''.  Some constructions.
[Equivalently, as it is an unsigned graph property:  Does $\Gamma$ have a cut $D=E(X,V\setm X)$ such that $\#X \approx \frac12\#V$ and $\#D \approx \frac12\#E$?]  
[Successors:  \xref{Santhi and Albert}{2015a}{Santhi.\ M \& J.\ James Albert}, \xref{Rozario Raj and Manoharan}{2016a}{P.\ Lawrence Rozario Raj \& R.\ Lawrence Joseph Manoharan}, \xref{Shobana and Vasuki}{2017a}{L.\ Shobana \& B.\ Vasuki}, \etal]
\dated{27 Jan 2013, 11 Mar 2017, rev 27 Dec 2020}
\annotctd{} [\xref{Delphy and Devaraj }{2011a}{Prem Delphy P \& Devaraj J} define ``signed cordiality'' based on edge signs.]
\dated{rev 26 Sept 2022} 
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\author{Goksen Bacak-Turan}						%BACAK-TURAN
\authorspace
\authref  See \hyperlink{R.\ Sundareswaran}{R.~Sundareswaran}.
\itemspace

\author{Constantin P.\ Bachas}						%BACHAS
\authorspace
\bsgitem{1984a}{}  Computer-intractibility of the frustration model of a spin glass.  \emph{J.\ Phys.\ A} 17 (1984), L709--L712.  
\MR{0763604} (85j:82043).  
\annot{} The frustration index decision problem on signed (3-dimensional) cubic lattice graphs is NP-complete.  [Proof is incomplete; completed and improved by \xref{Green}{1987a}{F.\ Green}.  Better result in \xref{Barahona}{1982a}{Francisco Barahona}.]
\par
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{F.\ Bachmann}						%BACHMANN
\authorspace
\authref  See \hyperlink{B.\ Fierro}{B.~Fierro}.
\itemspace

\authorx{Mathias Hudoba de Badyn}					%BADYN	%DE BADYN
\author{Mathias Hudoba de Badyn, Siavash Alemzadeh, \& Mehran Mesbahi}
\authorspace
\bsgitem{2017a}{}  Controllability and data-driven identification of bipartite consensus on nonlinear signed networks.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 3557--3562.  IEEE, 2017.
\arXiv{1709.06679}.  
\codes{\CSG}%
\itemspace

\authorx{Y.\ Bagheri}							%BAGHERI
\author{Y.\ Bagheri, A.R.\ Moghaddamfar, \& F.\ Ramezani}
\authorspace
\bsgitem{2020a}{}  The number of switching isomorphism classes of signed graphs associated with particular graphs.  \emph{Discrete Math.}\ 279 (2020), 25--33.
\MR{4092631}.  \Zbl{1439.05100}.  \arXiv{1909.06819}.				
\annot{} The number of switching isomorphism classes of signed $\Gamma$.  Thm.\ 3.2 on generalized Petersens:  For $\Gamma = \mathrm{GP}(n,k)$ with prime $n>5$ and $k \not\equiv \pm1 \text{ (mod }n)$, either $\Aut[\Gamma,\sigma] = \Aut\mathrm{GP}(n,k)$ or $|\Aut[\Gamma,\sigma]|\leq2$.  [An interesting dichotomy.]  All 36 classes are shown for $\mathrm{GP}(7,2)$.
\dated{29 Jul 2019, rev 27 Jul 2022} 
\codes{\CSG:~\CSw,~\CAut}%
\itemspace

\authorx{Chun-Hsiang Bai}						%BAI CH
\author{Chun-Hsiang Bai and Bang Ye Wu}
\authorspace
\bsgitem{2012a}{}  Finding the maximum balanced vertex set on complete graphs.  
In:  \emph{Proceedings of the 29th Workshop on Combinatorial Mathematics and Computation Theory} (Taipei, 2012), pp.\ 42--51.  
National Taipei College of Business, Taipei, Taiwan, 2012.  
URL \url{http://par.cse.nsysu.edu.tw/~algo/paper/paper_list12.htm}
\annot{} Algorithm for frustration number $l_0(K_n,\sigma)$.
\dated{5 Jun 2017} 
\par
\codes{\CSG:~\CKG:~\CFr:~\CAlgor}%
\itemspace

\author{G.\ David Bailey}						%BAILEY
\authorspace
\bsgitem{20xxa}{}  Inductively factored signed-graphic arrangements of hyperplanes.  Manuscript, ca.\ 1998? 
\annot{} Continues \xref{Edelman--Reiner}{1994a}{Paul H.\ Edelman \& Victor Reiner}.
\codes{\CSG:~\CGeom,~\CMatrd}%
\itemspace

\author{Keith Baker}							%BAKER
\authorspace
\authref  See \hyperlink{Julian O.\ Morrissette}{J.O.~Morrissette}.
\itemspace

\author{V.\ Balachandran}						%BALACHANDRAN
\authorspace
\bsgitem{1976a}{} An integer generalized transportation model for optimal job assignment in computer networks.  
\emph{Operations Res.}\ 24 (1976), 742--759.  
\MR{0439170} (55 \#12068).  \Zbl{356.90028}.
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{V.\ Balachandran \& G.L.\ Thompson}
\authorspace
\bsgitem{1975a}{} An operator theory of parametric programming for the generalized transportation problem:  I.  Basic theory.  II.  Rim, cost and bound operators.  III.  Weight operators.  IV.  Global operators.
\emph{Naval Res.\ Logistics Quart.}\ 22 (1975), 79--100, 101--125, 297--315, 317--339.  
\MR{0381706}--\MR{0381709} (52 \#2595--2598).  \Zbl{331.90048}--\Zbl{331.90051}.
\codes{\CGN:~\CMatrd}%
\itemspace

\authorx{R.\ Balakrishnan}						%BALAKRISHNAN
\author{R.\ Balakrishnan \& K.\ Ranganathan}
\authorspace
\bsgitem{2000a}{}  \emph{A Textbook of Graph Theory}.  Springer, New York, 2000. 
\MR{1729781} (2000j:\-05001).  \Zbl{938.05001}.
\annot{} \S10.6, ``Application to social psychology'':  Short introduction to balance in signed graphs.  \S10.7: Exercises on balance.
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\bsgitem{2012a}{}  \emph{A Textbook of Graph Theory}.  Second ed.  Springer, New York, 2012. 
\MR{2977757}.  \Zbl{1254.05001}.						
\annot{} \S1.11, ``Application to social psychology'':  Short introduction to balance in signed graphs.
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\author{R.\ Balakrishnan \& N.\ Sudharsanam}
\authorspace
\bsgitem{1982a}{}  Cycle-vanishing edge valuations of a graph.  \emph{Indian J.\ Pure Appl.\ Math.}\ 13 (1982), no.\ 3, 313--316.
\MR{0657670} (84d:05145).  \Zbl{485.05057}.					
\annot{} $f: E(\Gamma) \to \bbR$ is ``cycle-vanishing'' if $f(C) := \sum_{e\in C} f(e) = 0$ for every circle.  
Thm.\ 3:  $f$ is cycle-vanishing iff $f(S)=0$ for every series class of non-isthmus edges.  
Thm.\ 4:  $\dim\{\text{cycle-vanishing } f\} = \#E - {}$number of series classes of non-isthmus edges.  
Thm.\ 5:  Connected $\Gamma$ is 3-connected iff only $f=0$ is cycle vanishing. 
[Specialized to a sign-weighted graph $\Sigma$, ``cycle-vanishing'' means $\#E^+(C)=\#E^-(C)$ for every circle.  Thm.\ 3:  $\sigma$ is cycle-vanishing iff every series class of non-isthmus edges has evenly many edges, half positive and half negative.  Etc.  
[\Cf\ \xref{B.G.~Xu}{2009a}{Baogen Xu}, \xref{Vijayakumar}{2011a}{G.R.\ Vijayakumar} for generalization.]  
\dated{16 Oct 2011}
\par
\codes{\Csgw:~\CGen}%
\itemspace

\author{P.\ Balamuralidhar}						%BALAMURALIDHAR
\authorspace
\authref  See also \hyperlink{Hemant Kumar Rath}{H.K.~Rath}.
\authrefspace

\author{P.\ Balamuralidhar \& M.A.\ Rajan}
\authorspace
\bsgitem{2011a}{}  Signed graph based approach for on-line optimization in cognitive networks.  In:  \emph{2011 Third International Conference on Communication Systems and Networks} (COMSNETS, Bangalore, 2011).  IEEE, 2011. 
\codes{\CSD:~\CAlgor,~\CAppl}%
\itemspace

\author{Egon Balas}							%BALAS
\authorspace
\bsgitem{1966a}{} The dual method for the generalized transportation problem.  
\emph{Management Sci.}\ 12 (1966), no.\ 7 (March, 1966), 555--568.  
\MR{0189812} (32 \#7232).  \Zbl{142.16601} (142, p.\ 166a).
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\bsgitem{1981a}{}  Integer and fractional matchings.  In:  P.\ Hansen, ed., \emph{Studies on Graphs and Discrete Programming}, pp.\ 1--13.  North-Holland Math.\ Stud., 59.  Ann.\ Discrete Math., 11.  North-Holland, Amsterdam, 1981.  
\MR{0653814} (84h:90084).  
\annot{} Linear (thus ``fractional'', meaning half-integral) vs.\ integral programming solutions to maximum matching.  The difference of their maxima = $\frac 12$(max number of matching-separable vertex-disjoint odd circles).  Also noted (p.\ 12): (max) fractional matchings in $\Gamma$ correspond to (max) matchings in the double covering graph of $-\Gamma$.  
[\Question.  Does this lead to a definition of maximum matchings in signed graphs?]
\par
\codes{\Cpar,~\Cori:~\CInc,~\CGeom,~\CAlgor,~\Ccov,~\CCircles}%
\itemspace

\author{E.\ Balas \& P.L.\ Ivanescu [P.L.\ Hammer]} 
\authorspace
\bsgitem{1965a}{} On the generalized transportation problem.  \emph{Management Sci.}\  11 (1965), no.\ 1 (Sept., 1964), 188--202.  
\MR{0174395} (30 \#4599).  \Zbl{133.42505} (133, p.\ 425e).
\par
\codes{\CGN:~\CMatrd,~\CBal}%
\itemspace

\author{A.R.\ Balasubramanian}					%BALASUBRAMANIAN
\authorspace
\bsgitem{20xxa}{}  Generalized threshold arrangements.  Submitted.  \emph{}\ 
\arXiv{1904.08903}.  
\annot{} The arrangements have hyperplanes of the form $x_i+x_j = -l, -l+1, \ldots, m-1, m$ (integers) for $i<j$.
\dated{28 Jan 2020} 
\par
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\author{K.\ Balasubramanian}						%BALASUBRAMANIAN
\authorspace
\bsgitem{1988a}{}  Computer generation of characteristic polynomials of edge-weighted graphs, heterographs, and directed graphs.  \emph{J.\ Computational Chem.}\ 9 (1988), 204--211.  
\annot{} Here a ``signed graph'' means, in effect, an acyclically oriented graph $D$ along with the antisymmetric adjacency matrix $A_{\pm}(D) = A(+D \cup -D\inv)$, $D\inv$ being the converse digraph.  [That is, $A_{\pm}(D) = A(D) - A(D)\trans$.  The ``signed graphs'' are just acyclic digraphs with an antisymmetric adjacency matrix and, correspondingly, what we may call the `antisymmetric characteristic polynomial'.]  
Proposes an algorithm for the polynomial.  Observes in some examples a relationship between the characteristic polynomial of $\Gamma$  and the antisymmetric characteristic polynomial of an acyclic orientation.
\par
\codes{\CSD,~\Cwg:~\CEig:~\CInvar:~\CAlgor,~\CChem}%
\itemspace

\bsgitem{1991a}{}  Comments on the characteristic polynomial of a graph.  \emph{J.\ Computational Chem.}\ 12 (1991), 248--253.  
\MR{1093297} (92b:92057).  
\annot{} Argues (heuristically) that a certain algorithm is superior to another, in particular for the antisymmetric polynomial defined in \xrefd{1988a}{K.\ Balasubramanian}.
\par
\codes{\CSD:~\CEig:~\CInvar:~\CAlgor}%
\itemspace

\bsgitem{1992a}{}  Characteristic polynomials of fullerene cages.  \emph{Chem.\ Phys.\ Lett.}\ 198 (1992), 577--586.  
\annot{} Computed for graphs of six different cages of three different orders, in both ordinary and ``signed'' (see \xrefd{1988a}{K.\ Balasubramanian}) versions.  Observes a property of the ``signed graph'' polynomials [which is due to antisymmetry, as explained by P.W.\ Fowler (Comment on ``Characteristic polynomials of fullerene cages''.  \emph{Chem.\ Phys.\ Lett.}\ 203 (1993),  611--612)].
\par
\codes{\CSD:~\CEig:~\CInvar:~\CChem}%
\itemspace

\bsgitem{1994a}{}  Are there signed cospectral graphs?  \emph{J.\ Chem.\ Information Computer Sci.}\ 34 (1994), 1103--1104.  
\annot{} The ``signed graphs'' are as in \xrefd{1988a}{K.\ Balasubramanian}.  Simplified contents:  It is shown by example that the antisymmetric characteristic polynomials of two nonisomorphic acyclic orientations of a graph (see \xrefd{1988a}{K.\ Balasubramanian}) may be equal or unequal.  [Much smaller examples are provided by P.W.\ Fowler, Comment on ``Characteristic polynomials of fullerene cages''.  \emph{Chem.\ Phys.\ Lett.}\ 203 (1993),  611--612).]  
[\Question.  Are there examples for which the underlying (di)graphs are nonisomorphic?]  
[For cospectrality of other kinds of signed graphs, see \xref{Acharya, Gill, and Patwardhan}{1984a}{B.D.\ Acharya, M.K.\ Gill, \& G.A.\ Patwardhan} (signed $K_n$'s).]
\codes{\CSD:~\CEig:~\CInvar}%
\itemspace

\authorx{R.\ Balian}							%BALIAN
\author{R.\ Balian, J.M.\ Drouffe, \& C.\ Itzykson}
\authorspace
\bsgitem{1974a}{} Gauge fields on a lattice.  I.  General model.  \emph{Phys.\ Rev.\ D} 10 (1974), no.\ 10, 3376--3395.  
\annot{} Gain group $\SO(n)$ on a toroidal lattice graph (\S~C, ``Local invariance, gauge field, and minimal coupling), where $\SO(1)=\signnos$ (developed in \xrefd{1975a}{R.\ Balian, J.M.\ Drouffe, \& C.\ Itzykson}).
\dated{12 Aug 2012} 
\codes{\CSG:~\CPhys}%
\itemspace

\bsgitem{1975a}{} Gauge fields on a lattice.  II.  Gauge-invariant Ising model.  \emph{Phys.\ Rev.\ D} 11 (1975), no.\ 8, 2098--2103.  
\annot{} Dictionary:  ``Ising model'' = signed hypercubical lattice, ``gauge invariance'' = switching invariance, ``plaquette'' = quadrilateral.  The partition function depends on $p^+-p^-$ where $p^\epsilon =$ \# plaquettes with sign $\epsilon$ and sometimes also $\#E^+-\#E^-$.
\dated{12 Aug 2012}
\par
\codes{\CSG:~\CPhys,~\CSw,~\CFr}%
\itemspace

\author{M.L.\ Balinski}						%BALINSKI
\authorspace
\bsgitem{1970a}{}  On recent developments in integer programming.  
\emph{Proceedings of the Princeton Symposium on Mathematical Programming} (Princeton Univ., 1967), pp.\ 267--302.
Princeton Univ.\ Press, Princeton, N.J., 1970.
\MR{0437023} (55 \#9957).  \Zbl{222.90036}.
\annot{} Pp.\ 277--278 discuss integer programming problems on bidirected graphs in terms of the incidence matrix.
\codes{\Cori:~\Cinc:~\Cpar,~\CAlgor,~\CRef}%
\itemspace

\authorx{Igor Balla}							%BALLA
\author{Igor Balla, Felix Dr\"axler, Peter Keevash, \& Benny Sudakov}
\authorspace
\bsgitem{2018a}{}  Equiangular lines and spherical codes in Euclidean space.  \emph{Invent.\ math.}\ 211 (2018), 179--212.
\MR{3742757}.  \Zbl{1383.51035}.  \arXiv{1606.06620}.  
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\authorx{N\'uria Ballber Torres}						%BALLBER TORRES
\authorx{Nuria Ballber Torres}
\author{N\'uria Ballber Torres \& Claudio Altafini}
\authorx{Nuria Ballber Torres \& Claudio Altafini}
\authorspace
\bsgitem{2016a}{}  Drug combinatorics and side effect estimation on the signed human drug-target network.  \emph{BMC Systems Biol.}\ 10 (2016), art.\ 74, 12 pp.
\codes{\CSG:~\CCycles,~\Cbal}%
\itemspace

\author{Murad Banaji}						%BANAJI
\authorspace
\authref  See also \hyperlink{David Angeli}{D.~Angeli} and \hyperlink{Nicole Radde}{N.~Radde}.
\authrefspace

\bsgitem{2010a}{}  Graph-theoretic conditions for injectivity of functions on rectangular domains.  \emph{J.\ Math.\ Anal.\ Appl.}\ 370 (2010), 302--311.
\MR{2651147} (2011f:26012).  \Zbl{1227.26006}.					
\codes{\CSD}%
\itemspace

\author{Murad Banaji \& Gheorghe Craciun}
\authorspace
\bsgitem{2009a}{}  Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements.  \emph{Commun.\ Math.\ Sci.}\ 7 (2009), no.\ 4, 867--900. 
\MR{2604624} (2011i:05126).  \Zbl{1195.05038}.  \arXiv{0903.1190}.			
\codes{\CSG,~\CChem}%
\itemspace

\bsgitem{2010a}{}  Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems.  \emph{Adv.\ Appl.\ Math.}\  44 (2010), 168--184.
\MR{2576846} (2010m:80010).  \Zbl{1228.05204}.					
\annot{} Generalization of \xrefd{2009a}{Murad Banaji \& Gheorghe Craciun} to more general systems.
\dated{26 Oct 2011} 
\codes{\CSG,~\CChem}%
\itemspace


\author{Murad Banaji \& Carrie Rutherford}
\authorspace
\bsgitem{2011a}{}  $P$-matrices and signed digraphs.  \emph{Discrete Math.}\ 311 (2011), no.\ 4, 295--301.
\MR{3537201}.  \Zbl{1222.05080}.  \arXiv{1006.0152}.				
\codes{\CSD:~\CQM}%
\itemspace

\authorx{Afonso S.\ Bandeira}						%BANDEIRA
\author{Afonso S.\ Bandeira, Amit Singer, \& Daniel A.\ Spielman}
\authorspace
\bsgitem{2013a}{}  A Cheeger inequality for the graph connection Laplacian.  \emph{SIAM J.\ Matrix Anal.\ Appl.}\ 34 (2013), no.\ 4, 1611--1630.
\MR{3138103}.  \Zbl{1287.05081}.  \arXiv{1204.3873}.				
\annot{} Connection graph:  A real-weighted $O(\bbR,d)$-gain graph.	
\par
\codes{\CGG,~\CWG:~\CLap}%
\itemspace

\authorx{Debanjan Banerjee}						%BANERJEE
\author{Debanjan Banerjee \& Anita Pal}
\authorspace
\bsgitem{2015a}{}  Application of signed graph in decision making.  \emph{Int.\ J.\ Computers Technol.}\ 14 (2015), no.\ 6, 5825--5833.
\annot{} ``Character points'' := sum of edge signs, at a vertex or in $\Sigma$.  Positive = ``good''.  Tenuous applications.
\dated{11 Aug 2021} 
\par
\codes{\CSG,~\CSD,~\CVS:~\CKG,~\CAppl}%
\itemspace

\author{Subhashish Banerjee}						%BANERJEE
\authorspace
\authref  See \hyperlink{Bibhas Adhikari}{B.~Adhikari}.
\itemspace

\authorx{J{\o}rgen Bang-Jensen}					%BANG-JENSEN
\authorx{Jorgen Bang-Jensen}
\author{J{\o}rgen Bang-Jensen, St\'ephane Bessy, Bill Jackson, \& Matthias Kriesell}
\authorx{Jorgen Bang-Jensen, Stephane Bessy, Bill Jackson, \& Matthias Kriesell}
\authorspace
\bsgitem{2017a}{}  Antistrong digraphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 122 (2017), 68--90.
\MR{3575196}.  \Zbl{1350.05048}.  
\annot{} Antidirected trails, i.e., coherent in poise gains.  
Dictionary: ``even bicircular matroid'' = even-cycle matroid $\bfF(-\Gamma)$ (\cf\ \xref{Doob}{1973a}{Michael Doob}).
[\Cf\ antidirection matroid in \xref{Matthews}{1978c}{Laurence R.\ Matthews}.  \Question.  Does this generalize to bidirected graphs?]
\dated{10 Nov 2017, 30 May 2018} 
\codes{\Cgg:~\CStr,~\CMatrd}\codesctd{\Csg:~\Cpar:~\COri}%
\itemspace

\author{J{\o}rgen Bang-Jensen \& Gregory Gutin}
\authorx{Jorgen Bang-Jensen \& Gregory Gutin}
\authorspace
\bsgitem{1997a}{}  Alternating cycles and paths in edge-coloured multigraphs: A survey.  \emph{Discrete Math.}\ 165/166 (1997), 39--60.  
\MR{1439259} (98d:05080).  \Zbl{876.05057}.					
\annot{} A rich source for problems on bidirected graphs.  An edge 2-coloration of a graph becomes an all-negative bidirection by taking one color class to consist of introverted edges and the other to consist of extroverted edges.  An alternating path becomes a coherent path; an alternating circle becomes a coherent circle.  
[\emph{General} \Problem.  Generalize to bidirected graphs the results on edge 2-colored graphs mentioned in this paper.  (See esp.\ \S5.)  \Question.  To what digraph properties do they specialize by taking the underlying signed graph to be all positive?]  [See e.g.\ \xref{B\'ankfalvi and B\'ankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi} (\qv), \xref{Bang-Jensen and Gutin}{1998a}{Jorgen Bang-Jensen \& Gregory Gutin}, \xref{Das and Rao}{1983a}{Prabir Das \& S.B.\ Rao}, \xref{Grossman and H\"aggqvist}{1983a}{Jerrold W.\ Grossman \& Roland Haggkvist}, \xref{Mahadev and Peled}{1995a}{N.V.R.\ Mahadev \& U.N.\ Peled}, \xref{Saad}{1996a}{Rachid Saad}.] 
\codes{\Cpar:~\Cori:~\CPaths,~\CCircles}%
\itemspace

\bsgitem{1998a}{}  Alternating cycles and trails in $2$-edge-colored complete multigraphs.  \emph{Discrete Math.}\ 188 (1998), 61--72.
\MR{1630418} (99g:05072).  \Zbl{956.05040}.					
\annot{} The longest coherent trail, having degrees bounded by a specified degree vector, in a bidirected all-negative complete multigraph that satisfies an extra hypothesis.  Generalization of \xref{Das and Rao}{1983a}{Prabir Das \& S.B.\ Rao} and \xref{Saad}{1996a}{Rachid Saad}, thus ultimately of Thm.\ 1 of \xref{B\'ankfalvi and B\'ankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi} (\qv).  Also, a polynomial-time algorithm.
\codes{\Cpar:~\Cori:~\CPaths,~\CAlgor}%
\itemspace

\authorx{M.\ B\'ankfalvi}						%BANKFALVI
\authorx{M.\ Bankfalvi}
\author{M.\ B\'ankfalvi \& Zs.\ B\'ankfalvi}	
\authorx{M.\ Bankfalvi \& Zs.\ Bankfalvi}	
\authorspace
\bsgitem{1968a}{} Alternating Hamiltonian circuit in two-coloured complete graphs.
In:  P.\ Erd\H{o}s and G.\ Katona, eds., \emph{Theory of Graphs} (Proc.\ Colloq., Tihany, 1966), pp.\ 11--18.  Academic Press, New York, 1968.  
\MR{0233731} (38 \#2052).  \Zbl{159.54202}. 
\annot{} Let $\Beta$ be a bidirected $-K_{2n}$ which has a coherent $2$-factor.  (``Coherent'' means that, at each vertex in the $2$-factor, one edge is directed inward and the other outward.)  
Thm.\ 1:  $\Beta$ has a coherent Hamiltonian circle iff, for every $k \in \{2, 3, \ldots, n-2\}$, $s_k > k^2$, where $s_k :=$ the sum of the $k$ smallest indegrees and the $k$ smallest outdegrees.  
Thm.\ 2:  The number of $k$'s for which $s_k = k^2$ equals the smallest number $p$ of circles in any coherent $2$-factor of $\Beta$.  Moreover, the $p$ values of $k$ for which equality holds imply a partition of $V$ into $p$ vertex sets, each inducing $\Beta_i$ consisting of a bipartite [i.e., balanced] subgraph with a coherent Hamiltonian circle and in one color class only introverted edges, while in the other only extroverted edges.  
[\Problem.  Generalize these remarkable results to an arbitrary bidirected complete graph.  The all-negative case will be these theorems; the all-positive case will give the smallest number of cycles in a covering by vertex-disjoint cycles of a tournament that has any such covering.]  [See \xref{Bang-Jensen and Gutin}{1997a}{Jorgen Bang-Jensen \& Gregory Gutin} for further developments on alternating walks; also \xref{Busch, Jacobson, \etal\!}{2013a}{Arthur H.\ Busch, Michael S.\ Jacobson, Timothy Morris, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}, \xref{Busch, Mutar, and Slilaty}{20xxa}{Arthur Busch, Mohammed A.\ Mutar, \& Daniel Slilaty}.]
\codes{\Cpar:~\Cori:~\CCircles}%
\itemspace

\author{Zs.\ B\'ankfalvi} 						%BANKFALVI
\authorx{Zs.\ Bankfalvi}
\authorspace
\authref  See \hyperlink{M.\ Bankfalvi}{M.~B\'ankfalvi}.
\itemspace

\author{C.\ Bankwitz}						%BANKWITZ*
\authorspace
\bsgitem{1930a}{}  \"Uber die Torsionszahlen der alternierenden Knotes.  \emph{Math.\ Ann.}\ 103 (1930), 145--161.
\annot{} Introduces the sign-colored graph of a link diagram.  [Further work by numerous writers, e.g., S.\ Kinoshita \etal\ and esp.\ \xref{Kauffman}{1989a}{Louis H.\ Kauffman} and successors.]
\codes{\CKnot:~\CSGc}%
\itemspace

\authorx{Nikhil Bansal}						%BANSAL N
\author{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}
\authorspace
\bsgitem{2002a}{}   Correlation clustering.  In:  \emph{Proc.\ 43rd Ann.\ IEEE Sympos.\ Foundations of Computer Science (FOCS '02)}, pp.\ 238--247.
\Zbl{1089.68085}.  
\annot{} Preliminary version of \xrefd{2004a}{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}.
\codes{\CSG:~\CKG:~\CClu:~\CAlgor}%
\itemspace

\bsgitem{2004a}{}  Correlation clustering.  Theoretical Advances in Data Clustering.  \emph{Machine Learning} 56 (2004), no.\ 1-3, 89--113.
\Zbl{1089.68085}.  
\annot{} Clusterability index $Q$ [minimum number of inconsistent edges; see \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar} for notation] in signed complete graphs is NP-hard.  Polynomial-time algorithms for approximate optimal clustering: up to a constant factor from $Q$ (\S3); probably within $1-\epsilon$ of $\#E-Q$ for any $\epsilon$ (i.e., maximizing consistent edges within $1-\epsilon$) (\S4).  
\S3:  A 2-clustering within $3Q_2$ (Thm.\ 2).  A clustering within $c Q$ where $c \approx 20000$ (Thm.\ 13).  
\S4:  A clustering within $\epsilon n^2$ of $\#E-Q$ with high probability but slow in terms of $1/\epsilon$ (Thm.\ 15).  Asymptotically faster in terms of $1/\epsilon$ (Thm.\ 22).  The $1-\epsilon$ factor results from the fact that $\#E-Q = \binom{n}{2}-Q > \frac12\binom{n}{2}$ [so is not strong].  
\S6: ``Random noise''.  
\S7: ``Extensions'', considers edge weights in $[-1,1]$ (thus allowing incomplete graphs).  Thm.\ 23:  An unweighted approximation algorithm will also approximate this case, assuming ``linear cost'': $e$ costs $(1-w(e))/2$ if within a cluster and $(1+w(e))/2$ if between clusters.  
Thm.\ 24:  The problem for clustering that minimizes the total weight of $+$ edges outside clusters and $-$ edges within clusters (``minimizing disagreements'') is APX-hard.
[Improved in \xref{Charikar--Guruswami--Wirth}{2003a}{Moses Charikar, Venkatesan Guruswami, \& Anthony Wirth}, \xrefd{2005a}{Moses Charikar, Venkatesan Guruswami, \& Anthony Wirth}, \xref{Swamy}{2004a}{Chaitanya Swamy}.  Generalized in \xref{Demaine \etal{}}{2006a}{Erik D.\ Demaine, Dotan Emanuel, Amos Fiat, \& Nicole Immorlica}.]
\dated{22 Sept 2009} 
\codes{\CSG:~\CKG:~\CClu:~\CAlgor}%
\itemspace

\authorx{Bo Bao}							%BAO B
\author{Bo Bao, Rong Chen, \& Genghua Fan}
\authorspace
\bsgitem{2021a}{}  Circuit covers of signed Eulerian graphs.  \emph{Electronic J.\ Combin.}\ 28 (2021), no.\ 1, art.\ P1.14, 12 pp.
\MR{4245247}.  \Zbl{1456.05095}.  \arXiv{1910.09999}.  
\annot{} Assuming $\bfF(\Sigma)$ has no coloop, $E$ can be covered by frame circuits so each edge is in exactly 6 circuits.
\dated{28 Jan 2021} 
\codes{\CSG:~\Cflows}%
\itemspace

\author{R.B.\ Bapat}							%BAPAT RB
\authorspace
\authref  See also \hyperlink{Ranveer Singh}{R.~Singh}.
\authrefspace

\bsgitem{2010a}{}  \emph{Graphs and Matrices}.  Hindustan Book Agency, New Delhi, and Springer, London, 2010.
\MR{2797201} (2012f:05001).  \Zbl{1248.05002}.					
\annot{} \S2.6, ``$0-1$ Incidence matrix'':  The rank and related properties of the the unoriented incidence matrix.  [\Cf\ \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}.]
\dated{25 Aug 2011} 
\codes{\CPar:~\CInc:~\CExp}%
\itemspace

\author{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}
\authorspace
\bsgitem{1999a}{}  Generalized matrix tree theorem for mixed graphs.  \emph{Linear Multilinear Algebra} 46 (1999), 299--312.
\MR{1729196} (2001c:05091).  \Zbl{940.05042}.					
\annot{} Their ``mixed graph'' is a signed graph $\Sigma$: positive edges are called ``directed'' and negative edges ``undirected''.  The matrix-tree theorem is the unweighted case of \xref{Chaiken's}{1982a}{Seth Chaiken} all-minors theorem for signed graphs.  The technical formalism differs somewhat.  They point out that in case $U \cup W = V$, the minor is the sum of signed $\bar U \bar W$ matchings.  Dictionary:  ``$k$-reduced substructure'' $\cong$ independent set of rank $n-k$ in $\bfF(\Sigma)$;  ``quasibipartite'' = balanced.
Successor to \xref{Grossman, Kulkarni, and Schochetman}{1994a}{Jerrold W.\ Grossman, Devadatta M.\ Kulkarni, \& Irwin E.\ Schochetman} [\qv\ for more dictionary].
\codes{\Csg:~\CInc}%
\itemspace

\bsgitem{2000a}{}  Edge version of the matrix tree theorem for trees.  \emph{Linear Multilinear Algebra} 47 (2000), 217--229.
\MR{1785029} (2001d:05112).  \Zbl{960.05067}.					
\annot{} Successor to \xrefd{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}.  Their ``mixed tree'' $T$ is a signed tree as in \xrefd{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}.  Thm.\ 9 (simplified):  The minor of $\incid\trans \incid$ ($\incid$ is the incidence matrix of $\Sigma$) obtained by deleting rows corresponding to $E \subseteq E(\Sigma)$ and columns corresponding to $F \subseteq E(\Sigma)$ has determinant equal, up to sign, to the number of common SDR's of vertex sets of components of $T \setminus E$ and $T \setminus F$.  [Interesting, but edge signs are irrelevant because any tree switches to all positive.]
Dictionary:  ``substructure'' = subgraph allowing retention of edges incident to deleted vertices [thus they become loose or half edges].  [See \xrefd{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni} for more dictionary.]
\codes{\Csg:~\CInc}%
\itemspace

\author{R.B.\ Bapat \& E.\ Ghorbani}
\authorspace
\bsgitem{2014a}{}  Inverses of triangular matrices and bipartite graphs.  \emph{Linear Algebra Appl.}\ 447 (2014), 68--73.
\MR{3200207}.  \Zbl{1288.05152}.  \arXiv{1303.2177}.				
\annot{} Generalizes to rings the inverses of $A(T)$ for a tree; \cf\ \xref{Godsil}{1985a}{C.D.\ Godsil}.
\dated{7 Aug 2019} 
\codes{\Csg(\CGen):~\CAdj}%
\itemspace

\author{R.B.\ Bapat, D.\ Kalita, \& S.\ Pati}
\authorspace
\bsgitem{2012a}{}  On weighted directed graphs.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 1, 99--111.
\MR{2859913}.  \Zbl{1229.05198}.  
\annot{} They are complex unit gain graphs $\Phi$ with simple underlying graph.  Laplacian $\Lap(\Phi)$ is obtained in the usual way from $\incid(\Phi)$.  
\S2, ``$D$-similarity and singularity in weighted directed graphs'':  Thm.\ 8:  $\Lap(\Phi)$ is singular iff $\Phi \sim \|\Phi\|$ iff $\Phi$ (assumed connected) is balanced.  [\Cf\ \xref{Zaslavsky}{2003b}{Thomas Zaslavsky}, \S2.1 esp.\ Thm.\ 2.1(a), noting that $\rk \Lap(\Phi) = \rk\incid(\Phi) = \rk \bfF(\Phi)$.]  
\S3, ``Edge singularity of weighted directed graphs'':  Elementary results on frustration index, appearing less elementary because treated indirectly, through eigenvalues, rather than directly, through the graph.  Generalizing \xref{Y.Y.\ Tan and Fan}{2008a}{Ying Ying Tan \& Yi Zheng Fan} on signed graphs.
\S4, ``3-Colored digraphs and their singularity'':  Gains restricted to $\pm1, \mathrm{i}$.  Elementary results.
\annotctd{} Dictionary:  ``weighted directed graph'' = complex unit gain graph (\cf\ \xref{Reff}{2012a}{Nathan Reff}); ``mixed graph'' = signed graph; $D$-similarity'' [diagonal similarity] = switching equivalence, ``edge singularity'' = frustration index.
\dated{28 Oct 2011} 
\codes{\Cgg:~\CEig,~\CInc,~\CBal}%
\itemspace

\author{R.B.\ Bapat \& Devadatta M.\ Kulkarni}
\authorspace
\bsgitem{2000a}{}  Minors of some matrices associated with a tree.  
In:  \emph{Algebra and Its Applications} (Athens, Ohio, 1999), pp.\ 45--66.  Contemp.\ Math., Vol.\ 259.  American Math.\ Soc., Providence, R.I., 2000.
\MR{1778494} (2001h:05065).  \Zbl{979.05075}.					
\annot{} Concerns a ``mixed tree'', really an oriented signed tree without extroverted edges (see \xref{Bapat, Grossman, and Kulkarni}{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}).  The matrices are the incidence matrix $\incid$, the Laplacian matrix $\incid\,\incid\trans$, and the ``edge Laplacian'' $\incid\trans \incid$.  Partly expository.  New results concern Moore--Penrose inverses and their minor determinants.  [Since a ``mixed tree'' is switching equivalent to an ordinary unsigned tree, their results should be identical to those for ordinary trees except for multiplication by a $V \times V$ diagonal matrix with signs on the diagonal.]
\codes{\Csg:~\CInc}%
\itemspace

\author{Nadav S.\ Bar}						%BAR
\authorspace
\authref  See \hyperlink{Nicole Radde}{N.~Radde}.
\itemspace

\author{Francisco Barahona}						%BARAHONA
\authorspace
\bsgitem{1981a}{}  Balancing signed toroidal graphs in polynomial-time.  Unpublished manuscript, 1981.
\annot{} Given a $2$-connected $\Sigma$ whose underlying graph is toroidal, polynomial-time algorithms are given for calculating the frustration index $l(\Sigma)$ and the generating function of switchings $\Sigma^{\mu}$ by $\#E^-(\Sigma^{\mu})$.  The technique is to solve a Chinese postman ($T$-join) problem in the toroidal dual graph, $T$ corresponding to the frustrated face boundaries.  Generalizes \xrefd{1982a}{Francisco Barahona}.  
[See \xrefd{1990a}{Francisco Barahona}, p.\ 4, for a partial description.]
\codes{\CSG:~\CFr,~\CAlgor}%
\itemspace

\bsgitem{1982a}{}  On the computational complexity of Ising spin glass models.  \emph{J.\ Phys.\ A:  Math.\ Gen.}\ 15 (1982), 3241--3253.  
\MR{0684591} (84c:82022).  
\annot{} The frustration-index problem, that is, minimization of $\#E^-(\Sigma^{\zeta})$ over all switching functions $\zeta : V \to \{\pm 1\}$, for signed planar and toroidal graphs and subgraphs of $3$-dimensional grids.  Analyzed structurally, in terms of perfect matchings in a modified dual graph, and algorithmically.  $3$-dimensional is NP-hard, even when the grid has only 2 levels; the former are polynomial-time solvable even with weighted edges.  
\annotctd{} Also, the problem of minimizing $\#E^-(\Sigma^{\zeta}) + \sum_v \zeta(v)$ for planar grids (``$2$-dimensional problem with external magnetic field''), which is NP-hard.  [This corresponds to adding an extra vertex, positively adjacent to every vertex.]  
[See infinite analog in \xref{Istrail}{2000a}{Sorin Istrail}.]
\par
\codes{\CSG:~\CPhys,~\CFr,~\CFr(\CGen):~\CDual,~\CAlgor}%
\itemspace

\bsgitem{1982b}{}  Two theorems in planar graphs.  Unpublished manuscript, 1982.
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{1983a}{}  The max-cut problem on graphs not contractible to $K_5$.  \emph{Operations Res.\ Lett.}\ 2 (1983), no.\ 3, 107--111.
\MR{0717742} (84k:90048).  \Zbl{0525.90094}.  
\annot{} Thm.\ 3.2:  The real-weighted maximum cut problem is polynomial-time solvable for graphs not contractible to $K_5$.  [Frustration index $l(-\Sigma)$ is the special case of weight$(e) = -\sigma(e) = \pm1$.]
\dated{19 Dec 2014}
\par
\codes{\Csg,~\CWG:~\Cfr,~\CAlgor}%
\itemspace

\bsgitem{1983b}{}  On some weakly bipartite graphs.  \emph{Operations Res.\ Lett.}\ 2 (1983), no.\ 5, 239--242.
\MR{0733782} (85a:05072).  \Zbl{0549.90087}.  
\annot{} If the frustration number $l_0(-\Gamma) \leq 2$ (i.e., $-\Gamma \setm \{u,v\}$ is balanced for some $u,v \in V$), then $\Gamma$ is weakly bipartite (\cf\ \xref{Guenin}{2001a}{Bertrand Guenin}) and $l(-\Gamma)$ is polynomial-time computable.  [\Problem.  Characterize $\Sigma$ with $l_0\leq2$.  For $l_0 \leq 1$ see \xref{Zaslavsky}{1987c}{Thomas Zaslavsky} with $\Omega=\Sigma$.]
\dated{19 Dec 2014} 
\codes{\CPar,~\CSG:~\Cfr,~\CSw}%
\itemspace

\bsgitem{1990a}{}  On some applications of the Chinese Postman Problem.  
In:  B.\ Korte, L.\ Lov\'asz, H.J.\ Pr\"omel, and A.\ Schrijver, eds., \emph{Paths, Flows and VLSI-Layout}, pp.\ 1--16. 
Algorithms and Combin., Vol.\ 9.  Springer-Verlag, Berlin, 1990.  
\MR{1083374} (92b:90139).  \Zbl{732.90086}.					
\annot{} \S2:  ``Spin glasses.'' 
\codes{\CSG:~\CPhys,~\CFr:~\CExp}%
\annotctd{} \S5:  ``Max cut in graphs not contractible to $K_5$,'' pp.\ 12--13.  
\par
\codes{\Csg:~\Cfr:~\CExp}%
\itemspace

\bsgitem{1990b}{}  Planar multicommodity flows, max cut, and the Chinese Postman Problem.  
In: William Cook and Paul D.\ Seymour, eds., \emph{Polyhedral Combinatorics} (Proc.\ Workshop, Morristown, N.J., 1989),  pp.\ 189--202.  
DIMACS Ser.\ Discrete Math.\ Theor.\ Computer Sci., Vol.\ 1.  
Amer.\ Math.\ Soc.\ and Assoc.\ Comput.\ Mach., Providence, R.I., 1990.  
\MR{1105127} (92g:05165).  \Zbl{747.05067}.					
\annot{} Negative cutsets, where signs come from a network with real-valued capacities.  Dual in the plane to negative circles.  See \S2.
\par
\codes{\CSG:~\CDual:~\CBal,~\CAlgor}%
\itemspace

\author{Francisco Barahona \& Adolfo Casari}
\authorspace
\bsgitem{1988a}{}  On the magnetisation of the ground states in two-dimensional Ising spin glasses.  \emph{Comput.\ Phys.\ Commun.}\ 49 (1988), 417--421.
\MR{0945813} (89d:82004).  \Zbl{814.90132}.
\codes{\CSG:~\CState(\Cfr):~\CAlgor}%
\itemspace

\author{Francisco Barahona \& Michele Conforti}
\authorspace
\bsgitem{1987a}{}  A construction for binary matroids.  \emph{Discrete Math.}\ 66 (1987), 213--218.
\MR{0900044} (88g:05039).  \Zbl{0644.05017}.  
\par
\codes{\CSG:~\CMatrd}%
\itemspace

\author{Francisco Barahona, Martin Gr\"otschel, Michael J\"unger, \& Gerhard Reinelt}
\authorx{Francisco Barahona, Martin Grotschel, Michael Junger, \& Gerhard Reinelt}
\authorspace
\bsgitem{1988a}{}  An application of combinatorial optimization to statistical physics and circuit layout design.  \emph{Oper.\ Res.}\ 36 (1988), no.\ 3, 493--513.
\Zbl{646.90084}.  
\annot{} Frustration index of a weighted signed graph (Ising ground state; via minimum) is reduced to weighted max-cut.  The algorithm uses cutting planes on the cut polytope of the underlying graph, specifically applied to toroidal grids with an extra vertex.  Fractional solutions appear occasionally, especially for signed graphs.  Possible use of negative-circle constraints is mentioned.
\dated{18 Aug 2012} 
\codes{\Csg:~\CFr:~\CAlgor}%
\itemspace

\author{Francisco Barahona, Martin Gr\"otschel, \& Ali Ridha Mahjoub}
\authorx{Francisco Barahona, Martin Grotschel, \& Ali Ridha Mahjoub}
\authorspace
\bsgitem{1985a}{}  Facets of the bipartite subgraph polytope.  \emph{Math.\ Operations Res.}\ 10 (1985), 340--358.  
\MR{0793888} (87a:05123a).  \Zbl{578.05056}.					
\annot{} The polytope $P_B(\Gamma)$ is the convex hull in $\bbR^E$ of characteristic vectors of bipartite edge sets.  Various types of and techniques for generating facet-defining inequalities, thus partially extending the description of $P_B(\Gamma)$ from the weakly bipartite case (\xref{Gr\"otschel and Pulleyblank}{1981a}{M.\ Grotschel \& W.R.\ Pulleyblank}) in which all facets are due to edge and odd-circle constraints.  
[Some can be described best via signed graphs; see \xref{Poljak and Turz\'{\i}k}{1987a}{Svatopluk Poljak \& Daniel Turzik}.]  [A brief expository treatment of the polytope appears in \xref{Poljak and Tuza}{1995a}{Svatopluk Poljak \& Zsolt Tuza}.]
\codes{\Csg:~\Cpar:~\Cfr:~\CGeom}%
\itemspace

\author{Francisco Barahona \& Enzo Maccioni}
\authorspace
\bsgitem{1982a}{}  On the exact ground states of three-dimensional Ising spin glasses.  \emph{J.\ Phys.\ A:  Math.\ Gen.}\ 15 (1982), L611--L615.
\MR{0679090} (83k:82044).  
\annot{} Discusses a 3-dimensional analog of \xref{Barahona, Maynard, Rammal, and Uhry}{1982a}{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}.  There may not always be a combinatorial LP optimum; hence LP may not completely solve the problem.
\codes{\CSG:~\CPhys,~\CFr,~\CAlgor}%
\itemspace

\author{Francisco Barahona \& Ali Ridha Mahjoub}
\authorspace
\bsgitem{1986a}{}  On the cut polytope.  \emph{Math.\ Programming} 36 (1986), 157--173.  
\MR{0866986} (88d:05049).  \Zbl{616.90058}.					
\annot{} Call $P_{\text{\rm BS}}(\Sigma)$ the convex hull in $\bbR^E$ of characteristic vectors of negation sets (or ``balancing [edge] sets'') in $\Sigma$.  Finding a minimum-weight negation set in $\Sigma$ corresponds to a maximum cut problem, whence $P_{\text{\rm BS}}(\Sigma)$ is a linear transform of the cut polytope $P_{\text{\rm C}}(|\Sigma|)$, the convex hull of cuts.  Conclusions follow about facet-defining inequalities of $P_{\text{\rm BS}}(\Sigma)$.  See \S5: ``Signed graphs''.
\codes{\CSG:~\CFr:~\CGeom}%
\itemspace

\bsgitem{1989a}{}  Facets of the balanced (acyclic) induced subgraph polytope.  \emph{Math.\ Programming Ser.\ B} 45 (1989), 21--33.  
\MR{1017209} (91c:05178).  \Zbl{675.90071}.					
\annot{} The ``balanced induced subgraph polytope'' $P_{\text{\rm BIS}}(\Sigma)$ is the convex hull in $\bbR^V$ of incidence vectors of vertex sets that induce balanced subgraphs.  Conditions are studied under which certain inequalities of form $\sum_{i \in Y} x_i \leq f(Y)$ define facets of this polytope: in particular,  $f(Y) =$ max.\ size of balance-inducing subets of $Y$, $f(Y) = 1$ or $2$, $f(Y) = \#Y - 1$ when $Y = V(C)$ for a negative circle $C$, etc.
\codes{\CSG:~\CFr:~\CGeom,~\CAlgor}%
\itemspace

\bsgitem{1994a}{}  Compositions of graphs and polyhedra.  I:  Balanced induced subgraphs and acyclic subgraphs.  \emph{SIAM J.\ Discrete Math.}\ 7 (1994), 344--358.  
\MR{1285575} (95i:90056).  \Zbl{802.05067}.					
\annot{} More on $P_{\text{\rm BIS}}(\Sigma)$ (see \xrefd{1989a}{Francisco Barahona \& Ali Ridha Mahjoub}).  A balance-inducing vertex set in $\pm\Gamma =$ a stable set in $\Gamma$.  [See \xref{Zaslavsky}{1982b}{Thomas Zaslavsky} for a different correspondence.]  Thm.\ 2.1 is an interesting preparatory result:  If $\Sigma = \Sigma_1 \cup \Sigma_2$ where $\Sigma_1 \cap \Sigma_2 \cong \pm K_k$, then $P_{\text{\rm BIS}}(\Sigma) = P_{\text{\rm BIS}}(\Sigma_1) \cap P_{\text{\rm BIS}}(\Sigma_2)$.  
The main result is Thm.\ 2.2:  If $\Sigma$ has a $2$-separation into $\Sigma_1$ and $\Sigma_2$, the polytope is the projection of the intersection of polytopes associated with modifications of $\Sigma_1$ and $\Sigma_2$.  \S5: ``Compositions of facets'', derives the facets of $P_{\text{\rm BIS}}(\Sigma)$.
\par
\codes{\CSG:~\CGeom,~\CWG,~\CAlgor}%
\itemspace

\author{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}
\authorspace
\bsgitem{1982a}{\dag} Morphology of ground states of two-dimensional frustration model.  \emph{J.\ Phys.\ A:  Math.\ Gen.}\  15 (1982), 673--699.  
\MR{0642302} (83c:82045).  
\annot{} Treats many important aspects of the quantity $l := \min_{\zeta} \#E^-(\Sigma^{\zeta})$ [which equals the frustration index], over all switching functions $\zeta$ (``spin configurations $\sigma$'' in the paper) of a signed graph, mainly a signed planar graph.  
($\#E^-(\Sigma^{\zeta})$ is the paper's $\frac12(\#E+H)$, $H :=$ Hamiltonian.)
They maximize $-H = W^+ + W^- - W^{+-}$ where $W^+ + W^- :=$ \# unswitched positive edges $-$ \# unswitched negative edges and $W^{+-} :=$ \# switched positive edges $-$ \# switched negative edges.  Thus, $-H = \#E^+ - \#E^- = \#E - 2\#E^-$ after switching.  
Maximizing it $\iff$ minimizing $\#E^-$ over all $\zeta$.
\annotctd{} \S2: ``The frustration model as the Chinese postman's problem'', describes how to find $l$ when $|\Sigma|$ is planar, by solving a Chinese postman ($T$-join) problem in the dual graph, $T$ corresponding to the frustrated (i.e., negative) face boundaries.  The postman problem is solved by linear programming.  
[Solved independently by \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai}.]  
[\xref{Barahona}{1981a}{Francisco Barahona} generalizes to signed toroidal graphs.]
\annotctd{} \S3: ``Solution of the frustration problem by duality: rigidity''.  An edge is ``rigid'' if it has the same sign in every $\Sigma^\zeta$ that minimizes $\#E$ (such an $\zeta$ is a ``ground state'').  The endpoints of a rigid edge are called ``solidary''.  Rigid edges are found via the dual linear program.  The boundary contours of connected sets of frustrated faces play an important role.
\annotctd{} \S\S4--5: ``Numerical experimentation'' and ``Results'', for a randomly signed square lattice graph.  The proportion $x$ of negative edges strongly affects the properties; esp., there is significant long-range order below but not above $x \approx .15$.  
[See \xref{Deng and Abell}{2010a}{Hongzhong Deng \& Peter Abell} for numerical results on random signed graphs.]
\annotctd{} More general problems discussed are (1) allowing positive edge weights (due to variable bond strengths); (2) minimizing $\#E^-(\Sigma^\zeta) + c \sum_V \zeta(v)$, with $c \neq 0$ because of an external magnetic field.  Then one cannot expect the LP to have a combinatorial optimum.
\dated{20 Jan 2010}
\par
\codes{\CSG:~\CPhys,~\CFr,~\CFr(\CGen),~\CAlgor}%
\itemspace

\author{F.\ Barahona \& J.P.\ Uhry}
\authorspace
\bsgitem{1981a}{} An application of combinatorial optimization to physics.  \emph{Methods Operations Res.}\ 40 (1981), 221--224.  
\Zbl{461.90080}.  
\codes{\CSG:~\CPhys,~\CFr:~\CExp}%
\itemspace

\author{John S.\ Baras}						%BARAS
\authorspace
\authref  See \hyperlink{Guodong Shi}{G.-D.~Shi}.
\itemspace

\authorx{S.\ Barik}							%BARIK
\author{S.\ Barik, D.\ Kalita, S.\ Pati, \& G.\ Sahoo}
\authorspace
\bsgitem{2018a}{}  Spectra of graphs resulting from various graph operations and products: a survey.  \emph{Special Matrices} 6 (2018), 323--342.
\MR{3856591}.  \Zbl{1423.05097}.  
\annot{} The signless Laplacian $\Lap(-\Gamma)$ is one of the matrices surveyed.  No mention that it is signed-graphic.
\dated{29 Dec 2020} 
\par
\codes{\CPar:~\CLap:~\CEig:~\CExp}%
\itemspace

\author{J.\ Wesley Barnes}						%BARNES
\authorspace
\authref   See \hyperlink{Paul A.\ Jensen}{P.A.~Jensen}.
\itemspace

\author{Adriano Barra}						%BARRA
\authorspace
\bsgitem{2008a}{}  Fluctuations induce transitions in frustrated sparse networks.  \emph{Fluctuation Noise Lett.}\ 8 (2008), L341.
\arXiv{0911.5144}.  
\annot{} Physical quantities of a random signed graph, with states (``configurations'') $s \in \{\pm1\}^n$, studied by replication.  
\dated{29 Dec 2012} 
\par
\codes{\CPhys:~\CSG:~\CRand:~\CAdj}%
\itemspace

\authorx{Arun Kumar Baruah}						%BARUAH
\author{Arun Kumar Baruah \& Manoshi Kotoky}
\authorspace
\bsgitem{2015a}{}  Signed graph and its balance theory in transportation problem.  \emph{Int.\ J.\ Computer Appl.}\ 115 (2015), no.\ 12, 9--12.
\annot{} Confused; nothing of value.
\dated{13 Oct 2019} 
\codes{\CSG:~\CBal,~\CAppl}%
\itemspace

\author{Tamer Ba\c{s}ar}						%BASAR
\authorx{Tamer Basar}
\authorspace
\authref  See \hyperlink{Wei Chen}{W.~Chen} and \hyperlink{Xudong Chen}{X.-D.~Chen}.
\itemspace

\authorx{Nino Ba\v{s}i\'c}						%BASIC
\authorx{Nino Basic}
\author{Nino Ba\v{s}i\'c, Patrick W.\ Fowler, Toma\v{z} Pisanski, \& Irene Sciriha}
\authorspace
\bsgitem{2022a}{}  On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs.  \emph{Discuss.\ Math.\ Graph Theory} 42 (2022), no.\ 4, 1351--1382.
\MR{4445856}.  
\arXiv{2009.09018}.  
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Lowell Bassett} 						%BASSETT
\author{Lowell Bassett, John Maybee, \& James Quirk}
\authorspace
\bsgitem{1968a}{} Qualitative economics and the scope of the correspondence principle.  \emph{Econometrica} 36 (1968), 544--563.  
\MR{0237165} (38 \#5456).  \Zbl{217.26802} (217, p.\ 268b).				
\annot{} Lemma 3:  A square matrix with every diagonal entry negative is sign-nonsingular iff every cycle is negative in the associated signed digraph.  Thm.\ 4:  A square matrix with negative diagonal is sign-invertible iff all cycles are negative and the sign of any (open) path is determined by its endpoints.  And more.
\codes{\CQM:~\CSol,~\CSta:~\Csd}%
\itemspace

\author{Vladimir Batagelj}						%BATAGELJ
\authorspace
\authref  See also \hyperlink{Patrick Doreian}{P.~Doreian}, \hyperlink{Natasa Kejzar}{N.~Kej\v{z}ar}, and \hyperlink{Wouter de Nooy}{W.~de~Nooy}.
\authrefspace

\bsgitem{1990a}{}  [Closure of the graph value matrix.]  (In Slovenian.  English summary.)  \emph{Obzornik Mat.\ Fiz.}\ 37 (1990), 97--104.  
\MR{1074109} (91f:05058).  \Zbl{704.05035}.
\par
\codes{\CSG:~\CAdj,~\CBal,~\CClu}
\itemspace

\bsgitem{1994a}{}  Semirings for social networks analysis.  \emph{J.\ Math.\ Sociology} 19 (1994), 53--68.  
\Zbl{827.92029}.  
\codes{\CSG:~\CAdj,~\CBal,~\CClu}%
\itemspace

\bsgitem{1997a}{}  Notes on blockmodeling.  \emph{Social Networks} 19 (1997), 143--155.
\annot{} \S3, p.\ 6:  Predicates to use for searching out balanced or clusterable partitions.
\dated{10 Mar 2011} 
\codes{\CSG:~\CPsS,~\CAlgor}%
\itemspace

\author{V.\ Batagelj \& T.\ Pisanski}
\authorspace
\bsgitem{1979a}{}  On partially directed Eulerian multigraphs.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 25(39) (1979), 16--24.
\MR{0542818} (81a:05054).  \Zbl{418.05038}.					
\codes{\Csg:~\COri}%
\itemspace

\author{William H.\ Batchelder}						%BATCHELDER
\authorspace
\authref  See \hyperlink{Kalin Agrawal}{K.~Agrawal}.
\itemspace

\author{Christian Bauckhage}						%BAUCKHAGE
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Thierry-Pascal Baum}						%BAUM
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{Jan Baumbach}						%BAUMBACH
\authorspace
\authref  See \hyperlink{Sebastian Bocker}{S.~B\"ocker}.
\itemspace

\authorx{Andrei B\u{a}utu}						%BAUTU A
\authorx{Andrei Bautu}
\author{Andrei B\u{a}utu \& Elena B\u{a}utu}
\authorx{Andrei Bautu \& Elena Bautu}
\authorspace
\bsgitem{2007a}{}  Searching ground states of Ising spin glasses with particle swarms.  \emph{Rom.\ J.\ Phys.}\ 52 (2007), no.\ 3-4, 337--342.
\annot{} Experimental results for $l(\Sigma)$ compared with known results.
\dated{19 Aug 2012} 
\codes{\CSG,~\CPhys:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2007b}{}  Searching ground states of Ising spin glasses with genetic algorithms and binary particle swarm optimization.  In:  Natalio Krasnogor \etal, eds., \emph{Nature Inspired Cooperative Strategies for Optimization} (NICSO 2007, Int.\ Workshop, Acireale, Italy), pp.\ 85--94.  Stud.\ Comput.\ Intelligence, Vol.\ 129.  Springer, Berlin, 2008.
\annot{} Compares the two algorithms for $l(\Sigma)$.
\dated{19 Aug 2012} 
\par
\codes{\CSG,~\CPhys:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2009a}{}  Particle swarms in statistical physics.  In:  Aleksandar Lazinica, ed., \emph{Particle Swarm Optimization}, Ch.\ 4, pp.\ 77--88.  InTech, Rijeka, Croatia, and Shanghai, 2009.
\annot{} \S4, ``Binary particle swarm optimization and Ising spin glasses'':  The signed graph; spins and states; satisfied and frustrated edges; some history.  In particle swarm optimization, each vertex acts as a cell in a cellular automaton, learning probabilistically, seeking a most satisfied spin $\zeta(v)$ in order to minimize $\#E_v^-(\Sigma^\zeta)$.  [It seems that this local minimization suffers from the same potential instability as \xref{Mitra's}{1962a}{S.\ Mitra} deterministic local minimization, hence is not accurate.]
\dated{19 Aug 2012} 
\codes{\CSG,~\CPhys:~\CState,~\CFr:~\CAlgor:~\CExp}%
\itemspace

\author{Andrei B\u{a}utu, Elena B\u{a}utu, \& Henri Luchian}
\authorx{Andrei Bautu, Elena Bautu, \& Henri Luchian}
\authorspace
\bsgitem{2007a}{}  Particle swarm optimization hybrids for searching ground states of Ising spin glasses.  In:  Viorel Negru \etal, eds., \emph{SYNASC 2007: Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2007} (Timisoara, Romania), pp.\ 415--418.  IEEE Computer Soc., Los Alamitos, Calif., 2007.
\annot{} Particle swarm optimization combined with hill-climbing to find $l(\Sigma)$ (ground state of Ising model); a hybrid method is said to be promising. 
\dated{19 Aug 2012} 
\codes{\CSG,~\CPhys:~\CState(\Cfr):~\CAlgor}%
\itemspace

\bsgitem{2008a}{}  Searching ground states of Ising spin glasses with a tree bond-based representation.  In:  Viorel Negru \etal, eds., \emph{SYNASC 2008: 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (Timisoara, Romania), pp.\ 501--506.  IEEE Computer Soc., Los Alamitos, Calif., 2008.
\annot{} Bond-based representation means recording switched edge (``bond'') signs instead of vertex spins; \cf\ \xref{Pelikan and Hartmann}{2007a}{Martin Pelikan \& Alexander K.\ Hartmann}, \xrefd{2007b}{Martin Pelikan \& Alexander K.\ Hartmann}.  Here, a state $s: V \to \signnos$ is recorded as the signs of a spanning tree switched by $s$.  This has ambiguity [2, obviously].  Negating one tree edge implies a chain of spin changes; this ``may be considered a feature'' [and its implications could be interesting].  Computational experiments tested the implied algorithm.
\dated{19 Aug 2012} 
\par
\codes{\CSG,~\CPhys:~\CState(\Cfr):~\CAlgor}%
\itemspace

\author{Andrei B\u{a}utu \& Henri Luchian}
\authorspace
\bsgitem{2010a}{}  Particle swarm optimization with spanning tree representation for Ising spin glasses.  In: \emph{2010 IEEE Congress on Evolutionary Computation} (CEC 2010, Barcelona), pp.\ 1--6.  IEEE, 2010.  
\annot{} Applies \xref{B\u{a}utu, B\u{a}utu, and Luchian}{2008a}{Andrei Bautu, Elena Bautu, \& Henri Luchian}.  Shallower trees may produce better results due to the lesser effect of negating one tree edge.  Computational comparisons of this and other algorithms for ground state (i.e., frustration index).
\dated{19 Aug 2012} 
\codes{\CSG,~\CPhys:~\CFr:~\CAlgor}%
\itemspace

\author{Elena B\u{a}utu}						%BAUTU E
\authorx{Elena Bautu}						%BAUTU E
\authorspace
\authref  See \hyperlink{Andrei Bautu \& Elena Bautu}{A.~B\u{a}utu}.
\itemspace

\author{Sean R.B.\ Bearden}						%BEARDEN
\authorspace
\authref  See \hyperlink{Haik Manukian}{H.~Manukian}.
\itemspace

\authorx{Laurent Beaudou}						%BEAUDOU
\author{Laurent Beaudou, Florent Foucaud, \& Reza Naserasr}
\authorspace
\bsgitem{2019a}{}  Homomorphism bounds of signed bipartite $K_4$-minor-free graphs and edge-colorings of $2k$-regular $K_4$-minor-free multigraphs.  \emph{Discrete Appl.\ Math.}\ 261 (2019), 40--51.
\MR{3958228}.  \Zbl{1410.05083}.  \arXiv{1811.03807}.  \HAL{hal-01915269}.		
\codes{\CSG:~\CHom,~\CECol}%
\itemspace

\bsgitem{2020a}{}  Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$.  In:  Manoj Changat and Sandip Das, eds., \emph{Algorithms and Discrete Applied Mathematics} (CALDAM 2020, Hyderabad, 2020), pp.\ 184--196.  \LNCS 12016.  Springer, Cham, 2020.
\MR{4074908}.  \Zbl{1456.05117}.  \HAL{hal-03041342}.  
\annot{} Conference version of \xrefd{2022a}{Laurent Beaudou, Florent Foucaud, \& Reza Naserasr}.
\dated{28 Dec 2022} 
\codes{\Csg:~\CPar:~\CHom}%
\itemspace

\bsgitem{2022a}{}  Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$.  \emph{Discrete Appl.\ Math.}\ 319 (2022) 564--575.
\MR{4457824}.  \Zbl{1494.05043}.  \arXiv{1610.03685}.  \HAL{hal-03784719}.		
\codes{\Csg:~\CPar:~\CHom}%
\itemspace

\authorx{Matthias Beck}						%BECK M
\author{Matthias Beck \& Mela Hardin}
\authorspace
\bsgitem{2015a}{}  A bivariate chromatic polynomial for signed graphs.  \emph{Graphs Combin.}\ 31 (2015), no.\ 5, 1211--1221.
\MR{3386004}.  \Zbl{1327.05099}.  \arXiv{1204.2568}.				
\par
\codes{\CSG:~\CCol,~\CGeom}%
\itemspace

\author{Matthias Beck, Erika Meza, Bryan Nevarez, Alana Shine, \& Michael Young}
\authorspace
\bsgitem{2015a}{}  The chromatic polynomials of signed Petersen graphs.  \emph{Involve} 8 (2015), no.\ 5, 825--831.
\MR{3404660}.  \Zbl{1322.05073}.  \arXiv{1311.1760}.				
\annot{} Chromatic and zero-free chromatic polynomials of all six switching isomorphism types of signed Petersens (\cf\ \xref{Zaslavsky}{2012b}{Thomas Zaslavsky}) and all signed $K_n$'s for $n\leq5$.  Each switching isomorphism type has a different chromatic polynomial and each has a different zero-free polynomial.  Computer code in SAGE.
\dated{2 Nov 2013} 
\codes{\CSG:~\CCol:~\CInvar}%
\itemspace

\author{Matthias Beck \& Thomas Zaslavsky}
\authorspace
\bsgitem{2006a}{}  Inside-out polytopes.  \emph{Adv.\ Math.}\ 205 (2006), no.\ 1, 134--162.
\MR{2254310} (2007e:52017).  \Zbl{1107.52009}.  \arXiv{0309330}.			
\annot{} \S5: ``In which we color graphs and signed graphs.''  A geometric interpretation of signed graph coloring by lattice points and hyperplane arrangements unifies the chromatic and zero-free chromatic polynomials and gives immediate proofs of theorems on the chromatic polynomials and acyclic orientations.
\codes{\CSG:~\CCol:~\CGeom,~\CMatrd,~\CInvar,~\CBal}%
\itemspace

\bsgitem{2006b}{}  The number of nowhere-zero flows in graphs and signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 96 (2006), no.\ 6, 901--918.
\MR{2274083} (2007k:05084).  \Zbl{1119.05105}.  \arXiv{math/0309330}.			
\annot{} The nowhere-zero flow polynomial of a signed graph, for flows in an odd abelian group, and the integral nowhere-zero flow quasipolynomial with period 2.  [For even abelian groups see \xref{DeVos, Rollov\'a, and \v{S}\'amal}{2019a}{Matt DeVos, Edita Rollova, \& Robert Samal}.]
\codes{\CSG:~\CFlows:~\CGeom,~\CMatrd,~\CInvar,~\CBal}%
\itemspace

\bsgitem{2006c}{}  An enumerative geometry for magic and magilatin labellings.  \emph{Ann.\ Combin.}\ 10 (2006), no.\ 4, 395--413.
\MR{2293647} (2007m:05010).  \Zbl{1116.05071}.  \arXiv{math/0506315}.			
\annot{} In magic labellings of a bidirected graph, the labels are distinct positive integers; at each vertex the sum over entering edge ends equals that over departing edge ends.  Thms.\ (implicit):  The number of magic labellings is a quasipolynomial function of the magic sum, if the magic sum is prescribed.  It is also a quasipolynomial function of the upper bound on the labels, if an upper bound is prescribed.
\codes{\Cori:~\CGeom,~\CEnum}%
\annot{} \S5: ``Generalized exclusions.''  Complementarity rules in magic squares, etc., can be expressed by signed-graphic hyperplanes.
\par
\codes{\Csg:~\CLab:~\CGeom,~\CEnum}%
\itemspace

\bsgitem{2010a}{}  Six little squares and how their numbers grow.  \emph{J.\ Integer Sequences} 13 (2010), art.\ 10.6.2, 43 pp.
\MR{2659218} (2011j:05052).  \Zbl{1230.05062}.  \arXiv{1004.0282}.			
\annot{} \S3: ``Semimagic squares.''  Counts magic labellings of the extraverted $-K_{3,3}$ by an explicit geometrical solution.  Counted either by upper bound on the values or by magic sum.
\codes{\Cpar:~\CLab:~\Cinc,~\CGeom}%
\itemspace

\author{Richard Behr}						%BEHR
\authorspace
\bsgitem{2017a}{}  Edges and vertices in a unique signed circle in a signed graph.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 14 (2017), no.\ 3, 224--232.
\MR{3695979}.  \Zbl{1375.05122}.  \arXiv{1610.02107}.				
\annot{} In $\Sigma$, does edge $e$, or vertex $v$, lie in exactly one negative circle?  Exactly one positive circle?  The structure of $\Sigma$ and the corresponding edges or vertices are determined for each question.  
\dated{21 Dec 2017} 
\par
\codes{\CSG:~\CCircles:~\CStr}%
\itemspace

\bsgitem{2018a}{}  \emph{Edge Coloring and Special Edges of Signed Graphs}.  Doctoral dissertation, Binghamton University, 2018.
\MR{3818916} (no rev).  
\annot{} Ch.\ 2, ``Edge coloring signed graphs'': Vizing's edge-coloring theorem generalized to signed simple graphs.  Early version of \xrefd{2020a}{Richard Behr}.  Ch.\ 3, ``Special edges'':  Same as \xrefd{2017a}{Richard Behr}.
\dated{27 Apr 2018} 
\par
\codes{\CSG:~\CECol:~\Clg}\codesctd{\CSG:~\CStr}%
\itemspace

\bsgitem{2020a}{\dag}  Edge coloring signed graphs.  \emph{Discrete Math.}\ 343 (2020), no.\ 2, art.\ 111654, 15 pp.
\MR{4040037}.  \Zbl{1429.05058}.  \arXiv{1807.11465}.				
\annot{} Introduces edge coloring of signed graphs.  Vizing's Theorem generalizes to signed simple graphs: the number of proper edge colorings is $\chi'(\Sigma)\leq\Delta(\Sigma)+1$.  A proper edge coloring of $\Sigma$ is the same as a proper vertex coloring (\cf\ \xref{Zaslavsky}{1982b}{Thomas Zaslavsky}) of $-\Lambda(\Sigma)$, defined via bidirected graphs (\cf\ \xref{Zaslavsky}{20xxa}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}).  An opposite definition (``antiproper coloring'') is equivalent to balanced decomposition of $-\Sigma$ (\cf\ \xref{Zaslavsky}{1987b}{Thomas Zaslavsky}).  Remarks on total coloring.
[Independent partial proof by \xref{Zhang--Lu--Luo--Ye--Zhang}{2020a}{Li Zhang, You Lu, Rong Luo, Dong Ye, \& Shenggui Zhang} with results for signed planar graphs.]
\dated{26 Aug 2018, rev 2 Jul 2020} 
\codes{\CSG:~\CECol}\codesctd{\CSG:~\CLG:~\COri,~\CECol}%
\itemspace

\authorx{M.\ Behzad}
\author{M.\ Behzad \& G.\ Chartrand}					%BEHZAD
\authorspace
\bsgitem{1969a}{}  Line-coloring of signed graphs.  \emph{Elem.\ Math.}\ 24 (1969), 49--52.
\MR{0244098} (39 \#5415).  \Zbl{175.50302} (175, p.\ 503b).				
\annot{$\Lambda_{BC}$} Their line graph $\Lambda_{BC}(\Sigma)$ of a signed simple graph $\Sigma$ (not defined explicitly) is the line graph $\Lambda(|\Sigma|)$ with an edge negative when its two endpoints are negative edges in $\Sigma$.  They ``color'' as in \xref{Cartwright and Harary}{1968a}{Dorwin Cartwright \& Frank Harary} [i.e., clustering].  Characterized:  $\Sigma$ with colorable line graphs.  Found: the fewest colors for line graphs of signed trees, $K_n$, and $K_{r,s}$.	 
[For a more sophisticated kind of line graph see \xref{Zaslavsky}{1984c}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}.  For another line graph, see \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]
\par
\codes{\CSG:~\Clg:~\CClu}%
\itemspace

\authorx{Ghazaleh Beigi}						%BEIGI
\author{Ghazaleh Beigi, Jiliang Tang, \& Huan Liu}
\authorspace
\bsgitem{2016a}{}  Signed link analysis in social media networks.  In:  \emph{Proceedings of the Tenth International AAAI Conference on Web and Social Media} (ICWSM 2016, Cologne), pp.\ 539--542.  AAAI Press [Assoc.\ Advancement of Artificial Intelligence], Palo Alto, 2016.
\arXiv{1603.06878}.  
\codes{\CPsS:~\CSG,~\CAlgor}%
\itemspace

\authorx{Amos Beimel}						%BEIMEL
\author{Amos Beimel, Aner Ben-Efraim, Carles Padr\'o, \& Ilya Tyomkin}
\authorspace
\bsgitem{2014a}{}  Multi-linear secret-sharing schemes.  In:  Yehuda Lindell, ed., \emph{Theory of Cryptography} (Proc.\ 11th Theory of Cryptography Conf., TCC 2014, San Diego, 2014), pp.\ 394--418.  \LNCS 8349.  Springer, Berlin, 2014.
\MR{3183548}.  \Zbl{1326.94071}.  
\annot{} 	
[\Cf\ \xref{Ben-Efraim}{2016a}{Aner Ben-Efraim}.]
\codes{\Cgg:~\CMatrd}%
\itemspace

\authorx{Lowell W.\ Beineke}				 		%BEINEKE
\author{Lowell W.\ Beineke \& Frank Harary}
\authorspace
\bsgitem{1966a}{}  [As ``W.~Beineke and F.~Harary'']  Binary matrices with equal determinant and permanent.  \emph{Studia Sci.\ Math.\ Hungar.}\ 1 (1966), 179--183.  
\MR{0207582} (34 \#7397).  \Zbl{145.01505} (145, p.\ 15e).
\codes{\CSD}%
\itemspace

\bsgitem{1978a}{} Consistency in marked digraphs.  \emph{J.\ Math.\ Psychology} 18 (1978), 260--269.  
\MR{0522390} (80d:05026).  \Zbl{398.05040}. 					
\annot{} A ``marked digraph'' is a digraph $\vec D$ with signed vertices, $\vec S = \vec D,\mu)$ where $\mu:V \to \{+,-\}$.  It is ``consistent'' if all diwalks from $v$ to $w$ have the same sign $\mu(W)$.  The sign of a walk is the vertex sign product.  
Thm.\ 1.  Asssuming $\vec D$ is strongly connected, $\vec S$ is consistent iff every dicycle is positive.  [An important difference from signed graphs, where no restriction is needed.]  
Thm.\ 2.  $\vec S$ is consistent iff $V$ has a bipartition such that every arc with a positive tail lies within a set but no arc with a negative tail does so.  
Define $\sigma(\vec{uv} := \mu(u)$.  Thm.\ 3.  Asssuming $\vec D$ is strongly connected, this signed graph is balanced iff $\vec S$ is consistent.  
Thm.\ 4.  A vertex-signed tournament $\vec S$ is consistent iff:  When strongly connected, [it is all positive or] it has exactly two negative vertices $u,v$ and, deleting $uv$, $u$ is a source and $v$ is a sink.  When not strongly connected, it is consistent iff it is all positive, or it has one negative vertex which is a source or sink, or it has two negative vertices, one a source and the other a sink. 
Thm.\ 5.  $\vec D$ has $\mu \not\equiv +$ such that $(\vec D,\mu)$ is consistent (``markable'') iff $\exists\ \eset \subset V_0 \subset V$ such that, $\forall\ v$, all out-arcs from $v$, or none, go to $V_0$.  
\dated{16 Sept 2010}
\codes{\CVS}%
\itemspace

\bsgitem{1978b}{\dag} Consistent graphs with signed points.  \emph{Riv.\ Mat.\ Sci.\ Econom.\ Social.}\ 1 (1978), 81--88.  
\MR{0573718} (81h:05108).  \Zbl{493.05053}.					
\annot{} A graph (not necessarily simple) with signed vertices is ``consistent'' if every circle has positive sign product.  
Thm.\ 2.2:  $\Gamma$ with all negative vertices is consistent iff bipartite.  
Thm.\ 2.3:  3-connected vertices must have the same sign.  
Thm.\ 3.3:  Contracting an edge with positive endpoints preserves consistency and inconsistency.  
Further partial results.  
Open problem:  A full characterization of consistent vertex-signed graphs.  [For a good solution see \xref{Hoede}{1992a}{Cornelis Hoede}.  For the best solution see \xref{Joglekar, Shah, and Diwan}{2010a}{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan}.]
\dated{rev.\ 11 Sept 2010}
\codes{\CVS:~\CBal}%
\itemspace

\author{Lowell W.\ Beineke \& Robin J.\ Wilson, eds.}
\authorspace
\bsgitem{2009a}{}  \emph{Topics in Topological Graph Theory.}  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581536} (2011f:05005).  \Zbl{1170.05003}.  
\annot{} Several chapters discuss the use of gain graphs (as ``voltage graphs'') to construct surface embeddings and covering graphs.  Some mention edge signs \emph{in re} orientation.
Ch.\ 1: \xref{Gross and Tucker}{2009a}{Jonathan L.\ Gross \& Thomas W.\ Tucker}.  
Ch.\ 3: \xref{Gross}{2009a}{Jonathan L.\ Gross}.  
Ch.\ 8: \xref{Pisanski and \v{Z}itnik}{2009a}{Tomaz Pisanski \& Arjana Zitnik}.  
Ch.\ 9: \xref{Kwak and Lee}{2009a}{Jin Ho Kwak \& Jaeun Lee}.  
Ch.\ 10:  \xref{\v{S}ir\'a\v{n} and Tucker}{2009a}{Jozef Siran \& Thomas W.\ Tucker}.  
Ch.\ 11: \xref{Tucker}{2009a}{Thomas W.\ Tucker}.  
Ch.\ 12: \xref{A.T.\ White}{2009a}{Arthur T.\ White}.  
Ch.\ 13: \xref{Grannell and Griggs}{2009a}{M.J.\ Grannell \& T.S.\ Griggs}.  
\annotctd{} [Strangely for a book ``covering the full range of topological graph theory'' (p.\ xvii), orientation embedding of signed graphs is virtually ignored.  \Cf, e.g., \xref{Lins}{1985a}{Sostenes Lins}, \xref{\v{S}ir\'a\v{n} and \v{S}koviera}{1991a}{Jozef Siran \& Martin Skoviera}, and \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}, \xrefd{1993a}{Thomas Zaslavsky}.]
\dated{12 Jun 2013} 
\par
\codes{\CTop:~\CGG,~\CSG,~\CCov,~\CEnum:~\CExp}%
\itemspace

\author{Mohamed-Ali Belabbas}					%BELABBAS
\authorspace
\authref  See \hyperlink{Xudong Chen}{X.-D.~Chen}.
\itemspace

\authorx{Jacques B\'elair}						%BELAIR
\authorx{Jacques Belair}
\author{Jacques B\'elair, Sue Ann Campbell, \& P.\ van den Driessche}
\authorx{Jacques Belair, Sue Ann Campbell, \& P.\ van den Driessche}
\authorspace
\bsgitem{1996a}{}  Frustration, stability, and delay-induced oscillations in a neural network model.  \emph{SIAM J.\ Appl.\ Math.}\ 56 (1996), 245--255.
\MR{1372899} (96j:92003). \\ \Zbl{840.92003}.					
\annot{} The signed digraph of a square matrix is ``frustrated'' if it has a negative cycle.  Somewhat simplified: a negative cycle is necessary for there to be oscillation caused by intraneuronal processing delay.
	
	
\codes{\CSD:~\CQM,~\CRef}%
\itemspace

\author{Francesco Belardo}						%BELARDO
\authorspace
\authref  See also \hyperlink{Aida Abiad}{A.~Abiad}, \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Abdullah Alazemi}{A.~Alazemi}, and \hyperlink{Jianfeng Wang}{J.F.~Wang}.
\authrefspace

\bsgitem{2014a}{}  Balancedness and the least eigenvalue of Laplacian of signed graphs.  \emph{Linear Algebra Appl.}\ 446 (2014), 133--147.
\MR{3163133}.  \Zbl{1285.05112}.  
\codes{\CSG:~\CEig,~\CFr}%
\itemspace

\author{Francesco Belardo \& Maurizio Brunetti}
\authorspace
\bsgitem{2019a}{}  Connected signed graphs $L$-cospectral to signed $\infty$-graphs.  \emph{Linear Multilinear Algebra} 67 (2019), no.\ 12, 2410--2426.
\MR{4017722}.  \Zbl{1425.05067}.  
\annot{} All connected Laplacian-cospectral mates of unbalanced ``$\infty$-graphs'' (tight handcuffs).  They are certain bicyclic graphs.
\dated{18 Jan 2019} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Line graphs of complex unit gain graphs with least eigenvalue $-2$.  \emph{Electronic J.\ Linear Algebra} 37 (2021), 14--30.
\MR{4246513}.  \Zbl{1464.05170}.  
\annot{} Eigenvectors of $-2$ for $A(-\Lambda(\Phi)) = \Eta(\Phi)^T\Eta(\Phi)-2I$.  Dictionary:  ``line graph'' = spectral line graph $\Lambda_{\Spec}(\Phi) = -\Lambda(\Phi)$.
\dated{2, 11 Jan 2020} 
\codes{\CGG:~\CLG:~\CAdj:~\CEig}%
\itemspace

\author{Francesco Belardo, Maurizio Brunetti, Matteo Cavaleri, \& Alfredo Donno}
\authorspace
\bsgitem{2021a}{}  Constructing cospectral signed graphs.  \emph{Linear Multilinear Algebra}  69 (2021), no.\ 14, 2717--2732.
\MR{4306815}.  \Zbl{1472.05064}.  \arXiv{1908.02220}.				
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Francesco Belardo, Maurizio Brunetti, \& Adriana Ciampella}
\authorspace
\bsgitem{2018a}{}  Signed bicyclic graphs minimizing the least Laplacian eigenvalue.  \emph{Linear Algebra Appl.}\ 557 (2018), 201--233.
\MR{3848268}.  \Zbl{1396.05066}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2019a}{}  On the multiplicity of $\alpha$ as an $A_\alpha(\Gamma)$-eigenvalue of signed graphs with pendant vertices.  \emph{Discrete Math.}\  342 (2019), 2223--2233.
\MR{3952139}.  \Zbl{1418.05070}.  
\codes{\CSG:~\CAdj(\CGen):~\CEig}%
\itemspace

\bsgitem{2019b}{}  Edge perturbation on signed graphs with clusters: Adjacency and Laplacian eigenvalues.  \emph{Discrete Appl.\ Math.}\ 269 (2019), 130--138.
\MR{4016591}.  \Zbl{1421.05051}.  
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Godsil-McKay switching for mixed and gain graphs over the circle group.  \emph{Linear Algebra Appl.}\ 614 (2021), 256--269.
\MR{4209002}.  \Zbl{1459.05110}.  
\par
\codes{\Csg,~\CGG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021b}{}  Unbalanced unicyclic and bicyclic graphs with extremal spectral radius.  \emph{Czechoslovak Math.\ J.}\ 71 (2021), no.\ 2, 417--433.
\MR{4263178}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Francesco Belardo, Maurizio Brunetti, Nolan J.\ Coble, Nathan Reff, \& Howard Skogman}
\authorspace
\bsgitem{2022a}{}  Spectra of quaternion unit gain graphs.  \emph{Linear Algebra Appl.}\ 632 (2022), 15--49 [2021].
\MR{4319411}.  \Zbl{1478.05088}.  
\codes{\CGG:~\CAdj,~\CLap,~\CIncid:~\CEig}%
\itemspace

\author{Francesco Belardo, Maurizio Brunetti, \& Nathan Reff}
\authorspace
\bsgitem{2020a}{}  Balancedness and the least Laplacian eigenvalue of some complex unit gain graphs.  \emph{Discuss.\ Math.\ Graph Theory} 40 (2020), no.\ 2, 417--433.
\MR{4060993}.  \Zbl{1433.05185}.  
\codes{\CGG:~\CBal,~\CLap:~\CEig}%
\itemspace

\author{Francesco Belardo, Maurizio Brunetti, \& Vilmar Trevisan}
\authorspace
\bsgitem{2021a}{}  Locating eigenvalues of unbalanced unicyclic signed graphs.  \emph{Appl.\ Math.\ Comput.}\ 400 (2021), art.\ 126082.
\MR{4218433}.  
\codes{\CSG:~\CAdj:~\CEig:~\CAlgor}%
\itemspace

\author{Francesco Belardo, Sebastian M.\ Cioab\u{a}, Jack Koolen, \& Jianfeng Wang}
\authorx{Francesco Belardo, Sebastian M.\ Cioaba, Jack Koolen, \& Jianfeng Wang}
\authorspace
\bsgitem{2018a}{}  Open problems in the spectral theory of signed graphs.  \emph{Art Discrete Appl.\ Math.}\ 1 (2018), art.\ \#P2.10, 23 pp.
\MR{3997096}.  \Zbl{1421.05052}.  \arXiv{907.04349}.				
\annot{} Eigenvalue questions, almost all for $A(\Sigma)$, with a review of relevant knowledge.
\dated{13 Jun 2019} 
\codes{\CSG:~\CEig:~\CAdj,~\CKG}%
\itemspace

\author{Francesco Belardo, Enzo M.\ Li Marzi, \& Slobodan K.\ Simi\'c}
\authorx{Francesco Belardo, Enzo M.\ Li Marzi, \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2016a}{}  Signed line graphs with least eigenvalue $-2$: The star complement technique.  \emph{Discrete Appl.\ Math.}\ 207 (2016), 29--38.
\MR{3497981}.  \Zbl{1337.05051}.  
\annot{} Produces a nice basis, derived from $\Sigma$, for the eigenspace of $+2$ of $\Lambda(\Sigma)$.  
Dictionary:  \cf\ \xref{Belardo and Simi\'c}{2015a}{Francesco Belardo \& Slobodan K.\ Simic}.
\dated{11 Jan 2020} 
\par
\codes{\CSG:~\CLG:~\CAdj:~\CEig}%
\itemspace

\author{Francesco Belardo, Enzo M.\ Li Marzi, Slobodan K.\ Simi\'c, \& Jianfeng Wang}
\authorx{Francesco Belardo, Enzo M.\ Li Marzi, Slobodan K.\ Simic, \& Jianfeng Wang}
\authorspace
\bsgitem{2010a}{}  On the index of necklaces.  \emph{Graphs Combin.} 26 (2010), no.\ 2, 163--172.
\MR{2606492} (2011g:05171).  \Zbl{1231.05165}.  
\annot{} The largest eigenvalue of $A(G)$ for $G =$ chain or necklace of cliques, via $\Lap(-\Gamma)$ where $G=\Lambda(\Gamma)$.
\dated{16 Jan 2012} 
\codes{\Cpar:~\CLG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2011a}{}  Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 11, 2913--2920. 
\MR{2825291} (2012k:05221).  \Zbl{1221.05229}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Francesco Belardo \& Pawe{\l} Petecki}
\authorx{Francesco Belardo \& Pawel Petecki}
\authorspace
\bsgitem{2015a}{}  Spectral characterizations of signed lollipop graphs.  \emph{Linear Algebra Appl.}\ 480 (2015), 144--167.
\MR{3348518}.  \Zbl{1315.05083}.  
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Francesco Belardo, Pawe{\l} Petecki, \& Jianfeng Wang}
\authorx{Francesco Belardo, Pawel Petecki, \& Jianfeng Wang}
\authorspace
\bsgitem{2016a}{}  On signed graphs whose second largest Laplacian eigenvalue does not exceed 3.  \emph{Linear Multilinear Algebra} 64 (2016), no.\ 9, 1785--1799.
\MR{3509501}.  \Zbl{1341.05148}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Francesco Belardo, Toma\v{z} Pisanski, \& Slobodan K.\ Simi\'c}
\authorx{Francesco Belardo, Tomaz Pisanski, \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2016a}{}  On graphs whose least eigenvalue is greater than $-2$.  \emph{Linear Multilinear Algebra} 64 (2016), no.\ 8, 1570--1582.
\MR{3503369}.  \Zbl{1341.05149}.  
Erratum. \emph{Ibid.} 64 (2016), no.\ 8, I.
\MR{3503376}.  
\annot{} \S2, ``Line graphs and their generalizations'':  Adjacency eigenvalues of spectral line graph $\Lambda_{\Spec}(\Sigma)$ compared to Laplacian eigenvalues of $\Sigma$ by standard method.  Conj.\ 4.4 on signed graphs.
Dictionary:  \cf\ \xref{Belardo and Simi\'c}{2015a}{Francesco Belardo \& Slobodan K.\ Simic}.
\dated{4, 11 Jan 2020} 
\par
\codes{\CSG:~\COri:~\CAdj,~\CLap,~\CLG:~\CEig}%
\itemspace

\author{Francesco Belardo, Irene Sciriha, \& Slobodan K. Simi\'c}
\authorx{Francesco Belardo, Irene Sciriha, \& Slobodan K. Simic}
\authorspace
\bsgitem{2016a}{}  On eigenspaces of some compound signed graphs.  \emph{Linear Algebra Appl.}\ 509 (2016), 19--39.
\MR{3546399}.  \Zbl{1346.05161}.  
\codes{\CSG:~\CAdj,~\CLap,~\CLG}%
\itemspace

\author{Francesco Belardo \& Slobodan K.\ Simi\'c}
\authorx{Francesco Belardo \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2015a}{}  On the Laplacian coefficients of signed graphs.  \emph{Linear Algebra Appl.}\ 475 (2015), 94--113.
\MR{3325220}.  \Zbl{1312.05078}.  
\annot{$\Lambda_{\Spec}$} \S2:  Adjacency eigenvalues of spectral line graph $-\Lambda_S(\Sigma)$ compared to Laplacian eigenvalues of $\Sigma$ by standard method.  \S3:  Coefficients of Laplacian characteristic polynomial.  \xref{Ded\`o}{1981a}{Ernesto Dedo}-style matrix-tree theorem.  \S4, ``Laplacian coefficients of signed unicyclic graphs''.
Dictionary:  ``oriented edge'' = positive edge, ``unoriented edge'' = negative edge, ``line graph'' $\cL(\Sigma_\eta)$ = reduced negated line graph $-\overline\Lambda(\Sigma,\eta)$, ``subdivision graph'' $\cS(\Sigma,\eta)$ = signed vertex-edge incidence graph.  
\dated{5, 11 Jan 2020} 
\codes{\CSG:~\CLap,~\CLG}%
\itemspace

\author{Francesco Belardo, Zoran Stani\'c, \& Thomas Zaslavsky}
\authorx{Francesco Belardo, Zoran Stanic, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2023a}{}  Total graph of a signed graph.  \emph{Ars Math.\ Contemp.}\ 3 (2023) [2022], art.\ P1.02, 17 pp.
\MR{4518838}.  
\arXiv{1908.02001}.							
\annot{} Total graph defined to have matrix properties.  Two versions: combinatorial $\Tau_C(\Sigma)$ (with line graph $\Lambda(\Sigma)$ by \xref{Zaslavsky}{20xxa}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}, \etal) and spectral $\Tau_S(\Sigma)$ (with $-\Lambda(\Sigma)$ by \xref{Belardo and Simi\'c}{2015a}{Francesco Belardo \& Slobodan K.\ Simic}).  Frustration of line and total graphs.  Spectra of total graph and iterated and combined total graphs.  
(For a spectrally unsuitable definition \cf\ \xref{Sinha--Garg}{2011b}{Deepa Sinha \& Pravin Garg}.)
\dated{1 Apr 2022} 
\codes{\CSG:~\CLG,~\CLG(\CGen):~\CAdj:~\CEig}%
\itemspace

\author{Francesco Belardo \& Yue Zhou}
\authorspace
\bsgitem{2016a}{}  Signed graphs with extremal least Laplacian eigenvalue.  \emph{Linear Algebra Appl.}\ 497 (2016), 167--180.
\MR{3466641}.  \Zbl{1331.05135}.  
\annot{} Eigenvalues are $\lambda_n\leq\cdots\leq\lambda_1$.  Thms:  Min and max $\lambda_n(\Lap(\Sigma))$ over unbalanced [and connected] $\Sigma$ are attained by an unbalanced triangular lollipop and $-K_n$.
\dated{20 Mar 2016} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Hac\'ene Belbachir}						%BELBACHIR
\authorx{Hacene Belbachir}
\author{Hac\'ene Belbachir \& Imad Eddine Bousbaa}
\authorx{Hacene Belbachir \& Imad Eddine Bousbaa}
\authorspace
\bsgitem{2013a}{}  Translated Whitney and $r$-Whitney numbers: A combinatorial approach.  \emph{J.\ Integer Sequences} 16 (2013), art.\ 13.8.6, 7 pp.
\MR{3118323}.  \Zbl{1292.05050}.  
\annot{} Introduces $w_{m,0}(n,k)$ (first kind) and $W_{m,0}(n,k)$ (second kind) = \# of ways to permute (or partition) $[n]$ into $k$ cycles (or blocks); color all but the least element in each using $m$ colors.  Also ``translated Whitney--Lah numbers'' and ``translated $r$-Whitney numbers''.  Formulas and identities.
\annotctd{} [Improvement:  (1)  Let group $\fG$ have order $m$; use color set $\fG$, least element colored $\epsilon$; then $W_m(n,k,0) = W_k(\Latb Q_n(\fG)) =$ Whitney number of balanced-flat semilattice of frame matroid $\bfF(\fG K_n\full)$.  Proved by \xref{Dowling}{1973b}{Thomas A.\ Dowling} as amplified in \xref{Zaslavsky}{1991a}{Thomas Zaslavsky}.  Also \cf\ \xref{Remmel and Wachs}{2004a}{J.B.\ Remmel \& Michelle L.\ Wachs}.  The Belbachir--Bousbaa case is $\fG=\bbZ_m$.  (2)  \Conjecture.  $w_m(n,k,0) = w_k(\Latb Q_n(\fG)) = \pm$(\# of permutations labelled by $\fG$, the Belbachir--Bousbaa case being $\fG=\bbZ_m$.  By the same proof.]
\annotctd{} Dictionary:  ``translated Whitney numbers'' $[ \begin{smallmatrix} n \\ k \end{smallmatrix}]^{(m)}$, $\{ \begin{smallmatrix} n \\ k \end{smallmatrix}\}^{(m)}$ = $w_{m,0}(n,k)$, $W_{m,0}(n,k)$ [$0$-Whitney numbers; \cf\ \xref{Gyimesi and Nyul}{2018a}{Eszter Gyimesi \& Gabor Nyul}]; ``mutation'' = $m$-coloring; ``dominant'' = (here, for convenience) least.  
\annotctd{} [``Whitney'' should be ``Dowling''; Whitney numbers are more general.]
\dated{28 May 2018} 
\codes{\Cgg:~\Cmatrd:~\CInvar}%
\itemspace

\authorx{G.R.\ Belitskii}						%BELITSKII
\author{G.R.\ Belitski\u\i\ \& Yu.I.\ Lyubich}
\authorx{G.R.\ Belitskii \& Yu.I.\ Lyubich}
\authorspace
\bsgitem{1984a}{}  \emph{Norms of Matrices and Their Applications}.  (In Russian.)  Naukova Dumka, Kiev, 1984.
\MR{0799924} (87e:15002).  \Zbl{568.15016}.					
\annot{} See \xref{Al'pin}{2014a}{Yu.\ A.\ Al'pin}.
\dated{22 Feb 2021} 
\codes{\Cgg:~\CBal}%
\itemspace

\author{G.R.\ Belitskii \& Yu.I.\ Lyubich [G.R.\ Belitski\u\i\ \& Yu.I.\ Lyubich]}
\authorspace
\bsgitem{1984b}{}  \emph{Matrix Norms and Their Applications}.  Transl.\ A.\ Iacob from \xrefd{1984a}{G.R.\ Belitskii \& Yu.I.\ Lyubich}.  Operator Theory Adv.\ Appl., Vol.\ 36.  Birkh\"auser, Basel, 1988.
\MR{1015711} (90g:15003).  \Zbl{645.15019}.					
\codes{\Cgg:~\CBal}%
\itemspace

\authorx{A.\ Bellacicco}						%BELLACICCO
\author{A.\ Bellacicco \& V.\ Tulli}
\authorspace
\bsgitem{1996a}{}  Cluster identification in a signed graph by eigenvalue analysis.  
In: \emph{Matrices and Graphs: Theory and Applications to Economics} (full title \emph{Proceedings of the Conferences on Matrices and Graphs: Theory and Applications to Economics}) (Brescia, 1993, 1995), pp.\ 233--242.  World Scientific, Singapore, 1996.
\MR{1670488} (no rev).  \Zbl{914.65146}.						
\annot{} Signed (di)graphs (``spin graphs'') are defined.  The main concepts are ``dissimilarity'', ``balance'', and ``cluster'' are defined and propositions are stated.  Eigenvalues are mentioned.  [This may be an announcement.  There are no proofs.  It is hard to be sure what is being said.]
\codes{\CSD:~\CEig}%
\itemspace

\author{Joachim von Below}						%BELOW
\authorspace
\bsgitem{1994a}{}  The index of a periodic graph.  \emph{Results Math.}\ 25 (1994), 198--223.  
\MR{1273111} (95e:05081).  \Zbl{802.05054}.					
\annot{} Here a periodic graph [of dimension $m$] is defined as a connected graph $\Gamma = \tilde\Psi$ where $\Psi$ is a finite $\bbZ^m$-gain graph with gains contained in $\{\mathbf 0, \mathbf b_i, \mathbf b_i - \mathbf b_j\}$.  ($\mathbf b_1, \ldots, \mathbf b_m$ are the unit basis vectors of $\bbZ^m$.)  Let us call such a $\Psi$ a small-gain base graph for $\Gamma$.  Any $\tilde\Phi$, where $\Phi$ is a finite $\bbZ^m$-gain graph, has a small-gain base graph $\Psi$; thus this definition is equivalent to that of \xref{Collatz}{1978a}{L.\ Collatz}.  
The ``index'' $I(\Gamma)$, analogous to the largest eigenvalue of a finite graph, is the spectral radius of $A(\|\Psi\|)$ (here written $A(\Gamma, N)$) for any small-gain base graph of $\Gamma$.  
The paper contains basic theory and the lower bound $L_m = \inf \{I(\Gamma) : \Gamma {\text{\rm\ is $m$-dimensional}}\}$, where $1 = L_1, \sqrt{9/2} = L_2 \leq L_3 \leq \cdots$.
\codes{\CGG(\CCov):~\CEig}%
\itemspace

\author{Jean B\'enabou}						%BENABOU
\authorx{Jean Benabou}
\authorspace
\bsgitem{1996a}{}  Some geometric aspects of the calculus of fractions.  European Colloq.\ Category Theory (Tours, 1994).  \emph{Appl.\ Categ.\ Structures} 4 (1996), 139--165.
\MR{1406095} (97g:18007).  \Zbl{874.18007}.
\annot{} Morphisms of signed graphs are employed in category-theoretic constructions.
\codes{\CSG}
\itemspace

\author{Radel Ben-Av}						%BEN-AV
\authorspace
\authref  See \hyperlink{Daniel Kandel}{D.~Kandel}.
\itemspace

\authorx{Edward A.\ Bender}						%BENDER
\author{Edward A.\ Bender \& E.\ Rodney Canfield}
\authorspace
\bsgitem{1983a}{} Enumeration of connected invariant graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 34 (1983), 268--278.  
\MR{0714450} (85b:05099).  \Zbl{532.05036}.
\annot{} \S3:  ``Self-dual signed graphs,'' gives the number of $n$-vertex graphs that are signed, vertex-signed, or both; connected or not; self-isomorphic by reversing edge and/or vertex signs or not, for all $n \leq 12$.  Some of this appeared in \xref{Harary, Palmer, Robinson, and Schwenk}{1977a}{Frank Harary, Edgar M.\ Palmer, Robert W.\ Robinson, \& Allen J.\ Schwenk}.
\par
\codes{\CSG,~\CVS:~\CEnum}%
\itemspace

\author{Riccardo Benedetti}						%BENEDETTI
\authorspace
\bsgitem{1998a}{}  A combinatorial approach to combings and framings of $3$-manifolds.  
In: A.\ Balog, G.O.H.\ Katona, A.\ Recski, and D.\ Sa'sz, eds., \emph{European Congress of Mathematics} (Budapest, 1996), Vol.\ I, pp.\ 52--63.  
Progress in Math., Vol.\ 168.  Birkh\"auser, Basel, 1998.
\MR{1645797} (2000e:57033).  \Zbl{905.57018}.
\annot{} \S8, ``Spin manifolds'', hints at a use for decorated signed graphs in the structure theory of spin $3$-manifolds.
\codes{\Csg:~\CAppl:~\CExp}%
\itemspace

\author{Aner Ben-Efraim}						%BEN-EFRAIM
\authref  See also \hyperlink{Amos Beimel}{A.~Beimel}.
\authrefspace

\authorspace
\bsgitem{2016a}{}  Secret-sharing matroids need not be algebraic.  \emph{Discrete Math.}\ 339 (2016), 2136--2145.
\MR{3500143}.  \Zbl{1337.05016}.  \arXiv{403.6363}.					
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Joseph Ben Geloun}						%BEN GELOUN
\authorspace
\authref  See \hyperlink{Remi C.\ Avohou}{R.C.~Avohou}.
\itemspace

\authorx{Felipe Ben{\'\i}tez}						%BENITEZ
\authorx{Felipe Benitez}
\author{Felipe Ben{\'\i}tez, Julio Aracena, \& Christopher Thraves Caro}
\authorx{Felipe Benitez, Julio Aracena, \& Christopher Thraves Caro}
\authorspace
\bsgitem{20xxa}{}  The sitting closer to friends than enemies problem in the circumference.  Submitted.  \emph{}\ 
\arXiv{1811.02699}.  
\annot{} \Cf\ \xref{Kermarrec and Thraves}{2011a}{Anne-Marie Kermarrec \& Christopher Thraves}, \xrefd{2014a}{Anne-Marie Kermarrec \& Christopher Thraves}.
\codes{\CSG:~\CGeom,~\CKG}%
\itemspace

\author{Samia Ben Lamine}						%BEN LAMINE
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena} and \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\authorx{Curtis Bennett}		 				%BENNETT C
\author{Curtis Bennett \& Bruce E.\ Sagan}
\authorspace
\bsgitem{1995a}{}  A generalization of semimodular supersolvable lattices.  \emph{J.\ Combin.\ Theory Ser.\ A} 72 (1995), 209--231.  
\MR{1357770} (96i:05180).  \Zbl{831.06003}.					
\annot{} To illustrate the generalization, most of the article calculates the chromatic polynomial of $\pm K_n^{(k)}$ (called $\cD\cB_{n,k}$; this has half edges at $k$ vertices), builds an ``atom decision tree'' for $k = 0$, and describes and counts the bases of $\bfF(\pm K_n^{(k)})$ (called $\cD_n$) that contain no broken circuits.
\par
\codes{\CSG:~\CMatrd,~\CInvar,~\Ccol}%
\itemspace

\authorx{M.K.\ Bennett}						 %BENNETT MK*
\author{M.K.\ Bennett, Kenneth P.\ Bogart, \& Joseph E.\ Bonin}
\authorspace
\bsgitem{1994a}{}  The geometry of Dowling lattices.  \emph{Adv.\ Math.}\ 103 (1994), 131--161.  
\MR{1265790} (95b:05050).  \Zbl{814.51003}. 					
\annot{} Drawing an analogy between Desargues' and Pappus' theorems in projective spaces and similar incidence theorems in Dowling geometries.  [The rigorous avoidance of gain graphs makes the results less obvious than they could be.]
\codes{\Cgg:~\CMatrd,~\CGeom}%
\itemspace

\author{Moussa Benoumhani}						%BENOUMHANI
\authorspace
\bsgitem{1996a}{}  On Whitney numbers of Dowling lattices.  \emph{Discrete Math.}\ 159 (1996), 13--33.  
\MR{1415279} (98a:06005).  \Zbl{861.05004}.						
\annot{} \Cf\ \xref{Dowling}{1973b}{Thomas A.\ Dowling}.  Generating functions and identities for Whitney numbers of the first and second kinds, analogous to usual treatments of Stirling numbers.
\S2, ``Whitney numbers of the second kind'':  $W_m(n,k) := W_k(Q_n(\fG)) = W_k(\bfF(\fG K_n\full))$ where $m = \#\fG$.  E.g., Thm.\ 1:  $\sum_n W_m(n,k)z^n/n! = [(e^{mz}-1)/m]^k e^z/k!$.  Thm.\ 5:  $\sum_n W_m(n,k) u^{n-k} = m^{k+1} / ([1-u]/mu)_{k+1}$.
\S3, ``Whitney numbers of the first kind'':  $w_m(n,k)$ $:= w_k(\bfF(\fG K_n^\circ))$.  E.g., Thm.\ 10: 
$\sum_n w_m(n,k)z^n/n! = (1+mz)^{-1/m} \cdot \ln^k(1+mz) / k!m^k$.  Thm.\ 12 is a reciprocity relation between $w_m(n,k)$ and $s(n,k)$.  
\S4, ``The integers maximizing $W_m(n,k)$ and $w_m(n,k)$'':  Partial, complicated results.
\dated{30 Apr 2012}
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{1997a}{}  On some numbers related to Whitney numbers of Dowling lattices.  \emph{Adv.\ Appl.\ Math.}\ 19 (1997), 106--116.  
\MR{1453407} (98f:05004).  \Zbl{876.05001}.					
\annot{} Continuation of \xrefd{1996a}{Moussa Benoumhani}.  
\S2, ``Dowling polynomials'': $D_m(n,x) := \sum_k W_m(n,k) x^k$.  Generating function, recurrence, infinite series expression.
\S3 similarly studies $F_{m,1}(x) := \sum_k k! m^k W_m(n,k) x^k$ and $F_{m,1}(x) := \sum_k k! W_m(n,k) x^k$.
\S4, ``Log-concavity of $k! W_m(n,k)$''.  Deduced from real negativity of zeros.
\dated{1 May 2012}
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{1999a}{}  Log-concavity of Whitney numbers of Dowling lattices.  \emph{Adv.\ Appl.\ Math.}\ 22 (1999), 186--189.
\MR{1659426} (2000i:05008).  \Zbl{918.05003}.						
\annot{} Logarithmic concavity of Whitney numbers of the second kind is deduced by proving that their generating polynomial has only real zeros.  [\Cf\ \xref{Stonesifer}{1975a}{J.\ Randolph Stonesifer}, \xref{D\"ur}{1986a}{Arne Dur}, and \xref{Damiani, D'Antona, and Regonati}{1994a}{E.\ Damiani, O.\ D'Antona, \& F.\ Regonati}.]
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Julien Bensmail}						%BENSMAIL
\authorspace
\bsgitem{2019a}{}  On the 2-edge-coloured chromatic number of grids.  \emph{Australasian J.\ Combin.}\ 75 (2019), no.\ 3, 365--384.
\MR{4024807}.  \Zbl{1429.05059}.  \HAL{hal-02264958}.  
\annot{} The max homomorphic chromatic number of grids $\in \{8,9,10,11\}$, improving $\leq 12$ from \xref{Nesetril--Raspaud}{2000a}{Jaroslav Nesetril \& Andre Raspaud}.  [Improved to $\le 9$ in \xref{Dybizba\'nski}{20xxa}{Janusz Dybizbanski}.]  
Dictionary:  ``2-edge-coloured graph'' = signed graph, ``2-edge-coloured chromatic number'' = min order of homomorphism target.
\dated{3 Sept 2020} 
\codes{\CSG:~\CHom}%
\itemspace

\author{Julien Bensmail, Sandip Das, Soumen Nandi, Th\'eo Pierron, Sagnik Sen, \& \'Eric Sopena}
\authorx{Julien Bensmail, Sandip Das, Soumen Nandi, Theo Pierron, Sagnik Sen, \& Eric Sopena}
\authorspace
\bsgitem{2022a}{}  On the signed chromatic number of some classes of graphs.  \emph{Discrete Math.}\ 345 (2022), no.\ 2, art.\ 112664, 20 pp.
\MR{4339573}.  
\arXiv{2009.12059}.  \HAL{hal-02947399}.						
\annot{} 
Dictionary:  ``signed chromatic number'' = homomorphic chromatic number (not chromatic number).
\codes{\CSG:~\CHom:~\CInvar}%
\itemspace

\author{Julien Bensmail, Fran\c{c}ois Dross, Nacim Oijid, \& \'Eric Sopena}
\authorx{Julien Bensmail, Francois Dross, Nacim Oijid, \& Eric Sopena}
\authorspace
\bsgitem{2022a}{}  Generalising the achromatic number to Zaslavsky's colourings of signed graphs.  \emph{Theor.\ Computer Sci.}\ 923 (2022), 196--221.
\MR{4436569}.  
\arXiv{2109.13627}.  \HAL{hal-0335582}.						
\codes{\CSG:~\CCol}%
\itemspace

\author{Julien Bensmail, Soumen Nandi, Mithun Roy, \& Sagnik Sen}
\authorspace
\bsgitem{2020a}{}  Classification of edge-critical underlying absolute planar cliques for signed \linebreak graphs.  \emph{Australasian J.\ Combin.}\ 77 (2020), no.\ 1, 117--135.
\MR{4100300}.  \Zbl{1444.05062}.  \HAL{hal-01919007}.  
\codes{\CSG:~\CHom}%
\itemspace

\bsgitem{20xxa}{}  On homomorphisms of planar signed graphs and absolute cliques.  Submitted.  \emph{}\ 
\HAL{hal-01919007}.						
\codes{\CSG:~\CHom}%
\itemspace

\author{Josh Bentley}						%BENTLEY
\authorspace
\authref  See \hyperlink{Aimei Yang}{A.-M.~Yang}.
\itemspace

\author{C.\ Benzaken} 						%BENZAKEN
\authorspace
\authref  See also \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}.
\authrefspace

\author{C.\ Benzaken, S.C.\ Boyd, P.L.\ Hammer, \& B.\ Simeone} 
\authorspace
\bsgitem{1983a}{} Adjoints of pure bidirected graphs.  Proc.\ Fourteenth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1983).  \emph{Congressus Numer.}\ 39 (1983), 123--144.  
\MR{0734537} (85e:05077).  \Zbl{537.05024}.
\par
\codes{\Csg:~\COri:~\CLG}%
\itemspace

\author{Cl.\ Benzaken, P.L.\ Hammer, \& B.\ Simeone} 
\authorspace
\bsgitem{1980a}{}  Some remarks on conflict graphs of quadratic pseudo-boolean functions.  
In:  L.\ Collatz, G.\ Meinardus, and W.\ Wetterling, eds., \emph{Konstruktive Methoden der finiten nichtlinearen Optimierung} (Tagung, Oberwolfach, 1980), pp.\ 9--30.  Int.\ Ser.\ Num.\ Math., 55.  Birkh\"auser, Basel, 1980.  
\MR{0626538} (83e:90096).  \Zbl{455.90063}.
\codes{\Cpar:~\Cfr}\codesctd{\Csg:~\COri:~\CLG}%
\itemspace

\author{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra} 
\authorspace
\bsgitem{1981a}{}  Threshold signed graphs.  Res.\ Rep.\ No.\ 237, IMAG, Universit\'e Grenoble, Grenoble, 1981.
\annot{} See \xrefd{1985a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}.
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{1985a}{}  Threshold characterization of graphs with Dilworth number two.  \emph{J.\ Graph Theory} 9 (1985), no.\ 2, 245--267.
\MR{0797513} (87d:05135).  \Zbl{583.05048}.					
\annot{} [They are identical to ``threshold signed graphs'', \xref{Calamoneri--Monti--Petreschi}{2018a}{Tiziana Calamoneri, Angelo Monti, \& Rossella Petreschi}.]  $\Gamma$ is a threshold signed graph if $\exists\ a: V \to \bbR$, $S, T \in \bbR$, such that $v_iv_j \in E$ iff $|a_i+a_j|\geq S$ or $|a_i-a_j|\geq T$.  
[Proposed signed graph $\Sigma$:  $-v_iv_j \in E$ iff $|a_i+a_j|\geq S$, $+v_iv_j \in E$ iff $|a_i-a_j|\geq T$.  Then $\Gamma$ = simplification of $|\Sigma|$.  \Question.  Is $\Sigma$ interesting?]
\dated{16 Jan 2012} 
\codes{\CVS,~\Csg}%
\itemspace

\author{Michele Benzi}						%BENZI
\authorspace
\authref  See \hyperlink{Ernesto Estrada}{E.~Estrada}.
\itemspace

\authorx{Claude Berge}						%BERGE*
\author{C.\ Berge \& J.-L.\ Fouquet}
\authorspace
\bsgitem{1997a}{}  On the optimal transversals of the odd cycles.  \emph{Discrete Math.}\ 169 (1997), 169--175.
\MR{1449714} (98c:05094).  \Zbl{883.05088}.					
\annot{} All-negative signed graphs in which the vertex frustration number equals the negative-circle vertex-packing number.  This is called the ``K\"onig property'' [since it is a vertex K\"onig-type property for negative circles].
Example: the line graphs of cubic bipartite graphs.  [\Problems.  Investigate for arbitrary signed and biased graphs.]	
\codes{\Cpar:~\CFr,~\CCircles}
\itemspace

\author{Claude Berge \& A.\ Ghouila-Houri}
\authorspace
\bsgitem{1962a}{}  \emph{Programmes, jeu et reseaux de transport}.  Dunod, Paris, 1962.  
\MR{0192912} (33 \#1137).  \Zbl{111.17302}.
\annot{} $2^e$ partie, Ch.\ IV, \S2:  ``Les reseaux de transport avec multiplicateurs.''  Pp.\ 223--229. 
\codes{\CGN:~\Cinc}%
\itemspace

\bsgitem{1965a}{}  \emph{Programming, Games and Transportation Networks}.  Methuen, London; Wiley, New York, 1965.  
\MR{0198964} (33 \#7114).  
\annot{} English edition of \xrefd{1962a}{Claude Berge \& A.\ Ghouila-Houri}.
\annotctd{} Part II, 10.2:  ``The transportation network with multipliers.'' Pp.\ 221--227.
\codes{\CGN:~\Cinc}%
\itemspace

\bsgitem{1967a}{}  \emph{Programme, Spiele, Transportnetze.}  B.G.\ Teubner Verlagsgesellschaft, Leipzig, 1967, 1969.  
\MR{0218106} (36 \#1195).  \Zbl{183.23905}, \Zbl{194.19803}.
\annot{} German edition(s) of \xrefd{1962a}{Claude Berge \& A.\ Ghouila-Houri}.
\codes{\CGN:~\Cinc}%
\itemspace

\author{Claude Berge \& Bruce Reed}
\authorspace
\bsgitem{1999a}{}  Edge-disjoint odd cycles in graphs with small chromatic number.  
Symposium \`a la M\'emoire de Fran\c{c}ois Jaeger (Grenoble, 1998).  \emph{Ann.\ Inst.\ Fourier (Grenoble)} 49 (1999), 783--786.
\MR{1703423} (2000f:05051).  \Zbl{923.05034}.					
\annot{} If $-\Gamma$ is an all-negative signed graph in which the frustration index equals the negative-circle edge-packing number for every subgraph, then $\chi(\Gamma) \leq 3$.
[\Problem\ 1.  Is it natural to state this bound in terms of the chromatic number of $-\Gamma$?  \Problem\ 2.  Generalize to arbitrary signed graphs.]	
\codes{\Cpar:~\CFr:~\CCircles}
\itemspace

\bsgitem{2000a}{}  Optimal packings of edge-disjoint odd cycles.  \emph{Discrete Math.}\ 211 (2000), 197--202.
\MR{1735345} (2000h:05161).  \Zbl{945.05048}.					
\annot{} An upper bound on the frustration index in terms of the negative-circle edge-packing number.
	
\codes{\Cpar:~\CFr:~\CCircles}
\itemspace

\authorx{Joseph Berger}	 					%BERGER
\author{Joseph Berger, Bernard P.\ Cohen, J.\ Laurie Snell, \& Morris Zelditch, Jr.}
\authorspace
\bsgitem{1962a}{}  \emph{Types of Formalization in Small Group Research}.  Houghton Mifflin, Boston, 1962.  
\annot{} See Ch.\ 2:  ``Explicational models.'' 
\codes{\CPsS}\codesctd{\CSG:~\CBal}\codesctd{\CRef}%
\itemspace

\author{Nantel Bergeron}						%BERGERON
\authorspace
\bsgitem{1991a}{}  A hyperoctahedral analogue of the free Lie algebra.  \emph{J.\ Combin.\ Theory Ser.\ A} 58 (1991), 256--278.
\MR{1129117} (93g:20015).  \Zbl{0759.17003}.  
\par
\codes{\Csg:~\CAlgeb,~\CGeom}%
\itemspace

\author{A.\ Nihat Berker}						%BERKER
\authorspace
\authref  See \hyperlink{Daniel Blankschtein}{D.~Blankschtein}.
\itemspace

\authorx{Abraham Berman}			 			%BERMAN
\author{Abraham Berman \& B.\ David Saunders}
\authorspace
\bsgitem{1981a}{} Matrices with zero line sums and maximal rank.  \emph{Linear Algebra Appl.}\ 40 (1981), 229--235.  
\MR{0629620} (82i:15029).  \Zbl{478.15013}.
\codes{\CQM,~\Csd:~\Cori}%
\itemspace


\authorx{Olivier Bernardi}						%BERNARDI
\author{Olivier Bernardi \& Guillaume Chapuy}
\authorspace
\bsgitem{2010a}{}  Counting unicellular maps on non-orientable surfaces.  In:  \emph{22nd International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC 2010), pp.\ 155--166.  Discrete Math.\ Theor.\ Computer Sci.\ Proc., AN. The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS), Nancy, 2010.
\MR{2673832} (2012m:05168).  \Zbl{1373.05042}.  
\codes{\Csg:~\CTop}%
\itemspace

\bsgitem{2011a}{}  Counting unicellular maps on non-orientable surfaces.  \emph{Adv.\ Appl.\ Math.}\ 47 (2011), 259--275.
\MR{2803802} (2012i:05012).  \Zbl{1234.05019}.  
\annot{} Counting one-face cellular orientation embeddings of all graphs of order $n$ in $U_h$ (sphere with $h$ crosscaps).  Exact formulas if all degrees are 1 and 3 (Cors.\ 8, 9); asymptotic (as $n\to\infty$) in general (Thm.\ 11).  
Dictionary:  ``twist'' = negative edge, ``flip'' of vertex = switching.
[\Cf\ \xref{\v{S}ir\'a\v{n} and \v{S}koviera}{1991a}{Jozef Siran \& Martin Skoviera}.]
\dated{3 Nov 2017} 
\codes{\Csg:~\CTop}%
\itemspace

\author{Gilles Bernot}						%BERNOT
\authorspace
\authref  See \hyperlink{Adrien Richard}{A.~Richard}.
\itemspace

\author{Daniel Irving Bernstein}						%BERNSTEIN
\authorspace
\bsgitem{2017a}{}  Completion of tree metrics and rank 2 matrices.  \emph{Linear Algebra Appl.}\ 533 (2017), 1--13.
\MR{3695897}.  \Zbl{06778046}.  \arXiv{1612.06797}.				
\codes{\Cgg}%
\itemspace

\authorx{Pascal Berthom\'e}						%BERTHOME
\authorx{Pascal Berthome}
\author{Pascal Berthom\'e, Raul Cordovil, David Forge, V\'eronique Ventos, \& Thomas Zaslavsky}
\authorx{Pascal Berthome, Raul Cordovil, David Forge, Veronique Ventos, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2009a}{}  An elementary chromatic reduction for gain graphs and special hyperplane arrangements.  \emph{Electronic J.\ Combin.}\ 16 (2009), no.\ 1, art.\ R121, 31 pp.
\MR{2546324} (2010k:05253).  \Zbl{1188.05076}.  \arXiv{1001.4216}.  \HAL{inria-00491020}.	
\annot{} Calculating chromatic functions (which satisfy deletion-contraction for zero-gain edges and equal $0$ if there is a balanced loop) by eliminating or adding identity-gain edges.  Application to integral, modular, and zero-free chromatic polynomials of the Shi, Linial, Catalan, and intermediate hyperplane arrangements via their gain graphs [\cf\ \xref{Stanley}{1999a}{Richard P.\ Stanley}].  [See \xref{Ardila}{2007a}{Federico Ardila} for some of the corresponding coboundary and Tutte polynomials.]
\codes{\CGG:~\CInvar,~\CGeom}%
\itemspace

\authorx{Evgeny Bespalov}						%BESPALOV
\author{E.A.\ Bespalov}
\authorspace
\bsgitem{2014a}{}  On switching nonseparable graphs with switching separable subgraphs.  \emph{Siberian Electronic Math.\ Rep.}\ 11 (2014), 988--998.
\MR{3488478}.  \Zbl{1326.05118}.  
\annot{} \Cf\ \xref{D.S.\ Krotov}{2010a}{D.S.\ Krotov}.  Classifies $\Gamma$ for which deleting single vertices is insufficient.
\dated{31 Jul 2018} 
\codes{\Ctg:~\CSw:~\CStr}%
\itemspace

\authorx{Evgeny Bespalov \& Denis Krotov}
\author{E.A.\ Bespalov \& D.S.\ Krotov}
\authorspace
\bsgitem{2016a}{}  On a test for the switching separability of graphs modulo $q$.  (In Russian.)
\emph{Sibirsk.\ Mat.\ Zh.}\ 57 (2016), no.\ 1, 10--24.
\bsgitem{2016b}{}  On one test for the switching separability of graphs modulo $q$.  (English trans.)  \emph{Siberian Math.\ J.}\ 57 (2016), no.\ 1, 7--17.
\MR{3499848}.  \Zbl{1338.05211}.  \arXiv{1412.2947}.				
\annot{} For $g: E(K_n) \to \bbF_q^+$, let $E(\Gamma_g):=E(K_n)\setminus g\inv(0)$.  For $f:V\to\bbF_q$, let $\delta f(vw):=f(v)+f(w)$, i.e., $\delta f:=\Eta(-\Gamma)f$.  A ``switching'' of $\Gamma_g$ (a nonstandard switching) = any $\Gamma_{g+\delta f}$.  ``Separable'' = nontrivially disconnected; \cf\ \xref{D.S.\ Krotov}{2010a}{D.S.\ Krotov}.  ``Switching separable'' = switches to a separable graph.  Thm.\ 1 classifies $\Gamma$ that are switching inseparable but with all single-vertex deletions switching separable: $\exists$ iff $q$ even and odd $n>4$; all are determined in Prop.\ 3.
\dated{31 Jul 2018} 
\codes{\Cpar:~\CSw(\CGen),~\Cincid:~\CStr}%
\itemspace

\author{Ouahiba Bessouf}						%BESSOUF
\authorspace
\bsgitem{1999a}{}  \emph{Th\'eor\`eme de Menger dans les graphes biorient\'es}.  Th\`ese de Magister, Fac.\ des Math\'ematiques, Universit\'e des Sciences et de la Technologie Houari Boumediene, Alger, 1999.
\codes{\CSG:~\COri,~\CStr}%
\itemspace

\author{Ouahiba Bessouf \& Abdelkader Khelladi}
\authorspace
\bsgitem{2018a}{}  New concept of connection in bidirected graphs.  \emph{RAIRO--Oper.\ Res.}\ 52 (2018), 351--357.
\MR{3817469}.  \Zbl{1398.05119}.  
\codes{\CSG:~\COri:~\CStr}%
\itemspace

\author{Ouahiba Bessouf, Abdelkader Khelladi, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2019a}{}  Transitive closure and transitive reduction in bidirected graphs.  \emph{Czechoslovak Math.\ J.}\ 69 (2019), no.\ 2, 295--315.
\MR{3959945}.  
\arXiv{1610.00179}.							
\annot{} Fundamentals, generalizing from digraphs.  Complications arise.  Matroidal aspects: quasibalance and closure.
\dated{4 Jun 2019} 
\par
\codes{\CSG:~\COri:~\CStr}%
\itemspace

\bsgitem{20xxb}{}  New concept of connection in signed graphs.  Under revision.  
\arXiv{1708.01689}.							
\codes{\CSG:~\CStr}%
\itemspace

\author{St\'ephane Bessy}						%BESSY
\authorspace
\authref  See \hyperlink{Jorgen Bang-Jensen}{J.~Bang-Jensen}.
\itemspace

\author{Kenchappa S.\ Betageri}					%BETAGERI
\authorspace
\authref  See also \hyperlink{V.S.\ Shigehalli}{V.S.~Shigehalli}.
\authrefspace

\bsgitem{2016a}{}  The reduced color energy of graphs.  \emph{J.\ Computer Math.\ Sci.}\ 7 (2016), 13--20.  
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CAdj:~\CEig}%
\itemspace

\author{K.S.\ Betageri \& G.H.\ Mokashi}
\authorspace
\bsgitem{2016a}{}  A note on reduced color energy of graphs.  \emph{J.\ Computer Math.\ Sci.}\ 7 (2016), no.\ 4, 203--212.  
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CAdj:~\CEig}%
\itemspace

\author{Nadja Betzler}						%BETZLER
\authorspace
\authref  See \hyperlink{Falk Huffner}{F.~H\"uffner}.
\itemspace

\author{D.\ Bharathi}							%BHARATHI
\authorspace
\authref  See \hyperlink{Shaik Sajana}{S.~Sajana}.
\itemspace

\author{Mushtaq A.\ Bhat}						%BHAT MA
\authorspace
\authref  See also \hyperlink{S.\ Pirzada}{S.~Pirzada} and \hyperlink{Tahir Shamsher}{T.~Shamsher}.
\authrefspace

\bsgitem{2015a}{}  \emph{Energy of Graphs and Digraphs}.  Ph.D.\ thesis, Univ.\ of Kashmir, Srinagar, 2015.
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2017a}{}  Energy of weighted digraphs.  \emph{Discrete Appl.\ Math.}\ 223 (2017), 1--14. 
\MR{3627295}.  \Zbl{1471.05039}.  
\codes{\CSG,~\CWG:~\CEig}%
\itemspace

\author{Mushtaq A.\ Bhat \& S.\ Pirzada}
\authorspace
\bsgitem{2015a}{}  On equienergetic signed graphs.  \emph{Discrete Appl.\ Math.}\ 189 (2015), 1--7.
\MR{3348023}.  \Zbl{1316.05055}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2016a}{}  Spectra and energy of bipartite signed digraphs.  \emph{Linear Multilinear Algebra} 64 (2016), no.\ 9, 1863--1877.
\MR{3509506}.  \Zbl{1341.05150}.  \arXiv{1501.00572}.				
\par
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{2017a}{}  Unicyclic signed graphs with minimal energy.  \emph{Discrete Appl.\ Math.}\ 226 (2017), 32--39.
\MR{3659378}.  \Zbl{1365.05118}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Mushtaq A.\ Bhat, U.\ Samee, \& S.\ Pirzada}	
\authorspace
\bsgitem{2018a}{}  Bicyclic signed graphs with minimal and second minimal energy.  \emph{Linear Algebra Appl.}\ 551 (2018), 18--35.
\MR{3797933}.  \Zbl{1416.05130}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Pradeep G.\ Bhat}						%BHAT PG
\author{Pradeep G.\ Bhat \& Sabitha D'Souza}
\authorx{P.G.\ Bhat \& S.\ D'Souza}
\authorspace
\bsgitem{2013a}{}  Energy of binary labeled graphs.  \emph{Trans.\ Combin.}\ 2 (2013), no.\ 3, 53--67.
\MR{3509506}.  \Zbl{1302.05048}.  
\annot{} The authors introduce a ``label [adjacency] matrix'' $A_l(\Gamma,\zeta)$ (see below) where $\zeta: V\to\{0,1\}$ and $a,b,c$ are distinct real numbers.  They investigate the characteristic polynomial $\phi(\lambda)$ and energy $E_l := \sum |\lambda_i|$ where $\lambda_i$ are the eigenvalues.  Thms.\ 2.1, 2.2:  Top four coefficients of $\phi$.  Thms.\ 2.3, 2.4:  Properties of $E_l$.  Thm.\ 2.5:  Eigenvalue upper bound.  
\S3, ``Label energies of some families of graphs'':  $K_n$, $K_{1,n}$, $K_{r,s}$, double star.  Thm.\ 3.4:  If $K_n$ has $m$ 0-labelled vertices, the eigenvalues are $-a$ (multiplicity $m-1$), $-b$ (multiplicity $n-m-1$), and the roots of a quadratic.
\annotctd{} [\Problem.  Define a ``generic adjacency matrix'' $A_l(\Beta)$ of a bidirected simple graph $\Beta$ by $a_{l;ij} = a$ if $v_iv_j$ is extraverted, $b$ if $v_iv_j$ is introverted, $c$ if $v_iv_j$ is positive, and $0$ otherwise, where $a,b,c$ are generic numbers, indeterminates, etc.  (For a non-simple graph, sum the values of edges $v_iv_j$.)  Given a vertex-signed graph $(\Gamma,\zeta)$, bidirect it by $\tau(v_i,v_iv_j):=\zeta(v_i)$; that is, every vertex is a source (if $\zeta(v_i)=-$) or sink $\zeta(v_i)=+$).  Note that $\{0,1\}\cong\{+,-\}$.  Does this matrix of a bidirected graph have interesting properties?]
\dated{3 Oct 2013} 
\codes{\CVS:~\CEig}\codesctd{\Csg:~\Cori:~\CEig}%
\itemspace

\bsgitem{2015a}{}  Color Laplacian energy of a graph.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 18 (2015), no.\ 3, 321---330.
\MR{3410554}.  \Zbl{1337.05036}.  
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\bsgitem{2017a}{}  Color signless Laplacian energy of graphs.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 14 (2017), 142--148.
\MR{3701601}.  \Zbl{1372.05126}.  
\annot{} Laplacian energy of $-\Sigma$, \cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\dated{22 Dec 2018} 
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\authorx{R.N.\ Bhatt}							%BHATT
\author{R.N.\ Bhatt \& A.P.\ Young}
\authorspace
\bsgitem{1985a}{}  Search for a transition in the three-dimensional $\pm J$ Ising spin-glass.  \emph{Phys.\ Rev.\ Lett.}\ 54 (1985), no.\ 9, 924--927.
\codes{\CPhys,~\CSG:~\CFr,~\CState}%
\itemspace

\bsgitem{1988a}{}  Numerical studies of Ising spin glasses in two, three, and four dimensions.  \emph{Phys.\ Rev.\ Lett.}\ 37 (1988), no.\ 10, 5606--5614.
\codes{\CPhys,~\CSG:~\CFr,~\CState}%
\itemspace

\author{Bikash Bhattacharjya}						%BHATTACHARJYA
\authorspace
\authref  See \hyperlink{Deepak Sehrawat}{D.~Sehrawat}.
\itemspace

\authorx{Amitava Bhattacharya}					%BHATTACHARYA
\author{Amitava Bhattacharya, Uri N.\ Peled, \& Murali K.\ Srinivasan}
\authorspace
\bsgitem{2007a}{}  Cones of closed alternating walks and trails.  \emph{Linear Algebra Appl.}\ 423 (2007), no.\ 2-3, 351--365. 
\MR{2312413} (2008j:05132).  \Zbl{1115.05067}.
\annot{} The cone of Eulerian real-weighted subgraphs of a bidirected all-negative signed graph.
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\bsgitem{2009a}{}  The cone of balanced subgraphs.  \emph{Linear Algebra Appl.}\ 431 (2009), no.\ 1-2, 266--273.
\MR{2522574} (2010h:05226).  \Zbl{1169.05372}.					
\annot{} A ``balanced subgraph'' is an edge 2-colored graph where the red and blue degrees are equal at each vertex.  [I.e., a signed graph whose net degree $d^\pm(v)=0, \forall v$.  Equivalent to an all-negative signed graph, oriented so that every vertex has equal in- and out-degree, which is the all-negative case of an Eulerian bidirected graph.  P.D.\ Seymour, Sums of circuits, in \emph{Graph Theory and Related Topics}, pp.\ 341--355, Academic Press, New York, 1979, treated the all-positive case.]  
The problem is to describe the facets of the convex cone generated by Eulerian subgraphs of an all-negative bidirected graph.  [\Problem.  Solve for an arbitrary bidirected graph.]
\codes{\Csg:~\CPar,~\COri:~\CGeom}%
\itemspace

\authorx{Anindya Bhattacharya}					%BHATTACHARYA
\author{Anindya Bhattacharya \& Rajat K.\ De}
\authorspace
\bsgitem{2008a}{}  Divisive Correlation Clustering Algorithm (DCCA) for grouping of genes: detecting varying patterns in expression profiles.  \emph{Bioinformatics} 24 (2008), no.\ 11, 1359--1366.
\codes{\Csg:~\CClu:~\CAlgor,~\CBiol}%
\itemspace

\authorspace
\bsgitem{2010a}{}  Average correlation clustering algorithm (ACCA) for grouping of co-regulated genes with similar pattern of variation in their expression values.  \emph{J.\ Biomedical Informatics} 43 (2010), 560--568.
\codes{\Csg:~\CClu:~\CAlgor,~\CBiol}%
\itemspace

\author{Gora Bhaumik} 						%BHAUMIK
\authorspace
\authref  See \hyperlink{Paul A.\ Jensen}{P.A.~Jensen}.
\itemspace

\author{V.N.\ Bhave}							%BHAVE
\authorspace
\authref  See \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\authorx{Mani Bhushan}						%BHUSHAN
\author{Mani Bhushan \& Raghunathan Rengaswamy}
\authorspace
\bsgitem{2000a}{}  Design of sensor network based on the signed directed graph of the process for efficient fault diagnosis.  \emph{Ind.\ Eng.\ Chem.\ Res.}\ 39 (2000), 999--1019.
\annot{} Another application to fault diagnosis in chemical engineering, this one to location of sensors.
\codes{\CSD:~\CAppl}%
\itemspace

\author{Ginestra Bianconi}						%BIANCONI
\authorspace
\authref  See \hyperlink{Valerio Ciotti}{V.~Ciotti}.
\itemspace

\author{Christin Bibby}						%BIBBY
\authorspace
\bsgitem{2018a}{}  Representation stability for the cohomology of arrangements associated to root systems.  \emph{J.\ Algebraic Combin.}\ 48 (2018), 51--75.
\MR{3836246}.  \Zbl{1401.52033}.  \arXiv{1603.08131}.				
\annot{} Precursor of the marked Dowling posets of \xref{Bibby and Gadish}{2018a}{Christin Bibby \& Nir Gadish}.
\dated{1 Feb 2019} 
\codes{\Cgg:~\Cmatrd(\CGen)}%
\itemspace

\author{Christin Bibby \& Nir Gadish}
\authorspace
\bsgitem{2018a}{}  Combinatorics of orbit configuration spaces.  \emph{S\'em.\ Lotharingien Combin.}\ 80B (2018), art.\ 72, 11 pp. 
\MR{3940647}.  \Zbl{1444.55008}.  \arXiv{1804.06863}.				
\annot{} Introduces marked Dowling (``$S$-Dowling'') posets.  
	
The posets can be viewed as consisting of decorated flats of the frame matroid $\bfF(\fG K_n\full)$.  
Dowling's $Q_n(\fG) = \Lat \bfF(\fG K_n\full)$ is viewed as $\sF^\textb$ := \{closed, balanced subgraphs $B$ of $\bfF(\fG K_n\full)$, suitably ordered.  A marking is $m: V(B)^c \to S$.  The marked Dowling poset is $\{(N,m): N \in \sF^\textb\}$, suitably ordered.  
[More theory of marked Dowling posets in \xref{Delucchi--Girard--Paolini}{2019a}{Emanuele Delucchi, Noriane Girard, \& Giovanni Paolini}, \xref{Paolini}{2020a}{Giovanni Paolini}.]
\dated{1 Feb 2019} 
\codes{\Cgg:~\CMatrd(\CGen)}%
\itemspace

\authorx{I.\ Bieche}							%BIECHE
\author{I.\ Bieche, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}
\authorspace
\bsgitem{1980a}{} On the ground states of the frustration model of a spin glass by a matching method of graph theory.  
\emph{J.\ Phys.\ A:  Math.\ Gen.}\ 13 (1980), 2553--2576.  
\MR{0582907} (81g:82037).  
\annot{} The frustration index and ground states of a planar square grid graph can be found by matching in the dual graph.  [Solved for all planar graphs by \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai}, \xref{Barahona}{1982b}{Francisco Barahona}.]
\dated{29 Aug 2012}
\codes{\CSG:~\CPhys,~\CFr,~\CState(\Cfr),~\CAlgor}%
\itemspace

\author{Dan Bienstock}						%BIENSTOCK
\authorspace
\bsgitem{1991a}{}  On the complexity of testing for odd holes and induced odd paths.  \emph{Discrete Math.}\ 90 (1991), 85--92.  
\MR{1115733} (92m:68040a).  \Zbl{753.05046}.  
Corrigendum.  \emph{Ibid.}\ 102 (1992), 109.  
\MR{1168141} (92m:68040b).  \Zbl{760.05080}.						
\annot{} Given a graph.  Problem 1: Is there an odd hole on a particular vertex?  Problem 2:  Is there an odd induced path joining two specified vertices?  Problem 3:  Is every pair of vertices joined by an odd-length induced path?  All three problems are NP-complete.  
[Obviously, one can replace the graph by a signed graph and ``odd length'' by ``negative'' and the problems remain NP-complete.]
\codes{\Csg:~\CPar:~\CCircles,~\CPaths:~\CAlgor}%
\itemspace

\author{Norman Biggs} 						%BIGGS
\authorspace
\bsgitem{1974a}{} \emph{Algebraic Graph Theory}.  Cambridge Math.\ Tracts, No.\ 67.  Cambridge Univ.\ Press, London, 1974.  
\MR{0347649} (50 \#151).  \Zbl{284.05101}.					
\annot{} Ch.\ 19:  ``The covering graph construction.''  The covering graphs of gain graphs, with emphasis on automorphisms.  
Let $\Phi := (\Gamma,\phi)$ with gain group $\bbZ_2 E$ and $\phi(e) = e$.  Thm.\ 19.5:  If $\Gamma$ is $t$-transitive ($t\geq1$) [and connected], then $\tilde\Phi$ is vertex transitive [actually, $t$-transitive] and has $n-c(\Gamma)$ components (all isomorphic).  
[The number of components and the isomorphism of components of $\tilde\Phi$ require only connectedness of $\Phi$, because $\Aut\tilde\Phi$ acts transitively on each vertex fiber.]  
19A:  ``Double coverings.''  The signed covering graph of $-\Gamma$.  19B:  ``The Desargues graph.''  With $P :=$ Petersen graph, $\widetilde{-P}$ is the Desargues graph.
\dated{11 July 2009}
\annotctd{} [\xref{Tutte}{1967a}{W.T.\ Tutte} implicitly develops the double covering of an oriented $\Sigma$; it is a self-converse orientation of $\tilde\Sigma$.]
\codes{\CSG,~\CGG:~\CCov,~\CAut,~\Cbal}%
\itemspace

\bsgitem{1993a}{}  \emph{Algebraic Graph Theory}.  Second ed.  Cambridge Math.\ Library.  Cambridge Univ.\ Press, Cambridge, Eng., 1993.  
\MR{1271140} (95h:05105).  \Zbl{797.05032}.					
\annot{} As in \xrefd{1974a}{Norman Biggs}, but 19A, 19B have become Additional Results 19a, 19b.
\par
\codes{\CSG,~\CGG:~\CCov,~\CAut,~\Cbal}%
\itemspace

\bsgitem{1997a}{}  International finance.  
In: Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Graph Connections: Relationships between Graph Theory and other Areas of Mathematics}, Ch.\ 17, pp.\ 261--279.  The Clarendon Press, Oxford, 1997.
\MR{1634542} (99a:05001) (book).  \Zbl{876.90014}.
\annot{} A model of currency exchange rates in which no cyclic arbitrage is possible, hence the rates are given by a potential function.  [That is, the exchange-rate gain graph is balanced, with the natural consequences.]  Assuming cash exchange without accumulation in any currency, exchange rates are determined.  [\Cf\ \xref{Ellerman}{1984a}{David P.\ Ellerman}.]
\par
\codes{\CGG,~\Cgn:~\CBal:~\CExp}%
\itemspace

\author{Biju K}							%BIJU
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\authorx{Yonatan Bilu}						%BILU
\author{Yonatan Bilu \& Nathan Linial}
\authorspace
\bsgitem{2004a}{}  Ramanujan signing of regular graphs.  \emph{Combin.\ Probab.\ Comput.}\ 13 (2004), no.\ 6, 911--912.
\Zbl{1060.05040}.  
\annot{} \Conjecture\ 2 (based on \xrefd{2006a}{Yonatan Bilu \& Nathan Linial}).  Every $d$-regular Ramanujan graph can be signed so it has spectral radius $\leq 2\sqrt{d-1}$.  \Conjecture\ 3.  The same for every $d$-regular graph.  
Dictionary:  ``2-lift'' = signed covering graph.  
\dated{2 Mar 2011} 
\codes{\CSG:~\CEig,~\CCov}%
\itemspace

\bsgitem{2006a}{}  Lifts, discrepancy and nearly optimal spectral gap.  \emph{Combinatorica} 26 (2006), no.\ 5, 495--519.
\MR{2279667} (2008a:05160).  \Zbl{1121.05054}.					
\annot{} Reproves the eigenvalue theorem of \xref{Fowler}{2002a}{Patrick W.\ Fowler}.
\dated{2 Mar 2011, 13 Jan 2015} 
\codes{\CSG:~\CEig,~\CCov}%
\itemspace

\authorx{K.\ Binder}							%BINDER
\author{K.\ Binder \& A.P.\ Young}
\authorspace
\bsgitem{1986a}{}  Spin glasses: Experimental facts, theoretical concepts, and open questions.  \emph{Rev.\ Modern Phys.}\ 58 (1986), no.\ 4, 801--976.
\annot{} \S~III.F.2, ``Frustration and gauge invariance'':  A valuable summary of the state of knowledge and speculation at the time.  Signed graphs with spin set $\signnos$ (Ising spins) and $U(1)$ (``$XY$ spins'' = complex units).  Frustration is treated via girth circles (``plaquettes'') in lattice graphs, where the girth is 3 or 4 (triangular or square planar lattice).  Analytic solutions being too difficult, results are numerical, qualitative, or for ``simpler limiting cases''.  $XY$ spins show quantization (\cf\ \xref{Villain}{1977b}{Jacques Villain}).  For 3-dimensional lattices, plaquette duality leads to vector gains in a dual lattice, thence to closed paths of frustrated plaquettes.
\annotctd{} In Ch.\ IV, ``Mean-field theory'':  Complete-graph (``infinite range'') models.  \S~IV.A, ``Sherrington-Kirkpatrick model and replica-symmetric solutions'':  Ising models ($\fG=\signnos$).  \S~IV.H, ``Non-Ising models'':  Weighted edge signs are random variables.  Spins may be normalized vectors (\S1, ``Isotropic vector spin glasses in zero field'') or other.  \S3, ``Other models'':  ``$p$-spin couplings'' = $p$-uniform complete hypergraphs.  Energy valleys and their shapes.  Potts models (signed graphs, spins are multivalued).
\annotctd{} Dictionary:  ``site'' = vertex, ``bond'' = edge, ``state'' = function $s: V \to \fG$, ``spin'' = value $s(v)$, ``ferromagnetic'' = positive, ``antiferromagnetic'' = negative, ``quenched variable'' = constant (instead of random variable), ``gauge group'' = gain group, ``gauge transformation'' = switching, ``ground state'' = state minimizing $\#(E \setm E^{1_\fG}(\Phi^s))$.
\dated{17 Aug 2012} 
\codes{\CPhys:~\Csg,~\Cgg:~\CFr,~\CState(\Cfr),~\CSw,~\CExp,~\CRef}%
\itemspace

\authorx{B.D.\ Bingham}						%BINGHAM
\author{B.D.\ Bingham, D.D.\ Olesky, P.\ van den Driessche}
\authorspace
\bsgitem{2007a}{}  Potentially nilpotent and spectrally arbitrary even cycle sign patterns.  \emph{Linear Algebra Appl.}\ 421 (2007), 24--44.
\MR{2290683} (2007i:05112).  \Zbl{1112.05062}.  
\annot{} ``Even cycle pattern'' means the signed digraph has negative cycles of every even length (remark before Thm.\ 4.1).
\dated{13 Dec 2020} 
\codes{\Csd:~\CQM}%
\itemspace

\author{Robert E.\ Bixby} 						%BIXBY
\authorspace
\bsgitem{1981a}{} Hidden structure in linear programs.  
In:  Harvey J.\ Greenberg and John S.\ Maybee, eds., \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ Sympos., Boulder, Col., 1980), pp.\ 327--360; discussion, pp.\ 397--404.  Academic Press, New York, 1981.  
\MR{0617930} (82g:00016) (book).  \Zbl{495.93001} (book).
\codes{\CGN}%
\itemspace

\authorx{T\"urker B{\i}y{\i}ko{\u g}lu}					%BIYIKOGLU
\authorx{Turker Biyikoglu}
\author{T\"urker B{\i}y{\i}ko{\u g}lu \& Josef Leydold}
\authorx{Turker Biyikoglu \& Josef Leydold}
\authorspace
\bsgitem{2010a}{}  Semiregular trees with minimal Laplacian spectral radius.  \emph{Linear Algebra Appl.}\ 432 (2010), 2335--2341.
\MR{2599863} (2011b:05114).  \Zbl{1225.05139}.  
\annot{} In a semiregular tree $T$, all internal vertices have the same degree $d$.  Thm.\ 2:  Given $n, d \geq 3$, a semiregular $T$ minimizes $\lambda_1(\Lap(T))$ iff it is a caterpillar.  The proof is via $\Spec \Lap(-T)$, which $= \Spec \Lap(T)$ since a signed tree is balanced.
\dated{21 Jan 2012} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{T\"urker Biyiko{\u g}lu, Marc Hellmuth, \& Josef Leydold}
\authorx{Turker Biyikoglu, Marc Hellmuth, \& Josef Leydold}
\authorspace
\bsgitem{2009a}{\dag}  Largest eigenvalues of the discrete $p$-Laplacian of trees with degree sequences.  \emph{Electronic J.\ Linear Algebra}\ 18 (2009), 202--210.
\MR{2491656} (2010d:05089).  \Zbl{1169.05335}.  
\annot{} The $p$-Laplacian ($1<p<\infty$) generalizes the Laplacian matrix acting on vertex functions.  [Generalizing to signed graphs:] Define the $p$-Laplacian of $\Sigma$ by $\Delta_p(\Sigma)f(u) := \sum_{uv \in E}  \sgn[f(u) - \sigma(uv)f(v)]\cdot |f(u) - \sigma(uv)f(v)|^{p-1}.$  
Then $p=2$ gives $\Lap(\Sigma)$.]  The $p$-Laplacian of $\Gamma$ is $\Delta_p(+\Gamma)$ and its signless $p$-Laplacian is $\Delta_p(-\Gamma)$.  
Prop.\ 3.3 \etseq\ concern $\Delta_p(-\Gamma)$.
[Unlike with the Laplacian $\Lap$, switching does not preserve properties, so signs matter in a tree.]
[\Problem.  Generalize to signed graphs.]
\dated{21 Jan 2012} 
\codes{\Csg:~\Cpar:~\CEig:~\CGen}%
\itemspace

\authorx{Anders Bj\"orner}		 				%BJORNER
\authorx{Anders Bjorner}
\author{Anders Bj\"orner \& Bruce E.\ Sagan} 
\authorx{Anders Bjorner \& Bruce E.\ Sagan} 
\authorspace
\bsgitem{1996a}{}  Subspace arrangements of type $B_n$ and $D_n$.  \emph{J.\ Algebraic Combin.}\ 5 (1996), 291--314.  
\MR{1406454} (97g:52028).  \Zbl{864.57031}.						
\annot{} Lattices $\Pi_{n,k,h}$ (for $0 < h \leq k \leq n$) consisting of all spanning subgraphs of $\pm K_n^{\circ}$ that have at most one nontrivial component $K$, for which either $K$ is balanced and complete and $\#V(K) \geq k$, or $K$ is induced and $\#V(K) \geq h$.  (Also a generalization of this.)  Characteristic polynomial, homotopy and homology of the order complex, cohomology of the real complement. 
\codes{\CSG:~\CGeom,~\CMatrd(\CGen):~\CInvar,~\Ccol}%
\itemspace

\author{Anders Bj\"orner, Michel Las Vergnas, Bernd Sturmfels, Neil White, \& G\"un\-ter M.\ Ziegler}
\authorx{Anders Bjorner, Michel Las Vergnas, Bernd Sturmfels, Neil White, \& Gunter M.\ Ziegler}
\authorspace
\bsgitem{1993a}{}  \emph{Oriented Matroids}.  Encycl.\ Math.\ Appl., Vol.\ 46.  Cambridge Univ.\ Press, Cambridge, Eng., 1993.
\MR{1226888} (95e:52023).  \Zbl{773.52001}.
\annot{} The adjacency graph of bases of an oriented matroid is signed, using circuit signatures, to make the ``signed basis graph''.  See \S3.5, ``Basis orientations and chirotopes'', pp.\ 132--3.
\codes{\CMatrd:~\CSG}%
\itemspace

\author{Anders Bj\"orner \& Michelle L.\ Wachs}
\authorx{Anders Bjorner \& Michelle L.\ Wachs}
\authorspace
\bsgitem{2004a}{}  Geometrically constructed bases for homology of partition lattices of types $A$, $B$ and $D$.  \emph{Electronic J.\ Combin.}\ 11 (2004), no.\ 2, art.\ R3, 26 pp.
\MR{2120098} (2005m:52029).  \Zbl{1064.05151}.  
\annot{} \S9, ``Interpolating partition lattices'':  Homology of $\cL(\cH[\pm K_n^{(T)}])$ where $T$ is the set of vertices that have half edges.
	
\dated{12 Aug 2014} 
\par
\codes{\Csg:~\CGeom}%
\itemspace

\author{J.A.\ Blackman}						%BLACKMAN
\authorspace
\authref  See also \hyperlink{J.R.\ Goncalves}{J.R.~Gon{\c c}alves} and \hyperlink{J.\ Poulter}{J.~Poulter}..
\authrefspace

\bsgitem{1982a}{}  Two-dimensional frustrated Ising network as an eigenvalue problem.  \emph{Phys.\ Rev.\ B} 26 (1982), no.\ 9, 4987--4996.
\annot{} The signed square lattice graph, including effect of density of negative edges (``antiferromagnetic bonds'').  
\S III, ``Frustration'':  ``Local-mode'' eigenvectors of a Hermitian modification $D$ of a Pfaffian adjacency matrix correspond bijectively to negative squares (``frustrated plaquettes'').  Weight $>3$ of frustrated edge (compared to 1 of satisfied edges) alters the ground state [by switching; because the graph is 4-regular].  
\S IV, ``Frustration at higher density'':  Numerical studies of an $N^2$ grid with $N=50$ [small by today's standards] suggest a change of behavior at $p := l(\Sigma)/\#E \approx .045$ [a value that surely depends on 4-regularity].
Dictionary: ``gauge transformation'' = switching, ``gauge invariance'' = switching invariance, ``wrong bond'' = frustrated edge [in a ground state].  
\annotctd{} Continued in \xref{Blackman and Poulter}{1991a}{J.A.\ Blackman \& J.\ Poulter} and in \xref{Gon{\c c}alves, Poulter, and Blackman}{1997a}{J.R.\ Goncalves, J.\ Poulter, \& J.A.\ Blackman} and \xrefd{1998a}{J.A.\ Blackman, J.R.\ Goncalves, \& J.\ Poulter}.
\dated{17 May 2013} 
\par
\codes{\CPhys,~\CSG:~\CState(\Cfr),~\CEig,~\CWG}%
\itemspace

\author{J.A.\ Blackman, J.R.\ Gon{\c c}alves, \& J.\ Poulter}
\authorx{J.A.\ Blackman, J.R.\ Goncalves, \& J.\ Poulter}
\authorspace
\bsgitem{1998a}{}  Properties of the two-dimensional random-bond $\pm J$ Ising spin glass.  \emph{Phys.\ Rev.\ E} 58 (1998), no.\ 2, 1502--1507.
\annot{} Extension and refinement of \xref{Blackman and Poulter}{1991a}{J.A.\ Blackman \& J.\ Poulter}.
\dated{17 May 2013} 
\codes{\CPhys,~\CSG:~\CFr,~\CAdj}%
\itemspace

\author{J.A.\ Blackman \& J.\ Poulter}
\authorspace
\bsgitem{1991a}{}  Gauge-invariant method for the $\pm J$ spin-glass model.  \emph{Phys.\ Rev.\ B} 44 (1991), no.\ 9, 4374--4386.
\annot{} Square lattice graph with a definite proportion of negative edges.  \Cf\ \xref{Poulter and Blackman}{2001a}{J.\ Poulter \& J.A.\ Blackman} for triangular lattice.
\dated{16 Aug 2018} 
\codes{\CPhys:~\Csg:~\CFr}%
\itemspace

\authorx{Matthew Blair}						%BLAIR
\author{Matthew Blair, Rigoberto Fl\'orez, \& Antara Mukherjee}
\authorspace
\bsgitem{2021a}{}  Geometric patterns in the determinant Hosoya triangle.  \emph{Integers} 21 (2021), art.\ A90, 24 pp.
\MR{4321092}.  \Zbl{1490.11019}.  
\annot{} Signed quadrilaterals are a tool in \S3.1.
\dated{8 Nov 2022} 
\par
\codes{\CSG:~\CAppl}%
\itemspace

\authorx{Franco Blanchini}						%BLANCHINI
\author{Franco Blanchini, Elisa Franco, \& Giulia Giordano}
\authorspace
\bsgitem{2014a}{}  A structural classification of candidate oscillatory and multistationary biochemical systems.  \emph{Bull.\ Math.\ Biol.}\ 76 (2014), 2542--2569.
\MR{3266818}.  \Zbl{1329.92041}.  
\codes{\CSD,~\CBiol:~\CDyn}%
\itemspace

\authorx{Daniel Blankschtein}						%BLANKSCHTEIN
\author{Daniel Blankschtein, M.\ Ma, \& A.\ Nihat Berker}
\authorspace
\bsgitem{1984a}{}  Fully and partially frustrated simple-cubic Ising models: Landau-Ginzburg-Wilson theory.  \emph{Phys.\ Rev.\ B} 30 (1984), no.\ 3, 1362--1365.
\codes{\CPhys,~\CSG:~\CFr,~\Csw}%
\itemspace

\author{Daniel Blankschtein, M.\ Ma, A.\ Nihat Berker, Gary S.\ Grest, \& C.M.\ Soukoulis}
\authorspace
\bsgitem{1984a}{}  Orderings of a stacked frustrated triangular system in three dimensions.  \emph{Phys.\ Rev.\ B} 29 (1984), no.\ 9, 5250--5252.
\annot{} Physics of $(-L_3) \times (+P_m)$, consisting of $m$ all-negative triangular lattice layers $-L_3$, stacked vertically with vertical positive edges forming paths $P_m$ ($0 \ll m \leq \infty$).  The horizontal triangles are negative (the layers are ``totally frustrated'') while the vertical squares are positive.  Ground states ($\zeta: V \to \signnos$ that minmize $\#(E^\zeta)^-$) are ground states of $-L_3$ (\cf\ \xref{Wannier}{1950a}{G.H.\ Wannier}) repeated in every layer.
\dated{18 Jun 2012} 
\codes{\CPhys,~\CSG:~\CState(\Cfr),~\Csw}%
\itemspace

\author{Jaros{\l}aw B{\l}asiok}						%BLASIOK
\authorspace
\authref  See \hyperlink{Charalampos E.\ Tsourakakis}{C.E.~Tsourakakis}.
\itemspace

\author{Andreas Blass}						%BLASS
\authorspace
\bsgitem{1995a}{}  Quasi-varieties, congruences, and generalized Dowling lattices.  \emph{J.\ Algebraic Combin.}\ 4 (1995), 277--294.  
\MR{1346885} (96i:06012).  \Zbl{857.08002}.  
Errata.  \emph{Ibid.}\ 5 (1996), 167.  
\MR{1382046}.  \Zbl{857.08002}. 						
\annot{} Treats the generalized Dowling lattices of \xref{Hanlon}{1991a}{Phil Hanlon} as congruence lattices of certain quasi-varieties, in order to calculate characteristic polynomials and generalizations.
\codes{\CMatrd(\Cgg):~\CGen:~\CInvar}%
\itemspace

\author{Andreas Blass \& Frank Harary}
\authorspace
\bsgitem{1982a}{}  Deletion versus alteration in finite structures.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 7 (1982), 139--142.  
\MR{0685507} (84d:05087).  \Zbl{506.05038}. 						
\annot{} The theorem that deletion index = negation index of a signed graph (\xref{Harary}{1959b}{Frank Harary}) is shown to be a special case of a very general phenomenon involving hereditary classes of ``partial choice functions''.  
Another special case: deletion index = alteration index of a gain graph [an immediate corollary of \xref{Harary--Lindstr\"om--Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom}, Thm.\ 2].
\codes{\CSG,~\CGG:~\CBal,~\CFr}%
\itemspace

\author{Andreas Blass \& Bruce Sagan}
\authorspace
\bsgitem{1997a}{}  M\"obius functions of lattices.  \emph{Adv.\ Math.}\ 127 (1997), 94--123.  
\MR{1445364} (98c:06001).  \Zbl{970.32977}.  					
\annot{} \S3: ``Non-crossing $B_n$ and $D_n$''.  Lattices of noncrossing signed partial partitions.  Atoms of the lattices are defined as edge fibers of the signed covering graph of $\pm K_n^{\circ}$, thus corresponding to edges of $\pm K_n^{\circ}$.  [The ``half edges'' are perhaps best regarded as negative loops.]  The lattices studied, called $NCB_n, NCD_n, NCBD_n(S)$, consist of the noncrossing members of the Dowling and near-Dowling lattices of the sign group, i.e., $\Lat \bfF(\pm K_n^{(T)})$ for $T = [n], \eset, [n] \backslash S$, respectively.
\par
\codes{\CSG:~\CGeom,~\CMatrd(\CGen),~\CInvar,~\Ccov}%
\itemspace

\bsgitem{1998a}{}  Characteristic and Ehrhart polynomials.  \emph{J.\ Algebraic Combin.}\ 7 (1998), 115--126.  
\MR{1609889} (99c:05204).  \Zbl{899.05003}.						
\annot{} Signed-graph chromatic polynomials are recast geometrically by observing that the number of $k$-colorings equals the number of points of $\{-k, -k + 1, \ldots, k - 1, k\}^n$ that lie in none of the edge hyperplanes of the signed graph.  The interesting part is that this generalizes to subspace arrangements of signed graphs and, somewhat \emph{ad hoc}, to the hyperplane arrangements of the exceptional root systems.  
[\Cf\ \xref{Athanisiadis}{1996a}{Christos A.\ Athanasiadis}, \xref{Zaslavsky}{20xxi}{Thomas Zaslavsky}.  For applications see articles of \hyperlink{Bruce Sagan}{Sagan} and \hyperlink{Ping Zhang}{P.~Zhang}.]
\codes{\CSG,~\CGen:~\CMatrd(\CGen),~\CGeom:~\Ccol,~\CInvar}%
\itemspace

\author{Matthew Bloss}						%BLOSS
\authorspace
\bsgitem{2003a}{}  $G$-colored partition algebras as centralizer algebras of wreath products.  \emph{J.\ Algebra} 265 (2003), no.\ 2, 690--710. 
\MR{1987025} (2004e:20020).  \Zbl{1028.20007}.					
\annot{} Let $\fG$ denote any group.  
The algebra is $\bbC \Latb \bfF(\fG K_{2k}(U,W))$ where $\Latb \bfF(\fG K_{2k}(U,W)) =$ the semilattice of balanced flats of the Dowling lattice $Q_{2k}(\fG)$ on a set $V := U \cupdot W$ of $2k$ vertices, $U:=\{u_{1},\ldots,u_{k}\}$, and $W:=\{w_{1},\ldots,w_{k}\}$.  
\annotctd{} The definition requires a multiplication on $\Latb \bfF(\fG K_{2k}(U,W))$ which involves an indeterminate $x$.  For each balanced flat (equivalently, $\fG$-valued partition) $\alpha$ label its vertices $u_{\alpha i}:=u_i$, $w_{\alpha i}:=w_i$.  Define $\gamma := \alpha \cdot \beta$ by identifying $w_{\alpha i}$ with $u_{\beta _i}$ in $\alpha \cupdot \beta$ (call the result $\gamma'$), taking the closure in $\bfF(\fG K_{3k})$, multiplying by $x^m$ where $m :=$ \# of components of $\gamma'$ contained completely within the identified vertices, and deleting the identified vertices $w_{\alpha i}$.  Set $u_{\gamma i} := u_{\alpha i}$ and $w_{\gamma i} := w_{\beta i}$.  
\dated{20 Mar 2011} 
\codes{\Cgg:~\Cmatrd:~\CAlgeb}%
\itemspace

\author{Avrim Blum}							%BLUM
\authorspace
\authref  See \hyperlink{Nikhil Bansal}{N.~Bansal}.
\itemspace

\authorx{Rafa{\l} Bocian}						%BOCIAN
\authorx{Rafal Bocian}
\author{Rafa{\l} Bocian, Mariusz Felisiak, \& Daniel Simson}
\authorspace
\bsgitem{2013a}{}  On Coxeter type classification of loop-free edge-bipartite graphs and matrix morsifications.  
In:  Nikolaj Bj\"orner \etal, eds., \emph{15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2013, Timisoara, Romania, 2013), pp.\ 115--118.  IEEE, 2013.
\codes{\CSG}%
\itemspace

\bsgitem{2014a}{}  Numeric and mesh algorithms for the Coxeter spectral study of positive edge-bipartite graphs and their isotropy groups.  \emph{J.\ Comput.\ Appl.\ Math.}\ 259 (2014), part B, 815--827.
MR{3132846}.  \Zbl{1314.05196}.  
\codes{\CSG}%
\itemspace

\authorx{F.T.\ Boesch}						%BOESCH
\author{F.T.\ Boesch, X.\ Li [Xiao Ming Li], \& J.\ Rodriguez}
\authorspace
\bsgitem{1995a}{}  Graphs with most number of three point induced connected subgraphs.  \emph{ Discrete Appl.\ Math.}\ \ 59 (1995), no.\ 1, 1--10.
\MR{1326713} (96b:05073) (\qv).  \Zbl{835.05056}.
\annot{} Two-graphs and switching are mentioned.
\codes{\CTG,~\CSw}%
\itemspace

\authorx{Irina E.\ Bocharova}						%BOCHAROVA
\author{Irina E.\ Bocharova, Florian Hug, Rolf Johannesson, Boris D.\ Kudryashov, \& Roman V.\ Satyukov}
\authorspace
\bsgitem{2011a}{}  Some voltage graph-based LDPC tailbiting codes with large girth.  \emph{Information Theory (ISIT2011)} (Proc.\ 2011 IEEE Int.\ Sympos., St.\ Petersburg), pp.\ 732--736.  IEEE, 2011.
\arXiv{1108.0840}.  
\codes{\CGG:~\CCov}%
\itemspace

\bsgitem{2012a}{}  Searching for voltage graph-based LDPC tailbiting codes with large girth.  \emph{IEEE Trans.\ Information Theory} 58 (2012), no.\ 4, 2265--2279.
\MR{2951330}.  \arXiv{1108.0840}.  
\par
\codes{\CGG:~\CCov}%
\itemspace

\authorx{Sebastian B\"ocker}						%BOCKER
\authorx{Sebastian Bocker}
\author{Sebastian B\"ocker \& Jan Baumbach}
\authorx{Sebastian Bocker \& Jan Baumbach}
\authorspace
\bsgitem{2013a}{}  Cluster editing.  In:  Paola Bonizzoni, Vasco Brattka and Benedikt L\"owe, eds., \emph{The Nature of Computation: Logic, Algorithms, Applications} (Proc.\ 9th Conf.\ Computability in Europe (CiE 2013), Milan, 2013), pp. 33--44.  LNCS, Vol.\ 7921.  Springer, Berlin, 2013.
\MR{3102002}.  
\annot{$l_{\clu}$} Survey.  Cluster editing is equivalent to finding the frustration index, cluster index, or $p$-cluster index of a signed complete graph $\Sigma=(K_n,\sigma)$, often with such restrictions as negating at most $k$ edges at each vertex, or minimizing the total negated weight if edges are weighted.  
Dictionary:  ``editing'' = negating some edge signs, ``($p$)-cluster editing'' = changing the fewest signs to make $\Sigma^+$ into a union of ($p$) disjoint cliques; $2$-cluster editing'' = finding a minimum balancing edge set.
[The field seems to be unaware of signed graphs, balance, and clustering and that this is the special case of \xref{Davis}{1967a}{James A.\ Davis} with underlying complete graph.]  
\dated{4 Nov 2017} 
\codes{\Csg:~\Ckg:~\CClu:~\CAlgor:~\CExp,~\CBiol}%
\itemspace

\author{Sebastian B\"ocker, Falk H\"uffner, Anke Truss, \& Magnus Wahlstr\"om}
\authorx{Sebastian Bocker, Falk Huffner, Anke Truss, \& Magnus Wahlstrom}
\authorspace
\bsgitem{2009a}{}  A faster fixed-parameter approach to drawing binary tanglegrams.  
In: J.\ Chen and F.V.\ Fomin, eds., \emph{Parameterized and Exact Computation} (4th Int.\ Workshop, IWPEC 2009, Copenhagen), pp.\ 38--49.  \LNCS 5917.  Springer, Berlin, 2009.
\MR{2773930} (no rev).  \Zbl{1273.68157}.					
\annot{} The signed graph arises as a graph with edges labelled $=$ ($+$) or $\neq$ ($-$).  The ``Balanced Subgraph'' problem is to find a minimum balancing set.  The algorithm of \xref{H\"uffner, Betzler, and Niedermeier}{2007a}{Falk Huffner, Nadja Betzler, \& Rolf Niedermeier} is applied.  
\dated{6 Feb 2011} 
\codes{\Csg:~\Cfr:~\CAlgor:~\CAppl}%
\itemspace

\author{Alexander Bockmayr}						%BOCKMAYR
\authorspace
\authref  See also \hyperlink{Heike Siebert}{H.~Siebert}.
\authrefspace

\author{Alexander Bockmayr \& Heike Siebert}
\authorspace
\bsgitem{2013a}{}  Bio-logics: Logical analysis of bioregulatory networks.  In:  Andrei Voronkov and Christoph Weidenbach, eds.,  \emph{Programming Logics: Essays in Memory of Harald Ganzinger} (Workshop, Saarbr\"ucken, 2013), pp.\ 19--34.  \LNCS 7797.  Springer-Verlag, Heidelberg, 2013.
\MR{3084884}.  \Zbl{1383.92031}.  
\codes{\CSD:~\CDyn}%
\itemspace

\authorx{Hans L.\ Bodlaender}						%BODLAENDER
\author{Hans L.\ Bodlaender, Michael R.\ Fellows, Pinar Heggernes, Federico Mancini, Charis Papadopoulos, \& Frances Rosamond}
\authorspace
\bsgitem{2008a}{}  Clustering with partial information.  In:  Edward Ochma\'nski and Jerzy Tysz\-kie\-wicz, eds., \emph{Mathematical Foundations of Computer Science 2008} (Proc.\ 33rd Int.\ Symp., MFCS 2008, Tor\'un, Poland, 2008), pp.\ 144--155.  \LNCS, Vol.\ 5162.  Springer, Berlin, 2008. 
\MR{2539366} (2011e:\-05232).  \Zbl{1173.68596}.  
\annot{} Short version, without most proofs, of \xrefd{2010a}{Hans L.\ Bodlaender, Michael R.\ Fellows, Pinar Heggernes, Federico Mancini, Charis Papadopoulos, \& Frances Rosamond}.
\dated{17 Nov 2017} 
\codes{\CSG:~\CClu,~\CKG:~\CAlgor}%
\itemspace

\bsgitem{2010a}{}  Clustering with partial information.  \emph{Theor.\ Computer Sci.}\ 411 (2010), no.\ 7-9, 1202--1211.
\MR{2606055} (2011d:05345).  \Zbl{1213.05222}.  
\annot{} The ``fuzzy graphs'' (not related to fuzzy graph theory) are, in essence, signed simple graphs; ``fuzzy edges'' are non-edges, which in a ``normalization'' become signed edges, resulting in a signed $K_n$.	
\par
\codes{\CSG:~\CClu,~\CKG:~\CAlgor}%
\itemspace

\author{Hans L.\ Bodlaender \& Jurriaan Hage}
\authorspace
\bsgitem{2012a}{}  On switching classes, NLC-width, cliquewidth and treewidth.  \emph{Theor.\ Computer Sci.}\ 429 (2012), 30--35.
\MR{2901392}.  \Zbl{1242.05217}.  
\annot{} NLC-width and clique-width of a graph vary only linearly by switching, but tree-width can vary arbitrarily.  \Cf\ \xref{Hline\v{n}\'y \etal}{2008a}{Petr Hlineny, Sang-Il Oum, Detlef Seese, \& Georg Gottlob} for rank-width.  [\Problem.  Generalize the widths to signed graphs and see how they behave on switching classes.]
\dated{15 Jun 2022} 
\codes{\CTG:~\CSw}%
\itemspace

\authorx{Bernhard G.\ Bodmann}					%BODMANN
\author{Bernhard G.\ Bodmann \& Vern I.\ Paulsen}
\authorspace
\bsgitem{2005a}{}  Frames, graphs and erasures.  \emph{Linear Algebra Appl.}\ 404 (2005), 118--146.
\MR{2149656} (2006a:42047).  \Zbl{1088.46009}.  \arXiv{math/0406134}.			
\annot{} Develops \xref{Strohmer and Heath}{2003a}{Thomas Strohmer \& Robert W.\ Heath Jr.} and \xref{Holmes and Paulsen}{2004a}{Roderick B.\ Holmes \& Vern I.\ Paulsen}.  
\S4, ``Two-uniform frames and graphs'':  In Def.\ 4.1, the ``signature matrix'' $Q$ is $A(K_n,\phi)$, $(K_n,\phi)=$ real or complex unit gain graph and (Thm.\ 4.2) has exactly 2 distinct eigenvalues.  
Ex.\ 4.4, 4.5: $Q=A(K_\Gamma)$, $K_\Gamma=$ signed graph, from conference, skew-conference, and Hadamard matrices.  
Thm.\ 4.7:  For real $2$-uniform frames, $Q=A(K_\Gamma)$ where $\Gamma$ has 2 eigenvalues.  Frame equivalence = graph switching equivalence.  Hence, \# of inequivalent real 2-uniform frames = \# of switching classes = \# regular two-graphs.  
\S5, ``Graphs and error bounds''.   Def.\ 5.8:  $\cG^{(s)}_m=$ set of $(K_n,\sigma)$ with frustration index $s$.  Induced complete bipartite subgraphs (up to switching) hinder error bounds and have other significance.  Lemma 5.19:  $E_3=$ \# of non-triples of a regular two-graph.  
\S6: Specific examples.
\annotctd{} [Much literature follows, e.g.:  
\xref{Bodmann--Paulsen--Tomforde}{2009a}{Bernhard G.\ Bodmann, Vern I.\ Paulsen, \& Mark Tomforde}, 
\xref{Duncan, Hoffman, and Solazzo}{2010a}{David M.\ Duncan, Thomas R.\ Hoffman, \& James P.\ Solazzo}, \xref{Hoffman and Solazzo}{2012a}{Thomas R.\ Hoffman \& James P.\ Solazzo}, \xrefd{2018a}{Thomas R.\ Hoffman \& James P.\ Solazzo}.]
\dated{7 Aug 2018} 
\codes{\Csg,~\Cgg:~\Ckg:~\CTG:~\CAdj:~\CGeom,~\CAppl}%
\itemspace

\author{Bernhard G.\ Bodmann, Vern I.\ Paulsen, \& Mark Tomforde}
\authorspace
\bsgitem{2009a}{}  Equiangular tight frames from complex Seidel matrices containing cube roots of unity.  \emph{Linear Algebra Appl.}\ 430 (2009), 396--417.
\MR{2460526} (2010b:42040).  \Zbl{1165.42007}.  
\annot{} Adjacency matrices of cube-root-of-unity gain graphs on $K_n$.  
Dictionary: ``Seidel matrix'' = adjacency matrix of such a gain graph.
\dated{27 Apr 2012} 
\codes{\Cgg:~\Ckg:~\CGeom,~\Cadj}%
\itemspace

\author{T.B.\ Boffey} 							%BOFFEY
\authorspace
\bsgitem{1982a}{} \emph{Graph Theory in Operations Research}.  Macmillan, London, 1982.  
\Zbl{509.90053}.  
\annot{} Ch.\ 10:  ``Network flow:  extensions.''  10.1(g):  ``Flows with gains,'' pp.\ 224--226.  10.3:  ``The simplex method applied to network problems,'' subsection ``Generalised networks,'' pp.\ 246--250. 
\codes{\CGN:~\Cmatrd(\Cbases):~\CExp}%
\itemspace

\author{Kenneth P.\ Bogart}						%BOGART*
\authorspace
\authref  See \hyperlink{M.K.\ Bennett}{M.K.~Bennett}, \hyperlink{Joseph E.\ Bonin}{J.E.~Bonin}, and \hyperlink{Jeffrey R.\ Weeks}{J.R.~Weeks}.
\itemspace

\author{Petre Boldescu}						%BOLDESCU
\authorspace
\bsgitem{1970a}{}  Les th\'eor\`emes de Menelaus et Ceva dans un \'espace affine de dimension $n$.  [The theorems of Menelaus and Ceva in an $n$-dimensional affine space.]  (In Romanian.  French summary.)  \emph{An.\ Univ.\ Craiova Ser.\ a IV-a} 1 (1970), 101--106.
\MR{0333932} (48 \#12251).  \Zbl{275.50008}.
\annot{} Generalized Ceva [strengthened via gain graphs in \xref{Zaslavsky}{2003b}{Thomas Zaslavsky} \S2.6] and Menelaus theorems.  [\Problem.  Formulate, explain, generalize Boldescu's Menelaus generalization in terms of gain graphs.]
\par
\codes{\Cgg:~\CGeom}%
\itemspace

\authorx{Paolo Boldi}							%BOLDI
\author{Paolo Boldi \& Sebastiano Vigna}
\authorspace
\bsgitem{1996a}{}  On some constructions which preserve sense of direction.  In:  \emph{Proceedings of the 3rd International Colloquium on Structural Information and Communication Complexity} (SIROCCO'96, Siena, 1996), pp.\ 47--57.  Carleton Scientific, Waterloo, Ont., Canada, 1996.
\annot{} Preliminary version of \xrefd{2000a}{Paolo Boldi \& Sebastiano Vigna}.
\codes{\CGG:~\CCov:~\CAppl}%
\itemspace

\bsgitem{2000a}{}  Coverings that preserve sense of direction.  \emph{Inform.\ Processing Lett.}\ 75 (2000), no.\ 4, 175--180.
\MR{1783445} (2001d:05143).  \Zbl{997.05076}.  
\annot{} The sense-of-direction edge labelling is a gain function on a digraph whose covering graph preserves sense-of-direction under certain conditions.
\dated{2 Apr 2022} 
\codes{\CGG:~\CCov:~\CAppl}%
\itemspace

\author{Ethan D.\ Bolker} 						%BOLKER
\authorspace
\bsgitem{1977a}{} Bracing grids of cubes.  \emph{Environment and Planning B} 4 (1977), 157--172.  
\annot{} The elementary 1-cycles associated with circuits of $\bfF(-\Gamma)$ (``bicycles'') are crucial.  [Their first publication, I believe.]
\codes{\CECyc,~\Csg:~\Cmatrd}%
\itemspace

\bsgitem{1979a}{}  Bracing rectangular frameworks.  II.  \emph{SIAM J.\ Appl.\ Math.}\ 36 (1979), 491--503.  
\MR{0531610} (81j:73066b).  \Zbl{416.70010}.					
\annot{} The elementary 1-cycles associated with circuits of $\bfF(\Sigma)$ (``bicycles''), mostly for $\Sigma = -\Gamma$.  General signed graphs appear at Thm.\ 7, p.\ 505.  Dictionary: ``Signed bicycle'' = elementary 1-cycle (circulation) associated with a circuit.
\codes{\CECyc,~\CSG:~\CMatrd,~\Cinc}%
\itemspace

\author{Ethan D.\ Bolker \& Thomas Zaslavsky}
\authorspace
\bsgitem{2006a}{}  A simple algorithm that proves half-integrality of bidirected network programming.  \emph{Networks} 48 (2006), no.\ 1, 36--38.
\MR{2243932} (2007b:05098).  \Zbl{1100.05046}.					
\annot{} An idea of \xref{Bolker's}{1979a}{Ethan D.\ Bolker}, as developed by \xref{Bouchet}{1983a}{Andre Bouchet}, is turned into an algorithm slightly simpler than that of \xref{Appa and Kotnyek}{2006a}{Gautam Appa \& Balazs Kotnyek}.
\codes{\CSG:~\COri,~\CInc,~\CAlgor,~\CSw}%
\itemspace

\author{B\'ela Bollob\'as}						%BOLLOBAS
\authorx{Bela Bollobas}
\authorspace
\bsgitem{1978a}{}  \emph{Extremal Graph Theory}.  L.M.S.\ Monographs, Vol.\ 11.  Academic Press, London, 1978.
\MR{0506522} (80a:05120).  \Zbl{419.05031}.					
Repr.\ Dover Publications, Mineola, N.Y., 2004.
\MR{2078877} (2005b:05124).  \Zbl{1099.05044}					
\annot{} A rich source of problems: find interesting generalizations to signed graphs of questions involving even or odd circles, or bipartite graphs or subgraphs.
\codes{\Cpar:~\CXtreml}%
\annotctd{} \S3.2, Thm.\ 2.2, is \xref{Lov\'asz's}{1965a}{Laszlo Lovasz} characterization of the graphs having no two vertex-disjoint circles.  [\Problem.  Generalize to biased graphs having no two vertex-disjoint unbalanced circles, Lov\'asz's theorem being the contrabalanced case.]
\codes{\CGG:~\CCircles}%
\annotctd{} \S6.6, Problem 47, is the theorem on biparticity (all-negative vertex frustration number) from \xref{Bollob\'as, Erd\H{o}s, Simonovits, \& Szemer\'edi}{1978a}{B.\ Bollobas, P.\ Erdos, M.\ Simonovits, \& E.\ Szemeredi}.
\codes{\Cpar:~\CFr}%
\itemspace

\bsgitem{1998a}{}  \emph{Modern Graph Theory}.  Springer, New York, 1998.
\MR{1633290} (99h:05001).  \Zbl{902.05016}.						
\annot{} Sign-colored plane graphs in Ch.\ X, ``The Tutte polynomial'', \S6, ``Polynomials of knots and links'', pp.\ 368--370.  Little use is made of the signs.
\par
\codes{\CSGc:~\CKnot}%
\itemspace

\author{B.\ Bollob\'as, P.\ Erd\"os, M.\ Simonovits, \& E.\ Szemer\'edi}
\authorx{B.\ Bollobas, P.\ Erdos, M.\ Simonovits, \& E.\ Szemeredi}
\authorspace
\bsgitem{1978a}{}  Extremal graphs without large forbidden subgraphs.  
In: B.\ Bollob\'as, ed., \emph{Advances in Graph Theory} (Proc.\ Cambridge Combin.\ Conf., 1977), pp.\ 29--41.  Ann.\ Discrete Math., Vol.\ 3.  North-Holland, Amsterdam, 1978.
\MR{0499108} (80a:05119).  \Zbl{375.05034}.
\annot{} Thm.\ 9 asymptotically estimates upper bounds on frustration index and vertex frustration number for all-negative signed graphs with fixed negative girth.  [Sharpened by \xref{Koml\'os}{1997a}{Janos Komlos}.]
\codes{\Cpar:~\CFr}%
\itemspace

\author{B\'ela Bollob\'as \& Andr\'as Gy\'arf\'as}
\authorx{Bela Bollobas \& Andras Gyarfas}
\authorspace
\bsgitem{2008a}{}  Highly connected monochromatic subgraphs.  \emph{Discrete Math.}\ 308 (2008), no.\ 9, 1722--1725.
\MR{2392611} (2009b:05183).  \Zbl{1137.05025}.  
\annot{} See \xref{{\L}uczak}{2016a}{Tomasz Luczak}.
\dated{24 Jan 2016} 
\codes{\Csg:~\CStr}%
\itemspace

\author{Bela Bollob\'as, Luke Pebody, \& Oliver Riordan}
\authorx{Bela Bollobas, Luke Pebody, \& Oliver Riordan}
\authorspace
\bsgitem{2000a}{}  Contraction-deletion invariants for graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 80 (2000), 320--345.
\MR{1794697} (2001j:05055).  \Zbl{1024.05028}.						
\annot{} \S4, ``Coloured graphs''.	
\codes{\CSGc:~\CGen:~\CInvar}%
\itemspace

\author{Bela Bollob\'as \& Oliver Riordan}
\authorx{Bela Bollobas \& Oliver Riordan}
\authorspace
\bsgitem{1999a}{}  A Tutte polynomial for coloured graphs.  Recent Trends in Combinatorics (M\'atra\-h\'aza, 1995).  \emph{Combin.\ Probab.\ Comput.}\ 8 (1999), 45--93.
\MR{1684623} (2000f:05033).  \Zbl{926.05017}.  
\annot{} Discovers the fundamental relations for the commutative algebra underlying the parametrized Tutte polynomial of colored graphs.  \Cf\ \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}.
\codes{\CSGc:~\CGen:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{2002a}{}  A polynomial of graphs on surfaces.  \emph{Math.\ Ann.}\ 323 (2002), no.\ 1, 81--96.
\MR{1906909} (2003b:05052).  \Zbl{1004.05021}.  
\annot{} The polynomial is a deletion-contraction invariant of signed graphs with rotation systems (called ``ribbon graphs'').
\codes{\Csg:~\CTop:~\CInc}%
\itemspace

\author{S.\ Bolouki}							%BOLOUKI
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.

\authorx{Erik G.\ Boman}						%BOMAN
\author{Erik G.\ Boman, Doron Chen, Ojas Parekh, \& Sivan Toledo}
\authorspace
\bsgitem{2005a}{}  On factor width and symmetric $H$-matrices.  \emph{Linear Algebra Appl.}\ 405 (2005), 239--248.
\MR{2148173} (2006e:15024).  \Zbl{1098.15014}.					
\annot{} A real symmetric matrix $= \Eta(\Phi)\Eta(\Phi)\transpose$ for a real gain graph $\Phi$ with a link (called ``factor width 2'').  Thm.\ 9.  $A$ has factor width 2 iff it is a symmetric $H$-matrix with diagonal $\geq0$.
\dated{8 Mar 2011}
\par
\codes{\Cgg:~\CInc,~\CAdj}%
\itemspace

\author{Phillip Bonacich}						%BONACICH
\authorspace
\bsgitem{1999a}{}  An algebraic theory of strong power in negatively connected exchange networks.  
\emph{J.\ Math.\ Sociology} 23 (1999), no.\ 3, 203--224.
\Zbl{1083.91574}.  
\annot{} P.~214:  The distribution of power depends in part on whether $\incid(-\Gamma)$ has full rank, i.e., $\Gamma$ is bipartite (\cf\ \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}), where $\Gamma$ is the graph of potential exchanges.
\dated{13 Aug 2012}
\par
\codes{\Cpar:~\CInc,~\CPsS}%
\itemspace

\bsgitem{2007a}{}  Some unique properties of eigenvector centrality.  \emph{Social Networks} 29 (2007), 555--564.
\annot{} \S1.1.3, ``Uses of $c(\beta)$ and $x$ in signed graphs''.  
\dated{12 Sept 2010} 
\par
\codes{\CSG,~\CPsS:~\CEig}%
\itemspace

\author{Phillip Bonacich \& Paulette Lloyd}
\authorspace
\bsgitem{2004a}{}  Calculating status with negative relations.  \emph{Social Networks} 26 (2004), 331--338.
\annot{} Compares the dominant-eigenvector measure of centrality in $\Sigma$, $\Sigma^+$, and dense induced subgraphs, in a standard example.
\dated{22 Oct 2009} 
\codes{\CSG:~\CPsS:~\CEig}%
\itemspace

\authorx{Valerio Boncompagni}						%BONCOMPAGNI
\author{Valerio Boncompagni, Irena Penev, \& Kristina Vu\v{s}kovi\'c}
\authorx{Valerio Boncompagni, Irena Penev, \& Kristina Vuskovic}
\authorspace
\bsgitem{2017a}{}  Clique-cutsets beyond chordal graphs.  IX Latin-American Algorithms, Graphs, and Optimization Sympos., Marseille (LAGOS'17).  \emph{Electronic Notes Discrete Math.}\ 62 (2017), 81--86.
\MR{3746703}.  \Zbl{1383.05227}.  
\annot{} Extended abstract of \xrefd{2019a}{Valerio Boncompagni, Irena Penev, \& Kristina Vuskovic}.
\codes{\Csg:~\CStr}%
\itemspace

\bsgitem{2019a}{}  Clique-cutsets beyond chordal graphs.  \emph{J.\ Graph Theory} 91 (2019), no.\ 2, 192--246.
\MR{3948128}.  \Zbl{1414.05126}.  \arXiv{1707.03252}.				
\codes{\Csg:~\CStr}%
\itemspace

\authorx{J.A.\ Bondy}						%BONDY
\author{J.A.\ Bondy \& L.\ Lov\'asz}
\authorx{J.A.\ Bondy \& L.\ Lovasz}
\authorspace
\bsgitem{1981a}{}  Cycles through specified vertices of a graph.  \emph{Combinatorica} 1 (1981), 117--140.  
\MR{0625545} (82k:05073).  \Zbl{492.05049}.					
\annot{} If $\Gamma$ is $k$-connected [and not bipartite], then any $k$ $[k-1]$ vertices lie on an even [odd] circle.  [\Problem.  Generalize to signed graphs, this being the all-negative case.]  
\codes{\Csg:~\Cpar,~\CCircles}%
\itemspace

\author{J.A.\ Bondy \& M.\ Simonovits}
\authorspace
\bsgitem{1974a}{}  Cycles of even length in graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 16 (1974), 97--105.  
\MR{0340095} (49 \#4851).  \Zbl{283.05108}.					
\annot{} If a graph has enough edges, it has even circles of all moderately small lengths.  [\Problem\ 1.  Generalize to positive circles in signed graphs, this being the antibalanced (all-negative) case.  For instance, \Problem\ 2.  If an unbalanced signed simple graph has positive girth $\geq l$ (i.e., no balanced circle of length $< l$), what is its maximum size?  Are the extremal examples antibalanced?  Balanced?]
\codes{\Cpar:~\Cbal(\CCircles),~\CXtreml}%
\itemspace

\author{Joseph E.\ Bonin}						%BONIN
\authorspace
\authref   See also \hyperlink{M.K.\ Bennett}{M.K.~Bennett}.
\authrefspace

\bsgitem{1989a}{}  (as Joseph Edmond Bonin)  \emph{Structural Properties of Dowling Geometries and Lattices.}  Ph.D.\ thesis,. Dartmouth College, 1989.
\MR{2638328}.  
\codes{\Cgg:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{1993a}{}  Automorphism groups of higher-weight Dowling geometries.  \emph{J.\ Combin.\ Theory Ser.\ B} 58 (1993), 161--173. 
\MR{1223690} (94k:51005).  \Zbl{733.05027}, (\Zbl{789.05017}).  
\annot{} A weight-$k$ higher Dowling geometry of rank $n$,  $Q_{n,k}(\GF(q)^{\times})$, is the union of all coordinate $k$-flats of $\PG(n-1,q)$: i.e., all flats spanned by $k$ elements of a fixed basis.  If $k > 2$, the automorphism groups are those of $\PG(n-1,q)$ for $q > 2$ and are symmetric groups if $q = 2$.  \Cf\ \xref{Ravagnani}{2022a}{Alberto Ravagnani}.
\codes{\Cgg:~\CGen:~\CMatrd,~\CAut}%
\itemspace

\bsgitem{1993b}{}  Modular elements of higher-weight Dowling lattices.  \emph{Discrete Math.}\ 119 (1993), 3--11.  
\MR{1234055} (94h:05018).  \Zbl{808.06012}.						
\annot{} See definition in \xrefd{1993a}{Joseph E.\ Bonin}.  For $k > 2$ the only nontrivial modular flats are the projective coordinate $k$-flats and their subflats.  This gives some information about the characteristic polynomials [which, however, are still only partially known].  [\xref{Kung}{1996a}{Joseph P.S.\ Kung}, \S6, has further results.]
\par
\codes{\Cgg:~\CGen:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{1995a}{}  Automorphisms of Dowling lattices and related geometries.  \emph{Combin.\ Probab.\ Comput.}\ 4 (1995), 1--9.  
\MR{1336651} (96e:05039).  \Zbl{950.37335}. 
\annot{} The automorphisms of a Dowling geometry of a nontrivial group are the compositions of a coordinate permutation, switching, and a group automorphism.  A similar result holds, with two exceptions, if some or all coordinate points are deleted.  [A third exception is missed: the jointless Dowling geometry $Q^0_3(\bbZ_3)$.]  [\Cf\ \xref{Schwartz}{2002a}{Gary K.\ Schwartz}.]
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\bsgitem{1996a}{}  Open problem 6.  A problem on Dowling lattices.  
In:  Joseph E.\ Bonin, James G.\ Oxley, and Brigitte Servatius, eds., \emph{Matroid Theory} (Proc., Seattle, 1995), pp.\ 417--418.  Contemp.\ Math., Vol.\ 197.  Amer.\ Math.\ Soc., Providence, R.I., 1996.
\annot{} \Problem\ 6.1.  If a finite matroid embeds in the Dowling geometry of a group, does it embed in the Dowling geometry of some finite group?  [No; see \xref{Brooksbank, Qin, Robertson, and Seress}{2004a}{Peter Brooksbank, Hongxung Qin, Edmund Robertson, \& Akos Seress}.]
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2003a}{}  Strongly inequivalent representations and Tutte polynomials of matroids.  \emph{Algebra universalis} 49 (2003), 289--303.
\MR{2021388} (2004i:05027).  \Zbl{1090.05009}.  
\annot{} Dowling geometries are used to prove Prop.\ 1.1.
\dated{27 May 2018} 
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2006a}{}   Extending a matroid by a cocircuit.  \emph{Discrete Math.}\ 306 (2006), no.\ 8--9, 812--819.
\MR{2234987} (2006m:05045).  \Zbl{1090.05008}.					
\annot{} \S4 concerns Dowling lattices.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Joseph E.\ Bonin \& Kenneth P.\ Bogart}
\authorspace
\bsgitem{1991a}{}  A geometric characterization of Dowling lattices.  \emph{J.\ Combin.\ Theory Ser.\ A} 56 (1991), 195--202.  
\MR{1092848} (92b:05019).  \Zbl{723.05033}.  
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Joseph E.\ Bonin \& Joseph P.S.\ Kung}
\authorspace
\bsgitem{1994a}{}  Every group is the automorphism group of a rank-3 matroid.  \emph{Geom.\ Dedicata} 50 (1994), 243--246.  
\MR{1286377} (95m:20005).  \Zbl{808.05029}.  
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\bsgitem{2018a}{}  The $\cG$-invariant and catenary data of a matroid.  \emph{Adv.\ Appl.\ Math.}\ 94 (2018), 39--70.
\MR{3739496}.  \Zbl{1378.05020}.  \arXiv{1510.00682}.				
\annot{} Prop.\ 4.6. The Dowling matroids $Q_r(\fG)$ are an example.
\dated{10 Jan 2016} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Joseph E.\ Bonin \& William P.\ Miller}
\authorspace
\bsgitem{1999a}{}  Characterizing combinatorial geometries by numerical invariants.  \emph{European J.\ Combin.}\ 20 (1999), 713--724.
\MR{1730820} (2001a:51007).  \Zbl{946.05020}.					
\annot{} Dowling geometries are characterized amongst all simple matroids by numerical properties of large flats of ranks $\leq 7$ (Thm.\ 3.4); amongst all matroids by their Tutte polynomials.
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Joseph E.\ Bonin \& Hongxun Qin}
\authorspace
\bsgitem{2000a}{}  Size functions of subgeometry-closed classes of representable combinatorial geometries.  \emph{Discrete Math.}\ 224 (2000), 37--60.
\MR{1781284} (2001g:05031).  \Zbl{968.52009}.					
\annot{} Extremal matroid theory.  The Dowling geometry $Q_3(\GF(3)^{\times}) = \bfF(\pm K_3^\bullet)$ appears as an exceptional extremal matroid in Thm.\ 2.10.  The extremal subset of $\PG(n-1, q)$ that does not contain the higher-weight Dowling geometry $Q_{m,m-1}(\GF(q)^{\times})$ (see \xref{Bonin}{1993a}{Joseph E.\ Bonin}) is found in Thm.\ 2.14.
\codes{\CGG,~\CGen:~\CMatrd:~\CXtreml,~\CInvar}%
\itemspace

\authorx{C.\ Paul Bonnington}						%BONNINGTON
\author{C.\ Paul Bonnington \& Charles H.C.\ Little}
\authorspace
\bsgitem{1995a}{} \emph{The Foundations of Topological Graph Theory}.  Springer, New York, 1995.  
\MR{1367285} (97e:05071).  \Zbl{844.05002}.
\annot{} Signed-graph imbedding: see \S2.3, \S2.6 (esp.\ Thm.\ 2.4), pp.\ 44--48 (for the colorful 3-gem approach to crosscaps), \S3.3, and Ch.\ 4 (esp.\ Thms.\ 4.5, 4.6).
\codes{\Csg:~\CTop,~\Cbal}%
\itemspace

\author{Stefan Bornholdt}						%BORNHOLDT
\authorspace
\authref  See \hyperlink{Jorg Reichardt}{J.~Reichardt}.
\itemspace

\author{Bojana Borovi\'canin}						%BOROVICANIN
\authorx{Bojana Borovicanin}
\authorspace
\authref  See \hyperlink{Jianfeng Wang}{J.F.~Wang}.
\itemspace

\authorx{Endre Boros}						%BOROS
\author{E.\ Boros, Y.\ Crama, \& P.L.\ Hammer}
\authorspace
\bsgitem{1992a}{} Chv\`atal cuts and odd cycle inequalities in quadratic 0---1 optimization.  \emph{SIAM J.\ Discrete Math.}\ 5 (1992), 163--177.  
\MR{1157581} (93a:90043).  \Zbl{761.90069}.  
\annot{} \S4: ``Odd cycles [i.e., negative circles] in signed graphs.''  Main problem:  Find a minimum-weight deletion set in a signed graph with positively weighted edges.  Related problems:  A circle-covering formulation whose constraints correspond to negative circles.  A dual circle-packing problem.
\codes{\CSG:~\CFr,~\CGeom,~\CCircles,~\CAlgor}%
\itemspace

\author{Endre Boros, Vladimir A.\ Gurvich, \& Igor E.\ Zverovich}
\authorspace
\bsgitem{2010a}{}  Friendship two-graphs.  \emph{Graphs Combin.}\ 26 (2010), no.\ 5, 617--628.
\MR{2679935} (2011i:05106).  \Zbl{1228.05148}.  
\annot{} Oriented all-negative graphs in which every two vertices are joined by a unique coherent path.  (The authors describe this as alternating paths in an edge 2-colored graph.  The ``two-graph'' is the pair of monocolored graphs.)  
[\Problem.  Generalize to arbitrary bidirected graphs.]
[\Cf\ \xref{B\'ankfalvi and B\'ankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi} and \xref{Bang-Jensen and Gutin}{1997a}{Jorgen Bang-Jensen \& Gregory Gutin} for alternating walks.]  
\dated{25 Oct 2012} 
\codes{\Csg:~\Cori:~\CPaths}%
\itemspace

\author{Endre Boros \& Peter L.\ Hammer}
\authorspace
\bsgitem{1991a}{} The max-cut problem and quadratic 0---1 optimization; polyhedral aspects, relaxations and bounds.  \emph{Ann.\ Operations Res.}\ 33 (1991), 151--180.  
\MR{1140978} (92j:90049).  \Zbl{741.90077}. 
\annot{} Includes finding a minimum-weight deletion set (as in \xref{Boros, Crama, and Hammer}{1992a}{E.\ Boros, Y.\ Crama, \& P.L.\ Hammer}).  
\codes{\CSG,~\CWG:~\CFr:~\CGeom,~\CAlgor}
\itemspace

\author{Y.M.\ Borse}							%BORSE
\authorspace
\authref  See \hyperlink{Ganesh Mundhe}{G.~Mundhe}.
\itemspace

\author{Bart{\l}omiej Bosek}						%BOSEK
\authorx{Bartlomiej Bosek}
\authorspace
\authref  See \hyperlink{Marcin Anholcer}{M.~Anholcer}.
\itemspace

\authorx{J.-P.\ Bouchaud}						%BOUCHARD
\author{J.-P.\ Bouchaud, F.\ Krzakala, \& O.C.\ Martin}
\authorspace
\bsgitem{2003a}{}  Energy exponents and corrections to scaling in Ising spin glasses.  \emph{Phys.\ Rev.\ B} 68 (2003), art.\ 224404, 11 pp.
\annot{} Mostly, randomly weighted signed graphs (square and cubic lattices) with Gaussian signed weights.  
\S VII, ``Case of $+/-J$ couplings'':  Calculation experiments suggest unweighted signed graphs behave very differently from weighted ones.   ``[T]he local environment of a spin has no disorder out to finite distances: any sign of the Jij can be gauged [switched] away \dots''.  [That seems to mean imbalance can be switched away, which is wrong and casts doubt on the conclusions.]
\dated{28 Jan 2015} 
\codes{\CPhys:~\CSG,~\CWG}%
\itemspace

\author{Andr\'e Bouchet} 						%BOUCHET
\authorx{Andre Bouchet}
\authorspace
\bsgitem{1982a}{} Constructions of covering triangulations with folds.  \emph{J.\ Graph Theory} 6 (1982), 57--74.  
\MR{0644741} (83b:05057).  \Zbl{488.05032}.
\codes{\Csg:~\COri,~\CAppl(\CTop)}%
\itemspace

\bsgitem{1983a}{\dag} Nowhere-zero integral flows on a bidirected graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 34 (1983), 279--292.  
\MR{0714451} (85d:05109).  \Zbl{518.05058}.					
\annot{} Introduces nowhere-zero flows on signed graphs.  A connected, coloop-free signed graph has a nowhere-zero integral flow with maximum weight $\leq 216$.  The value $216$ cannot be replaced by $5$, but: \Conjecture (Bouchet): it can be replaced by $6$.  
[The bidirection is inessential; it is a device to keep track of the flow.]  
[For progress see \xref{Khelladi}{1987a}{Abdelkader Khelladi}, \xref{Z\'yka}{1987a}{Ondrej Zyka},  \xref{Xu and Zhang}{2005a}{Rui Xu \& Cun-Quan Zhang}, \xref{Raspaud and Zhu}{2011a}{Andre Raspaud \& Xuding Zhu}, \xref{Akbari, Daemi, \etal{}}{2015a}{S.\ Akbari, A.\ Daemi, O.\ Hatami, A.\ Javanmard, \& A.\ Mehrabian}, \xref{Wei, Tang, and Dan}{2014a}{Er Ling Wei, Wen Tang, \& Dong Ye}, \xref{Schubert and Steffen}{2015a}{Michael Schubert \& Eckhard Steffen}, \xref{DeVos, Li, Lu, \etal}{2021a}{Matt DeVos, Jiaao Li, You Lu, Rong Luo, Cun-Quan Zhang, \& Zhang Zhang}.  See \xref{Jensen and Toft}{1995a}{Tommy R.\ Jensen \& Bjarne Toft} for other contributions.]  
\annotctd{} A topological application is outlined.
\dated{ca.\ 1983}
\par
\codes{\CSG:~\CMatrd,~\COri,~\CFlows,~\CAppl(\CTop)}%
\itemspace

\author{Jean-Marie Bourjolly} 						%BOURJOLLY
\authorspace
\bsgitem{1988a}{}  An extension of the K\"onig--Egerv\'ary property to node-weighted bidirected {\linebreak} graphs.  \emph{Math.\ Programming} 41 (1988), 375--384.  
\MR{0955213} (90c:05161).  \Zbl{653.90083}.
\annot{} [See \xref{Sewell}{1996a}{E.C.\ Sewell}.]
	
\codes{\Csg:~\COri,~\CGG:~\CAlgor}%
\itemspace

\author{J.-M.\ Bourjolly, P.L.\ Hammer, \& B.\ Simeone} 
\authorspace
\bsgitem{1984a}{}  Node-weighted graphs having the K\"onig--Egerv\'ary property.  
Mathematical Programming at Oberwolfach II (Oberwolfach, 1983).  \emph{Math.\ Programming Stud.}\ 22 (1984), 44--63.
\MR{0774233} (86d:05099).  \Zbl{558.05054}.
\codes{\Cpar:~\Cori}%
\itemspace

\author{Jean-Marie Bourjolly \& William R.\ Pulleyblank}
\authorspace
\bsgitem{1989a}{}  K\"onig--Egerv\'ary graphs, 2-bicritical graphs and fractional matchings.  \emph{Discrete Appl.\ Math.}\ 24 (1989), 63--82.  
\MR{1011263} (90m:05069).  \Zbl{684.05036}.
\annot{} [It is hard to escape the feeling that we are dealing with all-negative signed graphs and that something here will generalize to other signed graphs.  Especially see Thm.\ 5.1.  Consult the references for related work.]  
\codes{\CPar;~\CRef}%
\itemspace

\author{Imad Eddine Bousbaa}						%BOUSBAA
\authorspace
\authref  See \hyperlink{Hacene Belbachir}{H.~Belbachir}.
\itemspace

\authorx{Nathan Bowler}						%BOWLER
\author{Nathan Bowler, Daryl Funk, \& Daniel Slilaty}
\authorspace
\bsgitem{2020a}{}  Describing quasi-graphic matroids.  \emph{European J.\ Combin.}\ 85 (2020), art.\ 103062, 26 pp.
\MR{4037634}.  \Zbl{1433.05066}.  \arXiv{1808.00489}.				
\annot{} \Cf\ \xref{Geelen--Gerards--Whittle}{2018a}{Jim Geelen, Bert Gerards, \& Geoff Whittle}.
	
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Garry S.\ Bowlin}						%BOWLIN
\authorspace
\bsgitem{2009a}{}  \emph{Maximum Frustration of Bipartite Signed Graphs}.  Doctoral dissertation, Binghamton University, 2009. 
\MR{2713583} (no rev).  
\annot{} Strong results on structure, bounds, and asymptotics of the generalized Gale--Berlekamp switching game, i.e., maximum frustration of a signed $K_{r,s}$ (\cf\ \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}), by a linear programming method.  Improves on \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer} (\qv), \xref{Gordon and Witsenhausen}{1972a}{Y.\ Gordon \& H.S.\ Witsenhausen}, \xref{Sol\'e and Zaslavsky}{1994a}{Patrick Sole \& Thomas Zaslavsky}. 
\dated{9 Sept 2010, 30 Oct 2011}
\codes{\CSG:~\CFr:~\CGeom}%
\itemspace

\bsgitem{2012a}{}  Maximum frustration in bipartite signed graphs.  \emph{Electronic J.\ Combin.}\ 19 (2012), no.\ 4, art.\ P10, 13 pp.
\MR{3001647}.  \Zbl{1266.05045}.  
\annot{} Maximum frustration is $l_{\max}(\Gamma) := \max_\sigma l(\Gamma,\sigma)$.  Thm.\ 27:  $l_{\max}(K_{l,r} = \frac12 lr \big( 1-2^{-(l-1)}\binom{l-1}{\lfloor(l-2)/2\rfloor} \big).$  It is attained uniquely if $2^{l-1}|r$ and not at all otherwise.  
\annotctd{} Thm.\ 31: $l_{\max}(K_{5,r}) = \lfloor\frac{25}{16}r\rfloor - \epsilon_r$ where $\epsilon_r\in\{0,1\}$, $=1$ iff $r \equiv 2,4,9,13 \mod 16$.  Thm.\ 33: $l_{\max}(K_{6,r}) = \lfloor\frac{66}{32}r\rfloor - \epsilon_r$ where $\epsilon_r\in\{0,1,2\}$ and depends on $r \mod 32$ if $r>6$.  Thm.\ 33: $l_{\max}(K_{7,r}) = \lfloor\frac{154}{64}r\rfloor - \epsilon_r$ where $\epsilon_r\in\{0,1\}$ and depends on $r \mod 64$ if $r>49$.   
\Question.  Is $\epsilon_r$ for fixed $l$ bounded by a linear function of $l$?  
\annotctd{} \Cf\ \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}.
\dated{21 Dec 2014}
\par
\codes{\CSG:~\CFr:~\CGeom}%
\itemspace

\author{Garry Bowlin \& Matthew G.\ Brin}
\authorspace
\bsgitem{2013a}{}  Coloring planar graphs via colored paths in the associahedra.  \emph{Int.\ J.\ Algebra Computation} 23 (2013), no.\ 6, 1337--1418.
\MR{3109450}.  \Zbl{1273.05060}.  \arXiv{1301.3984}.				
\par
\codes{\CSG:~\CBal}%
\itemspace

\author{John Paul Boyd} 						%BOYD
\authorspace
\bsgitem{1969a}{}  The algebra of group kinship.  \emph{J.\ Math.\ Psychology} 6 (1967), 139--167.  
Repr.\ in:  Samuel Leinhardt, ed., \emph{Social Networks:  A Developing Paradigm}, pp.\ 319--346.  Academic Press, New York, 1977.  
\Zbl{172.45501} (172, p.\ 455a).   
Erratum.  \emph{Ibid.}\ 9 (1972), 339.  
\Zbl{242.92010}.   
\codes{\CSG:~\CBal}%
\itemspace

\author{S.C.\ Boyd} 							%BOYD
\authorspace
\authref  See \hyperlink{C.\ Benzaken}{C.~Benzaken}.
\itemspace

\author{Durmus Bozkurt}						%BOZKURT
\authorspace
\authref  See \hyperlink{S.\ Burcu Bozkurt}{S.B.~Bozkurt}.
\itemspace

\authorx{S.\ Burcu Bozkurt}						%BOZKURT
\author{S.\ Burcu Bozkurt \& Durmus Bozkurt}
\authorspace
\bsgitem{2013a}{}  On the signless Laplacian spectral radius of digraphs.  \emph{Ars Combin.}\ 108 (2013), 193--200.
\MR{3060265}.  \Zbl{1289.05270}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{John Bramsen}						%BRAMSEN
\authorspace
\bsgitem{2002a}{}  Further algebraic results in the theory of balance.  \emph{J.\ Math.\ Sociology} 26 (2002), 309--319.
\Zbl{1014.05041}.  
\annot{} Algorithmic ideas for estimating $l(\Sigma)$.  Remarks on clusterability.  
\par
\codes{\CSH}\codesctd{\CSG:~\CFr:~\CAlgor;~\CClu}%
\itemspace

\author{Franz J.\ Brandenburg}						%BRANDENBURG
\authorspace
\bsgitem{2002a}{}  Cycles in generalized networks.  
In:  Lud\v{e}k Ku\v{c}era, ed., \emph{Graph-Theoretic Concepts in Computer Science} (28th Int.\ Workshop, WG 2002, \v{C}esk\'y Krumlov, Czech Rep., 2002), pp.\ 47--56.  \LNCS 2573.  Springer, Berlin, 2002. 
\MR{2062357} (no rev).  \Zbl{1022.90035}.					
\annot{} The effects of gainy and lossy cycles and negative cycles on cheapest flow from source or between two nodes.
\dated{21 Mar 2011} 
\codes{\CGN:~\CAlgor}%
\itemspace

\bsgitem{2003a}{}  Erratum: ``Cycles in generalized networks''.  
In:  Hans L.\ Bodlaender, ed., \emph{Graph-Theoretic Concepts in Computer Science} (29th Int.\ Workshop, WG 2003, Elspeet, The Netherlands, 2003), p.\ 383.  \LNCS 2880.  Springer-Verlag, Berlin, 2003. 
\MR{2080096} (no rev).  \Zbl{1255.90119}.					
\annot{} Results in \xrefd{2002a}{Franz J.\ Brandenburg} on cheapest flow from source are incorrect.
\dated{21 Mar 2011} 
\codes{\CGN:~\CAlgor}%
\itemspace

\author{Franz J.\ Brandenburg \& Mao-Cheng Cai}
\authorspace
\bsgitem{2009a}{}  Shortest path and maximum flow problems in networks with additive losses and gains.  In:  X.\ Deng, J.E.\ Hopcroft, and J.\ Xue, eds.,  \emph{Frontiers in Algorithmics: Third International Workshop} (FAW 2009, Hefei, China), pp.\ 4--15.  \LNCS 5598.  Springer-Verlag, Berlin, 2009.
\Zbl{1248.68211} (no rev).  
\annot{} See \xrefd{2011a}{Franz J.\ Brandenburg \& Mao-Cheng Cai}.
\codes{\Cgg:~\Cincid:~\CAlgor,~\Cmatrd}%
\itemspace

\bsgitem{2011a}{}  Shortest path and maximum flow problems in networks with additive losses and gains.  \emph{Theor.\ Computer Sci.}\ 412 (2011), no.\ 4-5, 391--401.
\MR{2778472} (2011k:68052).  \Zbl{1230.90045}.					
\annot{} Additive real gains.  The lift matroid is implicit.  Contrasts algorithmic complexity of additive with multiplicative gains.
\dated{30 May 2012} 
\par
\codes{\Cgg:~\Cincid:~\CAlgor,~\Cmatrd}%
\itemspace

\authorx{Benjamin Braun}						%BRAUN
\author{Benjamin Braun \& Sarah Crown Rundell}	
\authorspace
\bsgitem{2014a}{}  Hyperoctahedral Eulerian idempotents, Hodge decompositions, and signed {\linebreak} graph coloring complexes.  \emph{ Electronic J.\ Combin.}\ 21 (2014), no.\ 2, Paper 2.35, 21 pp.
\MR{3244801}.  \Zbl{1300.05091}.  \arXiv{1307.7323}.				
\codes{\CSG:~\CCol}%
\itemspace

\author{A.J.\ Bray}				 			%BRAY
\authorspace
\authref  See also \hyperlink{G.J.\ Rodgers}{G.J.~Rodgers}.
\authrefspace

\author{A.J.\ Bray, M.A.\ Moore, \& P.\ Reed}
\authorspace
\bsgitem{1978a}{}  Vanishing of the Edwards-Anderson order parameter in two- and three-dimen\-sion\-al Ising spin glasses.  
\emph{J.\ Phys.\ C:  Solid State Phys.}\ 11 (1978), 1187--1202.  
\annot{} Random edge signs on a hypercubic lattice.	
\dated{12 Aug 2012} 
\par
\codes{\CPhys:~\CSG:~\CRand,~\CFr}%
\itemspace

\authorx{Richard C.\ Brewster}						%BREWSTER
\author{Richard C.\ Brewster, Florent Foucaud, Pavol Hell, \& Reza Naserasr}
\authorspace
\bsgitem{2017a}{}  The complexity of signed graph and edge-coloured graph homomorphisms.  \emph{Discrete Math.}\ 340 (2017), 223--235.
\MR{3578819}.  \Zbl{1351.05099}.  \arXiv{1510.05502}.  \HAL{hal-01520743}.				
\codes{\CSG,~\CSGc:~\CHom,~\CAlgor}%
\itemspace

\author{Richard C.\ Brewster \& Timothy Graves}
\authorspace
\bsgitem{2009a}{}  Edge-switching homomorphisms of edge-coloured graphs.  \emph{Discrete Math.}\ 309 (2009), 5540--5546.
\MR{2567956} (2010k:05078).  \Zbl{1213.05066}.  
\par
\codes{\Cgg:~\CSw,~\CHom}%
\itemspace

\author{Richard C.\ Brewster, Arnott Kidner, \& Gary MacGillivray}
\authorspace
\bsgitem{20xxa}{}  $k$-colouring $(m, n)$-mixed graphs with switching.  Submitted.  \emph{}\ 
\arXiv{2203.08070}.  
\codes{\Cgg:~\CSw,~\CCol}%
\itemspace

\author{Richard C.\ Brewster \& Mark Siggers}
\authorspace
\bsgitem{2018a}{}  A complexity dichotomy for signed $\mathbf{H}$-colouring.  \emph{Discrete Math.}\ 341 (2018), 2768--2773.
\MR{3843264}.  \Zbl{1393.05108}.  
\codes{\CSG:~\CHom:~\CAlgor}%
\itemspace

\author{Matthew G.\ Brin}						%BRIN
\authorspace
\authref  See \hyperlink{Garry S.\ Bowlin}{G.S.~Bowlin}.
\itemspace

\authorx{Thomas Britz}						%BRITZ
\author{T.\ Britz, D.D.\ Olesky, \& P.\ Van Den Driessche}
\authorspace
\bsgitem{2004a}{}  Matrix inversion and digraphs: the one factor case.  \emph{Electronic J.\ Linear Algebra}\ 11 (2004), 115--131.
\MR{2111518} (2005m:15008).  \Zbl{1063.05089}.  
\codes{\Csd:~\CQM}%
\itemspace

\author{Hajo Broersma}						%BROERSMA
\authorspace
\authref  See \hyperlink{Dan Hu}{D.~Hu}.
\itemspace

\authorx{Jared C.\ Bronski}						%BRONSKI
\author{Jared C.\ Bronski \& Lee DeVille}
\authorspace
\bsgitem{2014a}{}  Spectral theory for dynamics on graphs containing attractive and repulsive interactions.  \emph{SIAM J.\ Appl.\ Math.}\ 74 (2014), no.\ 1, 83--105.
\MR{3158797}.  \Zbl{1332.05086}.  \arXiv{1303.0718}.				
\annot{} Bounds on the positive index of inertia, $n_+$, of a weighted graph, in terms of edge signs.
\dated{20 Mar 2016} 
\codes{\CSG,~\CWG:~\CAdj:~\CEig}%
\itemspace

\author{Jared C.\ Bronski, Lee Deville, \& Paulina Koutsaki}
\authorspace
\bsgitem{2015a}{}  The spectral index of signed Laplacians and their structural stability.  Manuscript, 2015.
\arXiv{1503.01069}.  
\annot{} How the positive index of inertia of fixed $\Sigma$ varies with edge weights.
\dated{20 Mar 2016} 
\codes{\CSG,~\CWG:~\CAdj:~\CEig}%
\itemspace

\authorx{Peter Brooksbank}						%BROOKSBANK
\author{Peter Brooksbank, Hongxung Qin, Edmund Robertson, \& \'Akos Seress}
\authorx{Peter Brooksbank, Hongxung Qin, Edmund Robertson, \& Akos Seress}
\authorspace
\bsgitem{2004a}{}  On Dowling geometries of infinite groups.  \emph{J.\ Combin.\ Theory Ser.\ A} 108 (2004), no.\ 1, 155--158.
\MR{2087311} (2005e:51014).  \Zbl{1056.51011}.						
\annot{} Solution of \xref{Bonin}{1996a}{Joseph E.\ Bonin}.  They produce a finite gain graph that has gains in no finite group.  Dictionary:  ``Dowling geometry'' = frame matroid of a gain graph [not an actual Dowling geometry, which would be impossible since a Dowling geometry determines its group; \cf\ \xref{Dowling}{1973b}{Thomas A.\ Dowling}].
\codes{\Cgg:~\CMatrd}%
\itemspace

\authorx{Andries E.\ Brouwer}						%BROUWER AE
\author{A.E.\ Brouwer, A.M.\ Cohen, \& A.\ Neumaier}
\authorspace
\bsgitem{1989a}{}  \emph{Distance-Regular Graphs}.  Ergeb.\ Math., Third Ser., Vol.\ 18.  Springer-Verlag, Berlin, 1989.
\MR{1002568} (90e:05001).  \Zbl{747.05073}.					
\annot{} \S1.5, ``Taylor graphs and regular two-graphs'':  Signed complete graphs appear in the form of double covers of the complete graph.  \S3.8, ``Graph switching, equiangular lines, and representations of two-graphs''.  \S7.6C, ``2-Transitive regular two-graphs''.
\codes{\CTG:~\Ckg,~\CGeom:~\CExp,~\CRef}%
\itemspace

\author{Andries E.\ Brouwer \& Willem H.\ Haemers}
\authorspace
\bsgitem{2012a}{}  \emph{Spectra of Graphs}.  Universitext.  Springer-Verlag, Berlin, 2012.
\MR{2882891}.  \Zbl{1231.05001}.						
\annot{} \S1.1, ``Matrices associated to a graph'': ``Laplace matrix'' = Laplacian matrix $\Lap(+\Gamma)$, from the ``directed [i.e., oriented] incidence matrix'' $\Eta(+\Gamma)$.  ``Signless Laplace matrix'' = Laplacian matrix $\Lap(-\Gamma)$, from the ``(undirected) [unoriented] incidence matrix'' $\Eta(-\Gamma)$ (with no $-1$s).  Many results employ $\Lap(-\Gamma)$, but signed graphs are ignored; e.g., see \S\S1.4.5, 14.4.3, ``Line graphs'' [\cf\ G.R.\ Vijayakumar \etal].  
\S1.8.2, ``Seidel switching'', defines the Seidel adjacency matrix $A(K_\Gamma)$ and its switching.  Ch.~10, ``Regular two-graphs''.
\annot{} $\Lap(-\Gamma)$ appears in:  Ch.\ 3: ``Eigenvalues and Eigenvectors of Graphs'',  \S15.3:  ``Other matrices with at most three eigenvalues''.  \S15.3.1: ``Few Seidel eigenvalues''; \S15.3.3: ``Three signless Laplace eigenvalues''.
\dated{19 Sept 2010, 23 Jan 2012} 
\codes{\CPar:~\CEig,~\Cincid,~\CTG,~\Csw}%
\itemspace

\authorx{Floor Brouwer}		 				%BROUWER F
\author{Floor Brouwer \& Peter Nijkamp}
\authorspace
\bsgitem{1983a}{}  Qualitative structure analysis of complex systems.  
In:  P.\ Nijkamp, H.\ Leitner, and N.\ Wrigley, eds., \emph{Measuring the Unmeasurable}, pp.\ 509--530.  Martinus Nijhoff, The Hague, 1983.  
\codes{\CQM,~\CSD:~\CSol,~\CSta:~\CExp}%
\itemspace

\authorx{Edward M.\ Brown}						%BROWN EM
\author{Edward M.\ Brown \& Robert Messer}
\authorspace
\bsgitem{1979a}{}  The classification of two-dimensional manifolds.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 255 (1979), 377--402.  
\MR{0542887} (80j:57007).  \Zbl{391.57010}, (\Zbl{414.57003}).
\annot{} Their ``signed graph'' we might call a type of Eulerian partially bidirected graph.  That is, some edge ends are oriented (hence ``partially bidirected''), and every vertex has even degree and at each vertex equally many edge ends point in and out (``Eulerian'').  More specially, at each vertex all or none of the edge ends are oriented.
\codes{\Csg:~\Cori:~\Cgen:~\CAppl}%
\itemspace

\authorx{Gerald G.\ Brown}		 				%BROWN GG
\author{Gerald G.\ Brown \& Richard D.\ McBride}
\authorspace
\bsgitem{1984a}{}  Solving generalized networks.  \emph{Management Sci.}\ 30 (1984), 1497--1523.  
\MR{0878883} (no rev).  \Zbl{554.90032}.
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{Gerald G.\ Brown, Richard D.\ McBride, \& R.\ Kevin Wood}
\authorspace
\bsgitem{1985a}{}  Extracting embedded generalized networks from linear programming problems.  \emph{Math.\ Programming} 32 (1985), no.\ 1, 11--31.
\MR{0787741} (86f:90090).  \Zbl{574.90060}.					
\annot{} Identifying largest embedded generalized network matrices (i.e., incidence matrices of real multiplicative gain graphs) in a matrix is NP-complete.  Heuristic algorithms for finding such embedded matrices and using them to speed up linear programming.
\dated{2 Oct 2009} 
\par
\codes{\CGN:~\CInc:~\CAlgor}%
\itemspace

\authorx{John Brown}						%BROWN J
\author{John Brown, Chris Godsil, Devlin Mallory, Abigail Raz, \& Christino Tamon}
\authorspace
\bsgitem{2013a}{}  Perfect state transfer on signed graphs.  \emph{Quantum Information Comput.}\ 13 (2013), no.\ 5-6, 511--530.
\MR{3076338}.  \arXiv{1211.0505}.  
\codes{\CSG:~\CCov}%
\itemspace

\authorx{Kenneth S.\ Brown}						%BROWN K
\author{Kenneth S.\ Brown \& Persi Diaconis}
\authorspace
\bsgitem{1998a}{}  Random walks and hyperplane arrangements.  \emph{Ann.\ Probab.}\ 26 (1998), 1813--1854.
\MR{1675083} (2000k:60138).  \Zbl{938.60064}.  
\annot{} The real hyperplane arrangement representing $-K_n$ is studied in \S3D.  It leads to a random walk on threshold graphs.
\codes{\Cpar:~\CGeom}%
\itemspace

\author{Thomas A.\ Brown} 						%BROWN TA
\authorspace
\authref  See also \hyperlink{Fred S.\ Roberts \& Thomas A.\ Brown}{F.S.~Roberts}.
\authrefspace

\author{T.A.\ Brown, F.S.\ Roberts, \& J.\ Spencer} 
\authorspace
\bsgitem{1972a}{} Pulse processes on signed digraphs: a tool for analyzing energy demand.  Rep.\ R-926-NSF, Rand Corp., Santa Monica, Calif., March, 1972.
\codes{\CSDw}%
\itemspace

\author{Thomas A.\ Brown \& Joel H.\ Spencer} 
\authorspace
\bsgitem{1971a}{}  Minimization of $\pm 1$ matrices under line shifts.  \emph{Colloq.\ Math.}\ 23 (1971), 165--171.  
\MR{0307944} (46 \#7059).  \Zbl{222.05016}.
\annot{} Asymptotic estimates for the Gale--Berlekamp switching game, i.e., $l(K_{r,s})$, the maximum frustration index of signatures of $K_{r,s}$.  [Improved by \xref{Gordon and Witsenhausen}{1972a}{Y.\ Gordon \& H.S.\ Witsenhausen} and \xref{Bowlin}{2009a}{Garry S.\ Bowlin}, \xrefd{2012a}{Garry S.\ Bowlin}.]  Also, exact values stated for $r \leq 4$ [extended by \xref{Sol\'e and Zaslavsky}{1994a}{Patrick Sole \& Thomas Zaslavsky} to $r=5$, which was corrected and generalized by \xref{Bowlin}{2009a}{Garry S.\ Bowlin}, \xrefd{2012a}{Garry S.\ Bowlin}].  
[\Cf\ also \xref{Fishburn and Sloane}{1989a}{P.C.\ Fishburn \& N.J.A.\ Sloane}, \xref{Carlson and Stolarski}{2004a}{Jordan Carlson \& Daniel Stolarski}, and \xref{Roth and Viswanathan}{2007a}{Ron M.\ Roth \& Krishnamurthy Viswanathan}, \xrefd{2008a}{Ron M.\ Roth \& Krishnamurthy Viswanathan} on Berlekamp's game, where $r=s$.]
\codes{\Csg:~\CFr}%
\itemspace

\author{William G.\ Brown, ed.}						%BROWN WG
\authorspace
\bsgitem{1980a}{}  \emph{Reviews in Graph Theory}.  4 vols.  American Math.\ Soc., Providence, R.I., 1980.  
\Zbl{538.05001}.  
\annot{} See esp.:  \S208: ``Signed graphs ($+$ or $-$ on each edge), balance'' (undirected and directed), Vol.\ 1, pp.\ 569--571.
\codes{\CSG,~\CSD}%
\itemspace

\author{Richard A.\ Brualdi}						%BRUALDI
\authorspace
\bsgitem{1976a}{}  Combinatorial properties of symmetric non-negative matrices.  In:  \emph{Colloquio Internazionale sulle Teorie Combinatorie} (Rome, 1973), Tomo II, pp.\ 99--120.  Atti dei Convegni Lincei, No.\ 17.  Accademia Nazionale dei Lincei, Rome, 1976.
\MR{0485437} (58 \#5275).  \Zbl{358.05013}.						
\annot{} Thm.\ 8.2.1 of \xrefd{2006a}{Richard A.\ Brualdi}.
\dated{13 Oct 2012} 
\codes{\Csg:~\Cpar:~\CAdj}%
\itemspace

\bsgitem{2006a}{}  \emph{Combinatorial Matrix Classes}.  Encyc.\ Math.\ Appl., Vol.\ 108.  Cambridge Univ.\ Press, Cambridge, Eng., 2006.
\MR{2266203} (2007k:05038).  \Zbl{1106.05001}.  
\annot{} \S8.2, ``Symmetric transportation polytopes'':  The vertices of the polytope of symmetric, non-negative matrices with given line sums (Thm.\ 8.2.1, due to \xref{Brualdi}{1976a}{Richard A.\ Brualdi}, \xref{Converse and Katz}{1975a}{George Converse \& M.\ Katz}, \xref{Lewin}{1977a}{Mordechai Lewin}) or bounded line sums (Thms.\ 8.2.6--8) correspond to the independent sets in the frame matroid $\bfF(-K_n)$.  
[\Problem.  Generalize to a polytope whose vertices are associated with independent sets in $\bfF(\pm K_n)$.  Possibly, the matrices have prescribed entry signs determining a signed graph.]
\dated{13 Oct 2012} 
\codes{\Csg:~\Cpar:~\CAdj}%
\itemspace

\bsgitem{2011a}{}  \emph{The Mutually Beneficial Relationship of Graphs and Matrices}.  CBMS Reg.\ Conf.\ Ser.\ Math., No.\ 115.  American Math.\ Soc., Providence, R.I., 2011 
\MR{2808017} (2012i:05159).  \Zbl{1218.05002}.					
\annot{} \S6.1, ``Sign-nonsingular matrices'':  Signed digraphs, called ``weighted digraphs'' of $(0,\pm1)$-matrices such that every matrix with that sign pattern is nonsingular.  \Cf\ esp.\ Maybee \etal, van den Driessche \etal
\dated{20 Nov 2011}
\codes{\CQM:~\CSol:~\Csd:~\CExp}%
\annotctd{} \S9.4, ``ASM patterns'':  Signed graphs appear in the study of patterns in alternating sign matrices.  \Cf\ \xref{Brualdi, Kiernan, \etal{}}{2013a}{Richard A.\ Brualdi, Kathleen P.\ Kiernan, Seth A.\ Meyer, \& Michael W.\ Schroeder}.
\dated{18 Nov 2011} 
\codes{\CSG:~\CExp}%
\itemspace

\author{Richard A.\ Brualdi, Kathleen P.\ Kiernan, Seth A.\ Meyer, \& Michael W.\ Schroeder}
\authorspace
\bsgitem{2013a}{}  Patterns of alternating sign matrices.  \emph{Linear Algebra Appl.}\ 438 (2013), no.\ 10, 3967--3990.
\MR{3034511}.  \Zbl{1281.15034}.  \arXiv{1104.4075}.				
\codes{\CSG}%
\itemspace

\author{Richard A.\ Brualdi \& Nancy Ann Neudauer}
\authorspace
\bsgitem{1997a}{}  The minimal presentations of a bicircular matroid.  \emph{Quart.\ J.\ Math.\ Oxford} (2) 48 (1997), 17--26. 
\MR{1439695} (97m:05065).  \Zbl{938.05023}.					
\annot{} Minimal transversal presentations of $\bfF(\Gamma,\eset)$, given $\Gamma$.
\codes{\CBic}%
\itemspace

\author{Richard A.\ Brualdi \& Herbert J.\ Ryser}
\authorspace
\bsgitem{1991a}{}  \emph{Combinatorial Matrix Theory}.  Encycl.\ Math.\ Appl., Vol.\ 39.  Cambridge Univ.\ Press, Cambridge, Eng., 1991.
\MR{1130611} (93a:05087).  \Zbl{746.05002}.					
\annot{} See \S7.5. 
\codes{\CQM:~\CSol,~\CSD,~\Cbal}\codesctd{\CExp,~\CRef}%
\itemspace

\author{Richard A.\ Brualdi \& Bryan L.\ Shader} 
\authorspace
\bsgitem{1991a}{}  On sign-nonsingular matrices and the conversion of the permanent into the determinant.  
In:  Peter Gritzman and Bernd Sturmfels, eds., \emph{Applied Geometry and Discrete Mathematics:  The Victor Klee Festschrift}, pp.\ 117--134.  DIMACS Ser.\ Discrete Math.\ Theor.\ Comput.\ Sci., Vol.\ 4.  American Math.\ Soc., Providence, R.I., 1991.
\MR{1116343} (92f:15003).  \Zbl{742.15001}.					
\annot{} \S1 reviews \xref{Seymour and Thomassen}{1987a}{Paul Seymour \& Carsten Thomassen}.  
Thm.\ 2.1:  If two sign-nonsingular $(0,1,-1)$-matrices have the same $0$'s (and total support), their signed digraphs are switching equivalent.  
\dated{12 Jun 2012} 
\par
\codes{\CQM,~\CSD:~\CSol:~\CExp}%
\itemspace

\bsgitem{1995a}{}  \emph{Matrices of Sign-Solvable Linear Systems}.  
Cambridge Tracts in Math., Vol.\ 116.  Cambridge Univ.\ Press, Cambridge, Eng., 1995.  
\MR{1358133} (97k:15001).  \Zbl{833.15002}.
\annot{} Innumerable results and references on signed digraphs are contained herein.
\codes{\CQM,~\CSD:~\CSol,~\CSta}\codesctd{\CExp,~\CRef,~\CAlgor}%
\itemspace

\author{Frank J.\ Bruggeman}						%BRUGGEMAN FJ
\authorspace
\authref  See \hyperlink{Boris N.\ Kholodenko}{B.N.~Kholodenko}.
\itemspace

\author{Jeroen Bruggeman}						%BRUGGEMAN J
\authorspace
\authref  See \hyperlink{Vincent Antonio Traag}{V.A.~Traag}.
\itemspace

\author{Michael Brundage}						%BRUNDAGE
\authorspace
\bsgitem{1996a}{}  \emph{From the Even-Cycle Mystery to the $L$-Matrix Problem and Beyond}.  M.S.\ thesis, Dept.\ of Mathematics, University of Washington, Seattle, 1996.  
URL (10/1997) \url{http://www.math.washington.edu/~brundage/evcy/}
\annot{} A concise expository survey.  Ch.\ 1: ``Even cycles in directed graphs''.  Ch.\ 2: ``$L$-matrices and sign-solvability'', esp.\ \S\ ``Signed digraphs''.  Ch.\ 3: ``Beyond'', esp.\ \S\ ``Balanced labellings'' (vertices labelled from the set $\{0, +1, -1\}$ so that from each vertex labelled $\epsilon \neq 0$ there is an arc to a vertex labelled $-\epsilon$) and \S\ ``Pfaffian orientations''.
\par
\codes{\CSD,~\CPar:~\CCircles,~\CSol,~\CAlgor,~\CVS:~\CExp,~\CRef}%
\itemspace

\author{Maurizio Brunetti}						%BRUNETTI
\authorspace
\authref  See also \hyperlink{Abdullah Alazemi}{A.~Alazemi}, \hyperlink{Francesco Belardo}{F.~Belardo}, and \hyperlink{Jianfeng Wang}{J.-F.~Wang}.
\authrefspace

\bsgitem{2020a}{}  Laplacian spectral properties of signed circular caterpillars.\  \emph{Theory Appl.\ Graphs} 7 (2020), no.\ 2, art.\ 1, 23 pp.
\MR{4119753}.  \Zbl{1447.05120}.  
\annot{} Unicycles with pendants that are $K_2$.  For girth 3, extremal spectral radius and first and second Zagreb indices.
\dated{15 Dec 2020} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Maurizio Brunetti, Matteo Cavaleri, \& Alfredo Donno}
\authorspace
\bsgitem{2019a}{}  A lexicographic product for signed graphs.  \emph{Australasian J.\ Combin.}\ 74 (2019), no.\ 2, 332--343.
\MR{3949477}.  \Zbl{1422.05060}.  
\annot{} I.e., composition $\Sigma[\Lambda]$.  Modified from \xref{Hameed and Germina's}{2012b}{Shahul Hameed K \& K.A.\ Germina} definition and said to respect switching (but see \xrefd{2019b}{Maurizio Brunetti, Matteo Cavaleri, \& Alfredo Donno}).  It has good eigenvalue properties.
\dated{20 Apr, 5 Jul 2019} 
\par
\codes{\CSG:~\CBal,~\CSw,~\CEig:~\CAdj,~\CLap}%
\itemspace

\bsgitem{2019b}{}  Erratum to the article [\xrefd{2019a}{Maurizio Brunetti, Matteo Cavaleri, \& Alfredo Donno}].  \emph{Australasian J.\ Combin.}\ 75 (2019), 256--258.
\MR{4008250}.  \Zbl{1429.05178}.  
\annot{} Their definition does not respect switching.  \Problem:  Find a definition that does.
\dated{2 Sept 2019} 
\codes{\CSG:~\CBal,~\CSw}%
\itemspace

\author{Maurizio Brunetti \& Adriana Ciampella}
\authorspace
\bsgitem{2023a}{}  Signed bicyclic graphs with minimal index.  \emph{Commun.\ Combin.\ Optim.}\ 8 (2023), no.\ 1, 207--241.
\MR{4503662}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Maurizio Brunetti \& Zoran Stani\'c}
\authorx{Maurizio Brunetti \& Zoran Stanic}
\bsgitem{2022a}{}  Unbalanced signed graphs with extremal spectral radius or index.  \emph{Computational Appl.\ Math.}\ (2022) 41 (2022), art.\ 118, 13 pp.
\MR{4400622}.  
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Ordering signed graphs with large index.  \emph{Ars Math.\ Contemp.}\ 22 (2022), article P4.05, 14 pp.
\MR{4498576}.  \Zbl{1497.05109}.  
\annot{} Largest max eigenvalue of $A(\Sigma)$ within $\Sigma$ that are connected, connected unbalanced, and signed complete.
\dated{17 Jan 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Michael Brusco}						%BRUSCO
\author{Michael Brusco, Patrick Doreian, Andrej Mrvar, \& Douglas Steinley}
\authorspace
\bsgitem{2011a}{}  Two algorithms for relaxed structural balance partitioning: linking theory, models, and data to understand social network phenomena.  \emph{Sociological Methods Res.}\ 40 (2011), no.\ 1, 57--87.
\MR{2758299} (no rev).  
\codes{\CSG:~\CPsS,~\CStr}%
\itemspace

\authorx{Thomas H.\ Brylawski}
\authorx{Tom Brylawski}
\author{Thomas H.\ Brylawski [Tom Brylawski]}				%BRYLAWSKI
\authorspace
\bsgitem{1975a}{}  A note on Tutte's unimodular representation theorem.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 52 (1975), 499--502.
\MR{0419271} (54 \#7294).  \Zbl{328.05017}.
\annot{} Implicitly, switching in the bipartite gain graph of a matrix.
\codes{\Cgg:~\Csw}%
\itemspace

\bsgitem{2000a}{}  A M\"obius identity arising from modularity in a matroid bilinear form.  \emph{J.\ Combin.\ Theory Ser.\ A}  91 (2000), 622--639.
\MR{1780040} (2002a:05059).  \Zbl{966.05014}.					
\annot{} \S5, ``$q$-Analogs from a M\"obius identity'': \S5.1, ``Dowling lattices'' (an example):  A complicated identity is derived from the M\"obius function of $Q_n(\bbF_q^\times)$.
\dated{26 Dec 2015} 
\codes{\Cgg:~\CMatrd,~\CInvar}%
\itemspace

\author{Thomas Brylawski \& James Oxley}
\authorspace
\bsgitem{1992a}{}  The Tutte polynomial and its applications.  In:  Neil White, ed., \emph{Matroid Applications}, Ch.\ 6, pp.\ 123--225.  Encycl.\ Math.\ Appl., Vol.\ 40.  Cambridge Univ.\ Press, Cambridge, Eng., 1992.
\MR{1165543} (93k:05060).  \Zbl{769.05026}.					
\annot{} \S6.4, ``The critical problem'', \S6.4.B, ``Minimal and tangential blocks'', pp.\ 171--172: Tangential blocks in Dowling geometries $Q_n(\GF(q)^\times)$, after \xref{Whittle}{1989a}{Geoff Whittle}.
\dated{16 Sept 2011}
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{J.A.\ Brzozowski}						%BRZOZOWSKI
\authorspace
\authref  See \hyperlink{C.-J.\ Shi \& J.A.\ Brzozowski}{C.J.~Shi}.
\itemspace

\authorx{Changjiang Bu}						%BU CJ
\author{Changjiang Bu \& Jiang Zhou}
\authorspace
\bsgitem{2012a}{}  Starlike trees whose maximum degree exceed 4 are determined by their Q-spectra.  \emph{Linear Algebra Appl.}\ 436 (2012), 143--151.
\MR{2859918} (2012j:05251).  \Zbl{1242.05160}.					
\annot{} A subdivided star is determined, among all graphs, by $\Spec \Lap(-\Gamma)$.  This completes work of \xref{Omidi}{2009a}{G.R.\ Omidi} and of \xref{Omidi and Vatandoost}{2010a}{Gholam R.\ Omidi \& Ebrahim Vatandoost}.
\dated{28 Nov 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012b}{}  Signless Laplacian spectral characterization of the cones over some regular graphs.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 9, 3634--3641.
\MR{2900741}.  \Zbl{1241.05069}.							
\annot{} 
Let $\Gamma = \Delta \vee K_1$, the join of $\Delta$ and a point.  
$\Spec \Lap(-\Gamma)$ determines $\Gamma$ if:  $\Gamma$ or $\Gamma^c$ is a matching; 2-regular $\Gamma$ has $n\geq11$; 2-regular $\Gamma^c$ is triangle-free.  [This implies graphs with the same $\Spec \Lap(-\Gamma)$.]
\annotctd{} Also, Thm.\ 3.6: $\Spec \Lap(-[K_{1,3}\vee\Gamma]) = \Spec \Lap(-[(C_3\cupdot K_1)\vee\Gamma])$ for any $\Gamma$.
\dated{28 Nov 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{[Miguel \'A.\ Valencia Bucio]}					%BUCIO
\authorspace
\authref  See \hyperlink{Miguel A.\ Valencia Bucio}{M.\'A.~Valencia~Bucio} (under `V').
\itemspace

\authorx{Fred Buckley}					 	%BUCKLEY
\author{Fred Buckley, Lynne L.\ Doty, \& Frank Harary}
\authorspace
\bsgitem{1988a}{}  On graphs with signed inverses.  \emph{Networks} 18 (1988), 151--157.  
\MR{0953918} (89i:05222).  \Zbl{646.05061}.
\annot{} ``Signed invertible graph'' [i.e., sign-invertible graph] = graph $\Gamma$ such that $A(\Gamma)^{-1} = A(\Sigma)$ for some signed graph $\Sigma$.  Finds two classes of such graphs.  Characterizes sign-invertible trees.  [\Cf\ \ \xref{Godsil}{1985a}{C.D.\ Godsil} and, for a different notion, \xref{Greenberg, Lundgren, and Maybee}{1984b}{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}.]
\par
\codes{\CSG:~\CAdj}%
\itemspace

\author{Fred Buckley \& Frank Harary}
\authorspace
\bsgitem{1990a}{}  \emph{Distance in Graphs}.  Addison--Wesley, Redwood City, Calif., 1990.
\MR{1045632} (90m:05002).  \Zbl{688.05017}.					
\annot{} Signed graphs and sign-invertible graphs (\xref{Buckley, Doty, and Harary}{1988a}{Fred Buckley, Lynne L.\ Doty, \& Frank Harary}): pp.\ 120--122.
\codes{\CSG:~\CAdj:~\CExp}%
\itemspace

\author{N.\ Buckley}							%BUCKLEY
\authorspace
\authref  See \hyperlink{C.\ Perina}{C.~Perina}.
\itemspace

\authorx{Vadim Bugaenko}						%BUGAENKO
\author{Vadim Bugaenko, Yonah Cherniavsky, Tatiana Nagnibeda, \& Robert Shwartz}
\authorspace
\bsgitem{2015a}{}  Weighted Coxeter graphs and generalized geometric representations of Coxeter groups.  \emph{Discrete Appl.\ Math.}\ 192 (2015), 17--27.
\MR{3354815}.  \Zbl{1319.05064}.  \arXiv{1304.0692}.  
\annot{} Weights are complex gains, or are signs.
	
\codes{\CGG,~\CSG:~\CBal}%
\itemspace

\author{Francesco Bullo}						%BULLO
\authorspace
\authref  See \hyperlink{Pedro Cisneros-Velarde}{P.~Cisneros-Velarde}, \hyperlink{Peng Jia}{P.~Jia} and \hyperlink{Wenjun Mei}{W.-J.~Mei}.
\itemspace

\authorx{Yurii Burman}						%BURMAN
\author{Yurii Burman, Andrey Ploskonosov, \& Anastasia Trofimova}
\authorspace
\bsgitem{2015a}{}  Matrix-tree theorems and discrete path integration.  \emph{Linear Algebra Appl.}\ 466 (2015), 64--82.
\MR{3278240}.  \Zbl{1310.15010}.  
\codes{\Cgg:~\CLap}%
\itemspace

\authorx{James R.\ Burns}				 		%BURNS
\author{James R.\ Burns \& Wayland H.\ Winstead}
\authorspace
\bsgitem{1982a}{}  Input and output redundancy.  \emph{IEEE Trans.\ Systems Man Cybernetics} SMC-12 (1982), no.\ 6, 785--793.  
\annot{} \S~IV:  ``The computation of contradictory redundancy.''  Summarized in modified notation:  In a signed graph, define $w^\epsilon_{ij}(r) =$ number of walks of length $r$ and sign $\epsilon$ from $v_i$ to $v_j$.  Define an adjacency matrix $A$ by $a_{ij} = w^+_{ij}(1) + w^-_{ij}(1)\theta$, where $\theta$ is an indeterminate whose square is 1.  Then $w^+_{ij}(r) + w^-_{ij}(r)\theta = (A^r)_{ij}$ for all $r\geq 1$.  [We should regard this computation as taking place in the group ring of the sign group, where the sign group is treated as $\{+1, \theta\}$.  The generalization to arbitrary gain graphs and digraphs is obvious.]  
Other sections also discuss signed digraphs [but have little mathematical content].
\codes{\CSD,~\Cgd:~\CAdj,~\CPaths}%
\itemspace

\author{Eugene Burnstein}						%BURNSTEIN
\authorspace
\authref  See \hyperlink{Robert B.\ Zajonc}{R.B.~Zajonc}.
\itemspace

\author{Martin Burtscher}						%BURTSCHER
\authorspace
\authref  See \hyperlink{Ghadeer Alabandi}{G.~Alabandi}.
\itemspace

\authorx{Arthur H.\ Busch}						%BUSCH
\author{Arthur H.\ Busch, Michael S.\ Jacobson, Timothy Morris, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}
\authorspace
\bsgitem{2013a}{}  Improved sufficient conditions for the existence of anti-directed hamiltonian cycles in digraphs.  \emph{Graphs Combin.}\ 29 (2013), 359--364.
\MR{3053585}.  \Zbl{1267.05159}.  
\annot{} Improvement of \xref{Diwan, Frye, Plantholt, and Tipnis}{2011a}{Ajit A.\ Diwan, Josh B.\ Frye, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}.
\dated{5 Jun 2017} 
\codes{\Cgg:~\CStr}\codesctd{\Csg:~\Cpar:~\COri}%
\itemspace

\author{Arthur Busch, Mohammed A.\ Mutar, \& Daniel Slilaty}
\authorspace
\bsgitem{2022a}{}  Hamilton cycles in bidirected complete graphs.  \emph{Contrib.\ Discrete Math.}\ 17 (2022), no.\ 2, 137--149.
\MR{4531948}.  
\annot{} A coherent cycle generalizes an alternating circle in an edge-2-colored graph and a directed cycle in a directed graph (\cf\ \xref{Bankfalvi and Bankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi}).  Strong connection is defined for a bidirected graph.  Thm.\ 3.3:  An oriented $\pm K_n$ ($n\geq3$) has a cycle of every length $3,\ldots,n$ iff it is strongly connected.  Thm.\ 4.4:  It has a Hamiltonian cycle iff it is strongly connected and has an alternating 2-factor.  \Cf\ \xref{Mutar}{2017a}{Mohammed A.\ Mutar}.
\dated{11 Aug 2018} 
\codes{\CSG:~\COri:~\CCycles}%
\itemspace

\authorx{F.C.\ Bussemaker}						%BUSSEMAKER
\author{F.C.\ Bussemaker, P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov}
\authorspace
\bsgitem{1991a}{}  Tables of signed graphs.
EUT Report 91-WSK-01.  Dept.\ of Math.\ and Computing Sci., Eindhoven University of Technology, Eindhoven, 1991.
\MR{1131079} (92g:05001).  
\codes{\CSG:~\CSw}%
\itemspace

\author{F.C.\ Bussemaker, D.M.\ Cvetkovi\'c, \& J.J.\ Seidel}
\authorx{F.C.\ Bussemaker, D.M.\ Cvetkovic, \& J.J.\ Seidel}
\authorspace
\bsgitem{1976a}{}  Graphs related to exceptional root systems.  
T.H.-Report 76-WSK-05, 91 pp.  Dept.\ of Math., Technological University Eindhoven, Eindhoven, The Netherlands, 1976.
\Zbl{338.05116}.  
\annot{} The 187 simple graphs with eigenvalues $\geq -2$ that are not (negatives of) reduced line graphs of signed graphs are found, with computer aid.  By \xref{Cameron, Goethals, Seidel, and Shult}{1976a}{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}, all are represented by root systems $E_d$, $d =  6,7,8$.  Most interesting is Thm.\ 2: each such graph is Seidel-switching equivalent to a line graph of a graph.  [\Problem.  Explain this within signed graph theory.]
\codes{\CLG:~\Cpar:~\CEig}%
\itemspace

\bsgitem{1978a}{}  Graphs related to exceptional root systems.  
In:  A.\ Hajnal and Vera T.\ S\'os, eds., \emph{Combinatorics} (Proc.\ Fifth Hungar.\  Colloq., Keszthely, 1976), Vol.\ 1, pp.\ 185--191.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 18.  North-Holland, Amsterdam, 1978.
\MR{0519264} (80g:05049).  \Zbl{392.05055}.					
\annot{} Announces the results of \xrefd{1976a}{F.C.\ Bussemaker, D.M.\ Cvetkovic, \& J.J.\ Seidel}.
\codes{\CLG:~\Cpar:~\CEig}%
\itemspace

\author{F.C.\ Bussemaker, R.A.\ Mathon, \& J.J.\ Seidel} 
\authorspace
\bsgitem{1979a}{} Tables of two-graphs.  TH-Report 79-WSK-05.  Dept.\ of Math., Technological University Eindhoven, Eindhoven, The Netherlands, 1979.  
\Zbl{439.05032}.  
\annot{} All $[K_n,\sigma]$, also cospectral pairs, and those that are regular, integral, or vertex-transitive, for $n \leq 9$, plus larger special types.
\dated{29 Dec 2020}
\codes{\CTG,~\CSw,~\CExp}\codesctd{\CTG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{1981a}{}  Tables of two-graphs.  In:  S.B.\ Rao, ed., \emph{Combinatorics and Graph Theory} (Proc.\ Sympos., Calcutta, 1980), pp.\ 70--112.  \LNM 885.  Springer-Verlag, Berlin, 1981.  
\MR{0655610} (84b:05055).  \Zbl{482.05024}.					
\annot{} ``The most important tables from'' \xrefd{1979a}{F.C.\ Bussemaker, R.A.\ Mathon, \& J.J.\ Seidel}. 
\codes{\CTG,~\CSw}\codesctd{\CTG:~\CAdj:~\CEig}%
\itemspace

\author{F.C.\ Bussemaker \& A.\ Neumaier}
\authorspace
\bsgitem{1992a}{}  Exceptional graphs with smallest eigenvalue $-2$ and related problems.  \emph{Math.\ Comput.}\ 59 (1992), 583--608.
\MR{1134718} (93a:05089).  \Zbl{770.05060}.  
\annot{} They are the antibalanced signed graphs with largest eigenvalue $-2$.  Also, largest eigenvalue around $-2$.  Two-graphs and work of \hyperlink{G.R.\ Vijayakumar}{Vijayakumar} \etal\ are mentioned.
\dated{29 Apr 2012} 
\codes{\CTG,~\CLG,~\CEig}%
\itemspace

\author{Steve Butler}							%BUTLER
\authorspace
\bsgitem{2010a}{}  Eigenvalues of $2$-edge coverings.  \emph{Linear Multilinear Algebra} 58 (2010), 413--423.
\MR{2663442} (2011g:05173).  \Zbl{1187.05047}.					
\annot{} Generalizing \xref{D'Amato}{1979a}{Susan S.\ D'Amato} and \xref{Bilu and Linial}{2006a}{Yonatan Bilu \& Nathan Linial}.  The ``signed graph'' $G$ is vertex-signed; it is a branched double cover of a signed graph $H$ whose edge signs are incorporated into weights.  The interesting new idea is the branching, wherein a vertex may be singly covered.  [May the branches correspond to half edges?]  Adjacency and normalized Laplacian spectra of $G$ are each obtained from those of $H$ and a modified $H$.  
\dated{9 Mar 2011} 
\codes{\CVS(\CGen:~\CEig)}\codesctd{\CSG:~\Ccov,~\CEig}%
\itemspace

\author{Steve Butler, Minerva Catral, H.\ Tracy Hall, Leslie Hogben, Xavier Mart\'{\i}nez-Rivera, Bryan Shader, \& Pauline van den Driessche}							%
\authorspace
\bsgitem{2017a}{}  The enhanced principal rank characteristic sequence for Hermitian matrices.  \emph{Electronic J.\ Linear Algebra} 32 (2017), 58--75.
\MR{3627007}.  \Zbl{1375.15057}.  
\annot{} Example:  Hermitian matrices of mixed graphs.  Gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.
\codes{\Cgg:~\CAdj}%
\itemspace

\authorx{Jesper Makholm Byskov}					%BYSKOV
\author{Jesper Makholm Byskov, Bolette Ammitzb{\o}ll Madsen, \& Bjarke Skjernaa}
\authorspace
\bsgitem{2005a}{}  On the number of maximal bipartite subgraphs of a graph.  \emph{J.\ Graph Theory} 48 (2005), no.\ 2, 127--132.
\MR{2110582} (2005h:05099).  \Zbl{1059.05045}.						
\annot{} Bounds on the number of maximal induced bipartite subgraphs.  [\Problem.  Generalize to maximal induced balanced subgraphs, equivalently minimal balancing sets of vertices, especially in a signed graph.]
\par
\codes{\Cpar:~\Cbal}%
\itemspace


\label{LETTER C} \hypertarget{LETTER C}{}	% LETTER  C


\authorx{S.\ Cabasino}						%CABASINO
\author{S.\ Cabasino, E.\ Marinari, P.\ Paolucci, \& G.\ Parisi}
\authorspace
\bsgitem{1988a}{}  Eigenstates and limit cycles in the SK model.  \emph{J.\ Phys.\ A}  21 (1988), no.\ 22, 4201--4210.
\MR{0983779} (89k:82070).  
\codes{\CPhys:~\CSG}%
\itemspace

\author{He Cai}							%CAI H
\authorspace
\authref  See \hyperlink{Ye Jiang}{Y.~Jiang}.
\itemspace

\authorx{Hongyan Cai}						%CAI HY
\author{Hongyan Cai, Qiang Sun, Yuanpei Wang, \& Guangjun Xu}
\authorspace
\bsgitem{20xxa}{}  Edge coloring of the signed Halin graph.  Submitted.  \emph{}\ 
\annot{} Thm.:  $\chi'(H,\sigma) = \Delta(H)$, $\forall\ \sigma$, where $H=$ Halin graph.
\dated{11 Feb 2022} 
\codes{\CSG:~\CECol}%
\itemspace

\author{Hongyan Cai, Qiang Sun, Guangjun Xu, \& Shanshan Zheng}
\authorspace
\bsgitem{2022a}{}  Edge coloring of the signed generalized Petersen graph.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 45 (2022), 647--661.
\MR{4391907}.  \Zbl{1485.05052}.  
\annot{} $\chi'(P(n,1))=\Delta(P(n,1))=3$ for $n\geq5$.  For $n=5,6$, $\chi'(P(n,2))=3$ usually but $=4$ in a few exceptions.
\dated{12 Feb 2022} 
\codes{\CSG:~\CECol}%
\itemspace

\authorx{Leishen Cai}						%CAI LS
\author{Leishen Cai \& Baruch Schieber}
\authorspace
\bsgitem{1997a}{}  A linear-time algorithm for computing the intersection of all odd cycles in a graph.  \emph{Discrete Appl.\ Math.}\ 73 (1997), 27--34.  
\MR{1431105} (97g:05149).  \Zbl{867.05066}.					
\annot{} By the negative-subdivision trick (subdividing each positive edge into two negative ones), the algorithm will find the intersection of all negative circles of a signed graph.
\codes{\CPar,~\Csg:~\CFr,~\CCircles:~\CAlgor}%
\itemspace

\author{Mao-cheng Cai}						%CAI MC
\authorspace
\authref  See  \hyperlink{Franz J.\ Brandenburg \& Mao-Cheng Cai}{F.J.~Brandenburg}.
\itemspace

\authorx{Qing Cai}							%CAI Q
\author{Qing Cai, Maoguo Gong, Lijia Ma, Shanfeng Wang, Licheng Jiao, \& Haifeng Du}
\authorspace
\bsgitem{2015a}{}  A particle swarm optimization approach for handling network social balance problem.  In:  \emph{2015 IEEE Congress on Evolutionary Computation (CEC)} (Sendai, Japan, 2015), pp.\ 3186--3991.
IEEE, 2015.
\annot{} ``Balance problem'' = clustering problem.  Degree of inclusterability.  
	
\annotctd{} As do many ``social balance'' articles, wrongly attributes \xref{Cartwright--Harary}{1956a}{Dorwin Cartwright \& Frank Harary} balance theory to Heider and incorrectly says the former only treats complete graphs.
\dated{27 Nov 2018} 
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Qing Cai, Maoguo Gong, Bo Shen, Lijia Ma, \& Licheng Jiao}
\authorspace
\bsgitem{2014a}{}  Discrete particle swarm optimization for identifying community structures in signed social networks.  \emph{Neural Networks} 58 (2014), 4--13.
\par
\codes{\CSG:~\CClu:~\CAdj,~\CAlgor}%
\itemspace

\authorx{Grant Cairns}						%CAIRNS
\author{Grant Cairns \& Yuri Nikolayevsky}
\authorspace
\bsgitem{2009a}{}  Generalized thrackle drawings of non-bipartite graphs.  \emph{Discrete Comput.\ Geom.}\ 41 (2009), no.\ 1, 119--134.
\MR{2470073} (2010a:05059).  \Zbl{1191.05032}.					
\annot{} Thm.\ 2:  $\Gamma$, connected and not bipartite, has a generalized thrackle drawing in the orientable surface of genus $g$ iff $-\Gamma$ has an orientation embedding in the nonorientable surface with demigenus $2g-1$.  [\Problem.  Generalize to all signed graphs.]
\codes{\Csg:~\CPar:~\CTop}%
\itemspace

\authorx{Tiziana Calamoneri}						%CALAMONERI
\author{Tiziana Calamoneri, Angelo Monti, \& Rossella Petreschi}
\authorspace
\bsgitem{2018a}{}  On dynamic threshold graphs and related classes.  \emph{Theor.\ Computer Sci.}\ 718 (2018), 46--57.
\MR{3775060}.  \Zbl{1388.68212}.  
\annot{} ``Threshold signed graph'' [not a signed graph]: a graph such that $(\exists\ S,T\in\bbR_{>0})(\exists\ a: V \to \bbR)$ $|a(v)|<\min(S,T)$ and $vw \in E \iff |a(v)+a(w)|\geq S$ or $|a(v)-a(w)|\geq T$.  [\Cf\ \xref{Benzaken, Hammer, and de Werra}{1985a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}, \xref{Calamoneri and Petreschi}{2014a}{T.\ Calamoneri \& R.\ Petreschi}.]
\codes{\CVS,~\Csg}%
\annotctd{} Uses the auxiliary signed graph of \xref{Hammer and Mahadev}{1985a}{P.L.\ Hammer \& N.V.R.\ Mahadev}.
\dated{22 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal}%
\itemspace

\author{T.\ Calamoneri \& R.\ Petreschi}
\authorspace
\bsgitem{2013a}{}  Graphs with Dilworth number two are pairwise compatibility graphs.  \emph{Electronic Notes Discrete Math.}\ 44 (2013), 31--38.
\codes{\CVS}%
\itemspace

\bsgitem{2014a}{}  On pairwise compatibility graphs having Dilworth number two.  \emph{Theor.\ Computer Sci.}\ 524 (2014), 34--40.
\MR{3163431} (\emph{q.v.}).  \Zbl{1283.05142}.  
Corrigendum.  \emph{Ibid.}\ 602 (2015), 158--159.
\MR{3399981} (no rev).  \Zbl{1283.05142}.  
\annot{} Defines ``threshold signed graphs''; \cf\ \xref{Calamoneri, Monti, and Petreschi}{2018a}{Tiziana Calamoneri, Angelo Monti, \& Rossella Petreschi}.
\codes{\CVS}%
\itemspace

\author{Kyle David Calderhead}						%CALDERHEAD
\authorspace
\bsgitem{2002a}{}  \emph{Variations on the Slope Problem}.  Doctoral dissertation, University of Minnesota, 2002.
\MR{2703263} (no rev).  
\annot{} Ch.~6, ``Type $B$ analogs'', introduces threshold signed graphs and applies signed graphs to the slopes problem (the minimum number of slopes of $n$ points in the plane) for centrally symmetric points.  A signed graph is threshold if its double cover is a threshold graph.
\codes{\CSG}%
\itemspace

\author{Laurence Calzone}						%CALZONE
\authorspace
\authref  See \hyperlink{Jean-Paul Comet}{J.-P.~Comet}.
\itemspace

\author{Ver\'onica Cambiazo}						%CAMBIAZO
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Peter J.\ Cameron}						%CAMERON
\authorspace
\authref  See also \hyperlink{Laszlo Babai}{L.~Babai} and \hyperlink{F.C.\ Bussemaker}{F.C.~Bussemaker}.
\authrefspace

\bsgitem{1977a}{} Automorphisms and cohomology of switching classes.  \emph{J.\ Combin.\ Theory Ser.\ B} 22 (1977), 297--298.  
\MR{0498227} (58 \#16382).  \Zbl{331.05113}, (\Zbl{344.05128}).
\annot{} The first step towards \xrefd{1977b}{Peter J.\ Cameron}, Thm.\ 3.1.
\codes{\CTG:~\CAut}%
\itemspace

\bsgitem{1977b}{\dag} Cohomological aspects of two-graphs.  \emph{Math.\ Z.}\ 157 (1977), 101--119.  
\MR{0505778} (58 \#21779).  \Zbl{353.20004}, (\Zbl{359.20004}).			
\annot{}Introducing the cohomological theory of two-graphs.  A two-graph $\tau$ is a $2$-coboundary in the complex of $\GF(2)$-cochains on $E(K_n)$.  [The $1$-cochains are the signed complete graphs, equivalently the graphs that are their negative subgraphs.  \Cf\ \xref{D.E.\ Taylor}{1977a}{D.E.\ Taylor}.]  Write $Z_i$, $Z^i$, $B^i$ for the $i$-cycle, $i$-cocycle, and $i$-coboundary spaces.  Switching a signed complete graph means adding a $1$-cocycle to it; a switching class of signed complete graphs is viewed as a coset of $Z^1$ and is equivalent to a two-graph.
\annotctd{} Take a group $\fG$ of automorphisms of $\tau$.  Special cohomology elements $\gamma \in H^1(\fG, B^1)$ and $\beta \in H^2(\fG, \tilde B^0)$ (where $\tilde B^0 = \{0, V(K_n)\}$, the reduced $0$-coboundary group) are defined.  
Thm.\ 3.1:  $\gamma = 0$ iff $\fG$ fixes a graph in $\tau$.  
Thm.\ 5.1:  $\beta = 0$ iff $\fG$ can be realized as an automorphism group of the canonical double covering graph of $\tau$ (viewing $\tau$ as a switching class of signed complete graphs).  
Conditions are explored for the vanishing of $\gamma$ (related to \xref{Harries and Liebeck}{1978a}{David Harries \& Hans Liebeck}) and $\beta$.
\annotctd{} $Z^1$ is the annihilator of $Z_1 =$ the space of even-degree simple graphs; the theorems of \xref{Mallows and Sloane}{1975a}{C.L.\ Mallows \& N.J.A.\ Sloane} follow immediately.  
More generally:  Lemma 8.2: $Z^i$ is the annihilator of $Z_i$.  Thm.\ 8.3: The numbers of isomorphism types of $i$-cycles and $i$-cocycles are equal, for $i = 1, \ldots, n-2$.
\annotctd{} \S8 concludes with discussion of possible generalizations, e.g., to oriented two-graphs (replacing $\GF(2)$ by $\GF(3)^\times$) and double coverings of complete digraphs (Thms.\ 8.6, 8.7).  [\Cf\ \xref{Moorhouse}{1995a}{G.\ Eric Moorhouse}.  A full ternary analog is developed in \xref{Cheng and Wells}{1984a}{Ying Cheng \& Albert L.\ Wells, Jr.}]
\par
\codes{\CTG:~\CSw,~\CAut,~\CEnum,~\CGeom}%
\itemspace

\bsgitem{1979a}{}  Cohomological aspects of $2$-graphs.  II.  
In: C.T.C.\ Wall, ed., \emph{Homological Group Theory} (Proc.\ Sympos., Durham, 1977), Ch.\ 11, pp.\ 241--244.  London Math.\ Soc.\ Lect.\ Note Ser.\ 36.  Cambridge Univ.\ Press, Cambridge, 1979.
\MR{0564428} (81a:05061).  \Zbl{461.20001}.
\annot{} Exposition of parts of \xrefd{1977b}{Peter J.\ Cameron} with a simplified proof of the connection between $\beta$ and $\gamma$.
\codes{\CTG:~\CAut,~\CEnum,~\CGeom,~\CExp}%
\itemspace

\bsgitem{1980a}{}  A note on generalized line graphs.  \emph{J.\ Graph Theory} 4 (1980), 243--245.  
\MR{0570359} (81j:05089).  \Zbl{403.05048}, (\Zbl{427.05039}).			
\annot{} [For generalized line graphs see \xref{Zaslavsky}{1984c}{Thomas Zaslavsky}.]  If two generalized line graphs are isomorphic, their underlying graphs and cocktail-party attachments are isomorphic, with small exceptions related to exceptional isomorphisms and automorphisms of root systems.  The proof, along the lines of \xref{Cameron, Goethals, Seidel, and Shult}{1976a}{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}, employs the canonical vector representation of the underlying signed graph.
\par
\codes{\Csg:~\CLG:~\CAut,~\CGeom}%
\itemspace

\bsgitem{1983a}{}  Automorphism groups of graphs.  In: Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Selected Topics in Graph Theory 2}, Ch.\ 4, pp.\ 89--127.  Academic Press, London, 1983. 
\MR{0797250} (86i:05079).  \Zbl{536.05037}.
\annot{} \S8, ``Switching'':  Graph switching, graph switching classes.  Existence of a ``representative'': a graph in a switching class that has the same automorphism group as the switching class.  
\S9, ``Digraphs'':  Switching classes of tournaments on pp.\ 117--118.  Switching a digraph means reversing all edges between $X \subseteq V$ and $X^c$.
\dated{27 Dec 2010}
\par
\codes{\CTG:~\CSw:~\CAut:~\CExp}%
\itemspace

\bsgitem{1994a}{}  Two-graphs and trees.  Graph Theory and Applications (Proc., Hakone, 1990).  \emph{Discrete Math.}\ 127 (1994), 63--74.  
\MR{1273592} (95f:05027).  \Zbl{802.05042}.
\annot{} Let $T$ be a tree.  Construction 1 (simplifying \xref{Seidel and Tsaranov}{1990a}{J.J.\ Seidel \& S.V.\ Tsaranov}):  Take all triples of edges such that none separates the other two.  This defines a two-graph on $E(T)$ [whose underlying signed complete graph is described by \xref{Tsaranov}{1992a}{S.V.\ Tsaranov}].  Construction 2:  Choose $X \subseteq V(T)$.  Take all triples of end vertices of $T$ whose minimal connecting subtree has its trivalent vertex in $X$.  The two-graphs $(V,\cT)$ that arise from these constructions are characterized by forbidden substructures, namely, the two-graphs of (1) $C_5$ and $C_6$; (2) $C_5$.  Also, trees that yield identical two-graphs are characterized.
\codes{\CTG}%
\itemspace

\bsgitem{1995a}{}  Counting two-graphs related to trees.  \emph{Electronic J.\ Combin.}\ 2 (1995), Research Paper 4.
\MR{1312733} (95j:05112).  \Zbl{810.05031}.					
\annot{} Counting two-graphs of the types constructed in \xrefd{1994a}{Peter J.\ Cameron}.
\codes{\CTG:~\CEnum}%
\itemspace

\bsgitem{2007a}{}  Orbit counting and the Tutte polynomial.  
In: Geoffrey Grimmett and Colin McDiarmid, eds., \emph{Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh}, pp.\ 1--10.  Oxford Lect.\ Ser.\ Math.\ Appl., Vol.\ 34.  Oxford Univ.\ Press, Oxford, 2007.  
\MR{2314558} (2008a:05043).  \Zbl{1122.05022}.
\itemspace

\author{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}
\authorspace
\bsgitem{1976a}{\dag\dag} Line graphs, root systems, and elliptic geometry.  \emph{J.\ Algebra} 43 (1976), 305--327.  
\MR{0441787} (56 \#182).  \Zbl{337.05142}.					
Repr.\ in \xref{Seidel}{1991a}{J.J.\ Seidel}, pp.\ 208--230.
\annot{} The essential idea is that graphs with least eigenvalue $\geq -2$ are represented by the angles of root systems.  It follows that line graphs are so represented.  
[Similarly, signed graphs with largest eigenvalue $\leq 2$ are represented by the inner products of root systems, as in Vijayakumar \emph{et al}.  These include the line graphs of signed graphs as in \xref{Zaslavsky}{1984c}{Thomas Zaslavsky}, since simply signed graphs are represented by $B_n$ or $C_n$ with a few exceptions.
The representation of ordinary graphs by all-negative signed graphs is motivated in \xref{Zaslavsky}{1984c}{Thomas Zaslavsky}.]
\par
\codes{\CLG:~\Csg:~\CEig,~\CGeom,~\CSw}%
\itemspace

\author{Peter J.\ Cameron, Bill Jackson, \& Jason D.\ Rudd}
\authorspace
\bsgitem{2008a}{}  Orbit-counting polynomials for graphs and codes.  \emph{Discrete Math.}\ 308 (2008), 920--930.
\MR{2378927} (2009e:05140).  \Zbl{1133.05030}.					
\codes{\Csg:~\CInvar:~\CFlows}
\itemspace

\author{Peter J.\ Cameron \& Charles R.\ Johnson}
\authorspace
\bsgitem{2006a}{}  The number of equivalence classes of symmetric sign patterns.  Int.\ Workshop Combin., Linear Algebra, Graph Coloring.  \emph{Discrete Math.}\ 306 (2006), no.\ 23, 3074--3077.
\MR{2273136} (2007j:05105).  \Zbl{1105.05034}.					
\annot{} The number of signatures of $K_n^\circ$, the complete graph with loops, under symmetry, switching, and negation (treated as totally nonzero symmetric sign pattern matrices) equals the number of switching isomorphism classes of signed complete graphs.  (\Cf\ \xref{Mallows and Sloane}{1975a}{C.L.\ Mallows \& N.J.A.\ Sloane} and \xref{Cameron}{1979a}{Peter J.\ Cameron}.)
\dated{12 Aug 2012, 16 Nov 2015} 
\par
\codes{\Csg:~\Csw,~\Ctg:~\CAdj:~\CEnum}%
\itemspace

\author{P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov} 
\authorspace
\bsgitem{1994a}{}  Signed graphs, root lattices, and Coxeter groups.  \emph{J.\ Algebra} 164 (1994), 173--209.  
\MR{1268332} (95f:20063).  \Zbl{802.05043}.					
\annot{} A generalized Coxeter group $\Cox(\Sigma)$ and a Tsaranov group $\Ts(\Sigma)$ are defined via Coxeter relations and an extra relation for each negative circle in $\Sigma$.  They generalize Coxeter groups of tree Coxeter graphs and the Tsaranov groups of a two-graph ($|\Sigma| = K_n$; see \xref{Seidel and Tsaranov}{1990a}{J.J.\ Seidel \& S.V.\ Tsaranov}).  
A new operation of ``local switching'' is introduced, which changes the edge set of $\Sigma$ but preserves the associated groups.
\annotctd{} \S2, ``Signed graphs'', proves some well-known properties of switching and reviews interesting data from \xref{Bussemaker, Cameron, Seidel, and Tsaranov}{1991a}{F.C.\ Bussemaker, P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov}.  
\S3, ``Root lattices and Weyl groups'': The ``intersection matrix'' $2I + A(\Sigma)$ is a hyperbolic Gram matrix of a basis of $\bbR^n$ whose vectors form only angles $\pi/2, \pi/3, 2\pi/3$.  To these vectors are associated the lattice $L(\Sigma)$ of their integral linear combinations and the Weyl group $W(\Sigma)$ generated by reflecting along the vectors.  $W$ is finite iff $2I + A(\Sigma)$ is positive definite (Thm.\ 3.1).  \Problem\ 3.6.  Determine which $\Sigma$ have this property.  
\S4 introduces local switching to partially solve \Problem\ 4.1: Which signed graphs generate the same lattice?  Results and some experimental data are reported.  All-negative signed graphs play a special role.  Definition of \emph{local switching at $v$}:  (1) switch so the edges at $v$ are positive, (2) divide the components of the negative subgraph of the neighborhood of $v$ into two halves $J, K$, (3) add negative edges joining all vertices of $J$ to all those of $K$, (4) negate all edges from $v$ to $J$, (5) reverse the switching in step (1).  
[See \xref{Isihara}{2007a}{Toru Ishihara} for more.]
\S6, ``Coxeter groups'': The relationship between the Coxeter and Weyl groups of $\Sigma$.  Cox($\Sigma$) is Cox($|\Sigma|$) with additional relations for antinegative (i.e., negative in $-\Sigma$) induced circles. 
\S7: ``Signed complete graphs''.
\S8: ``Tsaranov groups'' of signed $K_n$'s
\S9: ``Two-graphs arising from trees'' (as in \xref{Seidel and Tsaranov}{1990a}{J.J.\ Seidel \& S.V.\ Tsaranov}).
\annotctd{} Dictionary: 
``$(\Gamma, f)$'' = $\Sigma = (\Gamma, \sigma)$. 
``Fundamental signing'' = all-negative signing, giving the antibalanced switching class.  
``The balance'' of a cycle (i.e., circle) = its sign $\sigma(C)$; ``the parity'' = $\sigma(-C)$ where $-C = C$ with all signs negated.  
``Even'' = positive and ``odd'' = negative (referring to ``parity'').  
``The balance'' of $\Sigma$ = the partition of all circles into positive and negative classes $\cC^+$ and $\cC^-$; this is the bias on $|\Sigma|$ due to the signing and should not be confused with the customary meaning of ``balance'', i.e., all circles are positive.  
\annotctd{} [A more natural definition of the intersection matrix would be $2I - A$.  Then signs would be negative to those in the paper.  The need for ``parity'' would be obviated, ordinary graphs would correspond to all-positive signings (and those would be ``fundamental''), and the extra Coxeter relations would pertain to negative induced circles.]
\par
\codes{\CSG:~\CAdj,~\CEig,~\CGeom,~\CSw(\CGen),~\Clg}%
\itemspace

\author{Peter J.\ Cameron \& Sam Tarzi}
\authorspace
\bsgitem{2004a}{}  Switching with more than two colours.  \emph{European J.\ Combin.}\ 25 (2004), no.\ 2, 169--177.
\MR{2070538} (2005j:05059).  \Zbl{033.05038}.						
\annot{} The edges of $K_n$ are colored by $m$ colors.  Thm.: For $m>2$, the combined action of $\fS_n$ on vertices and $\fS_m$ on colors is transitive on $m$-edge-colored complete graphs for finite $n$ but not for infinite $n$.
\par
\codes{\CSGc:~\CGen:~\CSw}%
\itemspace

\author{P.J.\ Cameron \& Albert L.\ Wells, Jr.}
\authorspace
\bsgitem{1986a}{\dag} Signatures and signed switching classes.  \emph{J.\ Combin.\ Theory Ser.\ B} 40 (1986), 344--361.  
\MR{0842999} (87m:05115).  \Zbl{591.05061}.  
\codes{\CSG:~\CTG:~\CGen}%
\itemspace

\authorx{Federico Camia}						%CAMIA
\author{Federico Camia, Emilio De Santis, \& Charles M.\ Newman}
\authorspace
\bsgitem{2002a}{}  Clusters and recurrence in the two-dimensional zero-temperature stochastic Ising model.  \emph{Ann.\ Appl.\ Probab.}\ 12 (2002), no.\ 2, 565--580.
\MR{1910640} (2003h:60144).  \Zbl{1020.60094}.  
\codes{\CPhys,~\CVS:~\CRand}%
\itemspace

\author{Paul Camion}						%CAMION
\authorspace
\bsgitem{1963a}{}  Caracterisation des matrices unimodulaires.  \emph{Cahiers Centre \'Etudes Recherche Op\'er.}\ 5 (1963), 181--190. 
\MR{0179101} (31 \#3352).  \Zbl{124.00901}.
\annot{} Camion's signing algorithm (implicitly) finds a set of sign reversals to balance a bipartite signed graph.
\codes{\Csg:~\CBal}%
\itemspace

\bsgitem{1965a}{}  Characterization of totally unimodular matrices.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 16 (1965), 1068--1073.
\MR{0180568} (31 \#4802).  \Zbl{134.25201}.
\itemspace

\bsgitem{1968a}{}  Modules unimodulaires.  \emph{J.\ Combin.\ Theory} 4 (1968), 301--362.
\MR{0327576} (48 \#5918).  \Zbl{174.29504}.
\itemspace

\bsgitem{2006a}{}  Unimodular modules.  \emph{Discrete Math.}\ 306 (2006), no.\ 19-20, 2355--2382.
\MR{2261907} (2007e:05096).  \Zbl{1099.13021}.
\itemspace

\authorx{Massimo Campanino}						%CAMPANINO
\author{M.\ Campanino}
\authorspace
\bsgitem{1998a}{} Strict inequality for critical percolation values in frustrated random-cluster models.   \emph{}\ 
\MR{1670039} (2000b:60235).  \Zbl{926.60084}.  
\annot{} Compares the critical percolation values and the critical temperatures of a finite, positively edge-weighted, signed graph to those of the corresponding all-positive (``unfrustrated'', ``ferromagnetic'') weighted graph.  The graph is $\Lambda \subset \bbZ^d$ with a partition $\pi$ of the boundary $\partial\Lambda$ and all edges on each block of $\pi$.
Dictionary:  ``frustrated path'' = negative circle including a boundary edge; ``frustrated configuration'' = subgraph of $\Lambda$ having a negative circle including at least one boundary edge.
\dated{2 Apr 2013} 
\codes{\CPhys,~\CSG:~\CWG:~\CFr}%
\itemspace

\author{Sue Ann Campbell}						%CAMPBELL
\authorspace
\authref  See \hyperlink{Jacques Belair}{J.~B\'elair}.
\itemspace

\authorx{Manoel Campelo}						%CAMPELO
\author{Manoel Campelo \& G\'erard Cornu\'ejols}
\authorx{Manoel Campelo \& Gerard Cornuejols}
\authorspace
\bsgitem{2009a}{}  The Chv\'atal closure of generalized stable sets in bidirected graphs.  LAGOS'09  \emph{Electronic Notes Discrete Math.}\ 35 (2009), 89--95.
\MR{2579413} (no rev).  
\annot{} The generalized stable set polyhedron of $\Beta$ is (equivalent to) $\conv(\bbZ^n \cap \{0 \leq x \in \bbR^n : \incid(\Beta)x \leq b \}$ where $b \in \bbZ^m$, $m = \#E$.
Dictionary:  ``directed edge'' = positive, ``undirected edge'' = negative; ``odd cycle'' = negative circle.
\dated{9 Jun 2011} 
\codes{\Csg:~\Cori,~\CInc,~\CGeom}%
\itemspace

\author{Yves Candau}						%CANDAU
\authorspace
\authref  See \hyperlink{Nacim Ramdani}{N.~Ramdani}.
\itemspace

\author{E.\ Rodney Canfield} 						%CANFIELD
\authorspace
\authref  See \hyperlink{Edward A.\ Bender}{E.A.~Bender}.
\itemspace

\author{Ismail Naci Cangul}						%CANGUL
\authorspace
\authref  See \hyperlink{K.N.\ Prakasha}{K.N.~Prakasha}.
\itemspace

\author{Chun Zheng Cao}						%CAO CZ
\authorspace
\authref  See \hyperlink{Xiao Xin Zhu}{X.X.~Zhu}.
\itemspace

\author{D.S.\ Cao}							%CAO DS
\authorspace
\authref  See \hyperlink{Rodica Simion}{R.~Simion}.
\itemspace

\authorx{Fayun Cao}							%CAO FY
\author{Fayun Cao, Han Ren, \& Hanlin Chen}
\authorspace
\bsgitem{2019a}{}  New formulae for the bipartite vertex frustration and decycling number of graphs.  \emph{Appl.\ Math.\ Comput.}\ 347 (2019), 101--112.
\MR{3880070} (no rev).  \Zbl{1428.05148}.  
\annot{} For $-\Gamma$.
\codes{\Csg:~\CPar:~\CFr}%
\itemspace

\authorx{Meng-Yue Cao}						%CAO MY
\author{Meng-Yue Cao, Jack H.\ Koolen, Yen-Chi Roger Lin, \& Wei-Hsuan Yu}
\authorspace
\bsgitem{2022a}{}  The Lemmens-Seidel conjecture and forbidden subgraphs.  \emph{J.\ Combin.\ Theory Ser.\ A} 185 (2022) [2021], art.\ 105538, 28 pp.
\MR{4316717}.  \Zbl{1476.05109}.  \arXiv{2003.07511}.  
\annot{} Thm.\ 2.2:  If least eigenvalue of $A(K_n,\sigma)$ is $-5$ and $n\geq277$, then $\rk (A(K_n,\sigma)-5I) > \lceil 2n/3 \rceil$.  ``Switching graph'': a certain signed covering graph of $(K_n,\sigma)$.  Forbidden signed cliques for least eigenvalue $-5$.
\dated{13 Oct 2021} 
\codes{\Csg:~\CKG:~\CAdj:~\CEig,~\CSw}%
\itemspace

\author{Ming Cao}							%CAO M
\authorspace
\authref  See \hyperlink{Anton Proskurnikov}{A.~Proskurnikov} and \hyperlink{Dong Xue}{D.~Xue}.
\itemspace

\author{Andrea Capocci}						%CAPOCCI
\authorspace
\authref  See \hyperlink{Valerio Ciotti}{V.~Ciotti}.
\itemspace

\authorx{Chiara Cappello}						%CAPPELLO
\author{Chiara Cappello, Reza Naserasr, Eckhard Steffen, \& Zhouningxin Wang}
\authorspace
\bsgitem{20xxa}{\dag}  Critically 3-frustrated signed graphs.  Submitted.  \emph{}\ 
\arXiv{2304.10243}.  
\annot{} \Conjecture:  $\forall\ k$ $\exists$ finitely many irreducible critically $k$-frustrated signed graphs.  (Definition at \xref{Cappello and Steffen}{2022a}{Chiara Cappello \& Eckhard Steffen}.)  Several more detailed conjectures.  Partial results for $k=3$.
\dated{22 Apr 2023} 
\codes{\CSG:~\CFr}%
\itemspace

\author{Chiara Cappello \& Eckhard Steffen}
\authorspace
\bsgitem{2022a}{}  Frustration-critical signed graphs.  \emph{Discrete Appl.\ Math.}\ 322 (2022), 183--193.
\arXiv{2112.02664}.  
\annot{} $\Sigma$ is $k$-critical if $k=l(\Sigma) > l(\Sigma \setm e)$ for all edges.  Structure of some critical signed graphs, especially $2$-critical and certain signed cubic projective-planar graphs.  [Continued in \xref{Cappello--Naserasr--Steffen--Wang}{20xxa}{Chiara Cappello, Reza Naserasr, Eckhard Steffen, \& Zhouningxin Wang}.]
\dated{14 Feb 2022} 
\codes{\CSG:~\CFr}%
\itemspace

\author{Sergio Caracciolo}						%CARACCIOLO
\authorspace
\authref  See also \hyperlink{Matteo Palassini}{M~Palassini}.
\authrefspace

\author{S.\ Caracciolo, G.\ Parisi, \& N.\ Sourlas}
\authorspace
\bsgitem{1982a}{}  Variational real space renormalization group and its application to frustrated systems.  \emph{Nuclear Phys.\ B} 205 (1982), no.\ 3, 345--354.
\MR{0668812} (83g:82069).  \Zbl{968.82513}.					
\annot{} Approximation of physical quantities (free energy, critical temperature) on the triangular lattice with all-positive (``ferromagnetic''), all-negative (``antiferromagnetic''; ``fully frustrated''), and arbitrary (``randomly frustrated'') signs, by moving edges (``bonds'', ``couplings''), adjusting edge weights, and coarsening the lattice to get recursive formulas.  Also, a tentative analog for the square lattice with possible diagonals.
\dated{28 Mar 2013} 
\codes{\CPhys,~\Csg:~\CFr}%
\itemspace

\authorx{Alvaro Carbonero}						%CARBONERO
\author{Alvaro Carbonero, Janelle Domantay, \& Karen Guthrie}
\authorspace
\bsgitem{20xxa}{}  The optimization of signed trees.  Submitted.  \emph{}\ 
\arXiv{2109.01221}.  
\annot{} Which signed trees have a given degree set?  Some answers.
\dated{14 Feb 2022} 
\codes{\CSG}%
\itemspace

\author{Domingos M.\ Cardoso}						%CARDOSO
\authorspace
\authref  See also \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~de~Abreu} and \hyperlink{Ivan Gutman}{I.~Gutman}.
\authrefspace

\author{Domingos M.\ Cardoso, Drago\v{s} Cvetkovi\'c, Peter Rowlinson, \& Slobodan K.\ Simi\'c}
\authorx{Domingos M.\ Cardoso, Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2008a}{}  A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph.  \emph{Linear Algebra Appl.}\ 429 (2008), no.\ 11-12, 2770--2780.
\MR{2455532} (2009i:05145).  \Zbl{1148.05046}.					
\annot{} Thm.:  $\min_\Gamma \lambda_1(\Lap(-\Gamma))$, for connected, nonbipartite $\Gamma$ with $\#V=n$ is attained iff $\Gamma$ is $K_3$ with an attached path.	
[\Problem.  Generalize to connected, unbalanced signed graphs.]
\dated{4 Sept 2010} 
\par
\codes {\Csg:~\CPar:~\CEig}%
\itemspace

\authorx{Charles Carlson}						%CARLSON
\author{Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, \& Alexandra Kolla}
\authorspace
\bsgitem{2019a}{}  Spectral aspects of symmetric matrix signings.  In:  Peter Rossmanith \etal, eds., \emph{44th International Symposium on Mathematical Foundations of Computer Science}\ (MFCS 2019, Aachen), art.\ 81, 13 pp.  Leibniz Int.\ Proc.\ Informatics, Schloss Dagstuhl -- Leibniz-Zentrum f\"ur Informatik, Dagstuhl Publishing, Germany, 2019.
\MR{4008470}.  
\annot{} Extended abstract of \xrefd{2020a}{Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, \& Alexandra Kolla}.
\codes{\Csg:~\CQM:~\CStr,~\CAlgor}%
\itemspace

\bsgitem{2020a}{}  Spectral aspects of symmetric matrix signings.  \emph{Discrete Optimization} 37 (2020), art.\ 100582, 22 pp.
\MR{4095771}.  
\arXiv{1611.03624v2}.  
\annot{} Problem: sign a positively weighted graph (possibly with loops) to have a desired property such as invertibility or solvability, by forming Kronecker product $A(\Gamma,w) \circ A(\Sigma)$.  Or, also add new edges to get the property.  Algorithmic complexity, algorithms, and structural descriptions.
\annotctd{} Simpler, weaker initial version at \arXiv{1611.03624v1} [and with one result and proof duplicated independently by \xref{Akbari, Ghafari, Kazemian, and Nahvi}{2020a}{S.\ Akbari, A.\ Ghafari, K.\ Kazemian, \& M.\ Nahvi}].
\dated{21 Dec 2020} 
\codes{\Csg:~\CQM:~\CStr,~\CAlgor}%
\itemspace

\authorx{Jordan Carlson}						%CARLSON
\author{Jordan Carlson \& Daniel Stolarski}
\authorspace
\bsgitem{2004a}{}  The correct solution to Berlekamp's switching game.  \emph{Discrete Math.}\ 287 (2004), 145--150.
\MR{2094708} (2005d:05005).  \Zbl{1054.94023}.					
\annot{} The minimum frustration index of a signed $K_{n,n}$ for $n=10,11,12$ and bounds up to $20$.  Corrects and extends \xref{Fishburn and Sloane}{1989a}{P.C.\ Fishburn \& N.J.A.\ Sloane}.
\par
\codes{\Csg:~\Cfr}%
\itemspace

\author{Jaime Cartes}						%CARTES
\authorx{J.\ Cartes}
\authorspace
\authref  See \hyperlink{W.\ Lebrecht}{W.~Lebrecht}, \hyperlink{J.F.\ Valdes}{J.F.~Vald\'es}, and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\author{Dorwin Cartwright}						%CARTWRIGHT
\authorspace
\authref  See also \hyperlink{Terry C.\ Gleason}{T.C.~Gleason}, and \xref{Harary--Norman--Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} \etal
\authrefspace

\author{Dorwin Cartwright \& Terry C.\ Gleason} 
\authorspace
\bsgitem{1966a}{}  The number of paths and cycles in a digraph.  \emph{Psychometrika} 31 (1966), 179--199.  
\MR{0197195} (33 \#5377).  \Zbl{143.43702}.					
\annot{} Pp.\ 194ff., ``A generalization of the method'':  Digraph edges have gains in a (commutative) semiring.  A matrix method produces counts of paths and cycles of given lengths.  Remarks on pp.\ 194, 199:  The method can be applied to signed digraphs (unexplained) [the group ring $\bbZ[+,-]$ must be intended; matrix entry $= n_+(+)+n_-(-)$, $n_\epsilon =$ \# of $\epsilon$-paths/circles].
Dictionary:  ``generalized addition, multiplication'' = operations in the semiring; ``value matrix'' = adjacency matrix of the gain graph.
\dated{28 Apr 2017}
\codes{\Cgd(\CGen),~\Csd:~\CAdj,~\CPaths,~\CCircles}%
\itemspace

\author{Dorwin Cartwright \& Frank Harary}
\authorspace
\bsgitem{1956a}{} Structural balance:  a generalization of Heider's theory.  \emph{Psychological Rev.}\ 63 (1956), 277--293.  
Repr.\ in: Dorwin Cartwright and Alvin Zander, eds., \emph{Group Dynamics: Research and Theory}, second ed., pp.\ 705--726.  Harper and Row, New York, 1960.  
Also reprinted in:  Samuel Leinhardt, ed., \emph{Social Networks: A Developing Paradigm}, pp.\ 9--25.  Academic Press, New York, 1977.  
\annot{} Expounds \xref{Harary}{1953a}{Frank Harary}, \xrefd{1955a}{Frank Harary} with sociological discussion.  Proposes to measure imbalance by the proportion of balanced circles (the ``degree of balance'') or balanced circles of length $\leq k$ (``degree of $k$-balance'').
\codes{\CPsS,~\CSG:~\CBal,~\CFr}%
\itemspace

\bsgitem{1968a}{} On the coloring of signed graphs.  \emph{Elem.\ Math.}\ 23 (1968), 85--89.  
\MR{0233732} (38 \#2053).  \Zbl{155.31703} (155, p.\ 317c).				
\annot{} ``Coloring'' is clustering as in \xref{Davis}{1967a}{James A.\ Davis}.  Thm.\ 1 adds a bit to \xref{Davis}{1967a}{James A.\ Davis} in Thm.\ 3:  The clustering is unique $\iff$ all components of $\Sigma^+$ are adjacent.
\codes{\CSG:~\CClu}%
\itemspace

\bsgitem{1970a}{}  Ambivalence and indifference in generalizations of structural balance.  \emph{Behavioral Sci.}\ 15 (1970), no.\ 6, 497--513.  
\codes{\CSD:~\CGen:~\CBal}%
\itemspace

\bsgitem{1977a}{} A graph theoretic approach to the investigation of system-environment relationships.  \emph{J.\ Math.\ Sociology} 5 (1977), 87--111.  
\MR{0444117} (56 \#2477).  \Zbl{336.92026}.					
\par
\codes{\CSD:~\CClu}%
\itemspace

\bsgitem{1979a}{} Balance and clusterability: an overview.  
In: Paul W.\ Holland and Samuel Leinhardt, eds., \emph{Perspectives on Social Network Research} (Proc.\ Sympos., Dartmouth Coll.,  Hanover, N.H., 1975), Ch.\ 3, pp.\ 25--50.  Academic Press, New York, 1979.  
\codes{\CSG,~\CSD,~\CVS:~\CBal,~\CFr,~\CClu,~\CAdj:~\CExp}%
\itemspace

\author{Adolfo Casari} 						%CASARI
\authorspace
\authref  See \hyperlink{Francisco Barahona}{F.~Barahona}.
\itemspace

\author{Federico Castillo}						%CASTILLO
\authorspace
\authref  See \hyperlink{Federico Ardila}{F.~Ardila}.
\itemspace

\author{Paul A.\ Catlin}						%CATLIN*
\authorspace
\bsgitem{1979a}{}  Haj\'os' graph-coloring conjecture: variations and counterexamples.  \emph{J.\ Combin.\ Theory Ser.\ B} 26 (1979), 268--274.  
\MR{0532593} (81g:05057).  \Zbl{385.05033}, \Zbl{395.05033}.
\annot{} Thm.\ 2:  If $\Gamma$ is $4$-chromatic, $[-\Gamma]$ contains a subdivision of $[-K_4]$ (an ``odd-$K_4$'').  [\Question.  Can this possibly be a signed-graph theorem?  For instance, should it be interpreted as concerning the chromatic number of $-\Gamma$?]
\codes{\Cpar:~\Ccol}%
\itemspace

\author{Minerva Catral}						%CATRAL
\authorspace
\authref  See also \hyperlink{Steve Butler}{S.~Butler}.
\authrefspace

\author{M.\ Catral, D.D.\ Olesky, \& P.\ van den Driessche}
\authorspace
\bsgitem{2009a}{}  Allow problems concerning spectral properties of sign pattern matrices: A survey.  \emph{Linear Algebra Appl.}\ 430 (2009), no.\ 11-12, 3080--3094.
\MR{2517861} (2010i:15066).  \Zbl{1165.15009}.					
\annot{} $D$ is the signed digraph of a square sign-pattern matrix $S$.  Thm.\ 3.1:  If the spectrum of $A$ with signs $S$ is arbitrary, $D$ has positive and negative disjoint cycle unions of all orders.  Thm.\ 4.1:  If the inertia is arbitrary, $D$ has a positive and a negative loop and a negative digon.  
\dated{4 Nov 2011} 
\codes{\CSG:~\CQM,~\CExp}%
\itemspace

\author{Bogdan Cautis}						%CAUTIS
\authorspace
\authref  See \hyperlink{Christos Giatsidis}{C.~Giatsidis}.
\itemspace

\author{Matteo Cavaleri}						%CAVALERI
\authorspace
\authref  See also \hyperlink{Francesco Belardo}{F.~Belardo}.
\authrefspace

\author{Matteo Cavaleri, Daniele D'Angeli, \& Alfredo Donno}
\authorspace
\bsgitem{2021a}{\dag}  A group representation approach to balance of gain graphs.  \emph{J.\ Algebraic Combin.}\ 54 (2021), no.\ 1, 265--293.
\MR{4290425}.  \Zbl{1470.05071}.  \arXiv{2001.08490}.  
\codes{\CGG:~\CAdj:~\CAlgeb}%
\itemspace

\bsgitem{2021b}{}  Gain-line graphs via $G$-phases and group representations.  \emph{Linear Algebra Appl.}\ 613 (2021), 241--270.
\MR{4199816}.  \Zbl{1459.05163}.  \arXiv{2007.14839}.  
\par
\codes{\CGG:~\CLG:~\CAdj:~\CAlgeb,~\CEig}%
\itemspace

\bsgitem{2022a}{}  Characterizations of line graphs in signed and gain graphs.  \emph{European J.\ Combin.}\ 102 (2022), art.\ 103479, 19 pp.
\MR{4342436}.  
\arXiv{2101.09677}.  
\par
\codes{\CGG,~\CSG:~\CLG}%
\itemspace

\author{Matteo Cavaleri and Alfredo Donno}
\authorspace
\bsgitem{2022a}{}  On cospectrality of gain graphs.  \emph{Special Matrices} 10 (2022), 343--365. 
\MR{4436247}.  
\arXiv{2111.12428}.  
\par
\codes{\CSG,~\CGG:~\CAdj:~\CEig}%
\itemspace

\author{Matteo Cavaleri, Alfredo Donno, \& Stefano Spessato}
\authorspace
\bsgitem{20xxa}{}  Godsil-McKay switchings for gain graphs.  Submitted.  \emph{}\ 
\arXiv{2207.10986}.  
\par
\codes{\CGG:~\CAdj:~\CEig,~\CBal}%
\itemspace

\author{Michael S.\ Cavers}						%CAVERS
\authorspace
\bsgitem{2010a}{}  On reducible matrix patterns.  \emph{Linear Multilinear Algebra} 58 (2010), no.\ 2, 257--267.
\MR{2641538} (2011b:15072).  \Zbl{1189.15010}.					
\codes{\CSD:~\CQM}%
\itemspace

\author{M.\ Cavers, S.M.\ Cioab\u{a}, S.\ Fallat, D.A.\ Gregory, W.H.\ Haemers, S.J.\ Kirkland, J.J.\ McDonald, \& M.\ Tsatsomeros}
\authorx{M.\ Cavers, S.M.\ Cioaba, S.\ Fallat, D.A.\ Gregory, W.H.\ Haemers, S.J.\ Kirkland, J.J.\ McDonald, \& M.\ Tsatsomeros}
\authorspace
\bsgitem{2012a}{}  Skew-adjacency matrices of graphs.  \emph{Linear Algebra Appl.}\ 436 (2012), 4512--4529.
\MR{2917427}.  \Zbl{1241.05070}.  
\annot{} \S 4, ``Characteristic polynomials of skew-adjacency matrices'':  From $\Gamma$ form a signed digraph: each edge becomes a negative digon.  An orientation $\Gamma^\tau$ is a choice of one arc from each digon; thus, it is a signed digraph.  A ``skew adjacency matrix'' of $\Gamma$ is a matrix $S := A(\Gamma^\tau)$.  The characteristic polynomial $p_S(x)$ is odd/even for odd/even $n$.  Eqs.\ (7), (8) give formulas for the coefficients in terms of matchings and cycle signs.  
A ``generic skew-adjacency matrix'' of $\Gamma$ has variables $x_{ij} = x_{ji}$ instead of 1's in $S$.  
Thm.\ 4.2:  $\Spec S$ is unique iff $\Gamma$ has no even circles (an ``odd-circle graph'').
Dictionary:  ``cycle'' = circle, ``dicycle'' = cycle, ``$\sigma$'' = $\tau$, ``$\vec G(S)$'' = $\Gamma^\tau$.
\annotctd{} More results about odd-circle and bipartite (even-circle) graphs.  Lemma 6.3 implicitly switches $\Gamma^\tau$ through $S$.
[\Problem.  Generalize to bidirected graphs, thereby unifying symmetric and skew-symmetric adjacency matrices.  A negative edge becomes a positive digon, all-positive if oriented extraverted and all-negative if introverted.  The all-negative case is symmetric, the all-positive case is skew-symmetric.]
\dated{4 Jan 2013} 
\par
\codes{\Csd:~\CAdj,~\Csw}%
\itemspace

\author{Michael S.\ Cavers \& Shaun M.\ Fallat}
\authorspace
\bsgitem{2012a}{}  Allow problems concerning spectral properties of patterns.  \emph{Electronic J.\ Linear Algebra}\ 23 (2012), 731--754.
\MR{2966802}.  \Zbl{1251.15034}.  
\annot{} Signed digraphs are generalized to edges labelled by $0, +, -, +_0 \,(\ge0), -_0 \,(\le0), * \,(\ne0), \# \,(\text{real})$.
\dated{24 May 2013} 
\codes{\CQM:~\CSD(\CGen)}%
\itemspace

\author{Michael S.\ Cavers \& Kevin N.\ Vander Meulen}
\authorspace
\bsgitem{2005a}{}  Spectrally and inertially arbitrary sign patterns.  \emph{Linear Algebra Appl.}\ 394 (2005), 53--72.
\MR{2100576} (2005f:15008).  \Zbl{1065.15009}.					
\annot{} Lem.\ 5.1:  An inertially arbitrary sign pattern contains a negative digon.
\dated{5 Nov 2011} 
\codes{\CQM:~\Csd,~\Csw}%
\itemspace

\authorx{Nicol\`o Cesa-Bianchi}						%CESA-BIANCHI
\authorx{Nicolo Cesa-Bianchi}
\author{Nicol\`o Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella}
\authorx{Nicolo Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella}
\authorspace
\bsgitem{2012a}{}  A correlation clustering approach to link classification in signed networks.  In:  Shie Mannor, Nathan Srebro, and Robert C.\ Williamson, eds., \emph{Proceedings of the 25th Annual Conference on Learning Theory} (COLT 2012, Edinburgh, 2012), paper 34, 20 pp., electronic.  JMLR: Workshop and Conf.\ Proc., Vol.\ 23.  ACM, 2012.  URL \url{http://jmlr.csail.mit.edu/proceedings/papers/v23/}
\arXiv{1301.4769} (Full version, 22 pp.).  
\codes{\CSG:~\CClu}%
\itemspace

\bsgitem{2012b}{}  A linear time active learning algorithm for link classification.  In:  \emph{Proceedings of the Workshop on Mining and Learning with Graphs} (10th, MLG-2012, Edinburgh, 2012).  6 pp., electronic.  
URL (9/2015) \url{https://dtai.cs.kuleuven.be/events/mlg2012/papers/7\_Linear\_Nicolo.pdf} \\
\arXiv{1301.4767} (full version).  
\annot{} Very short version of \xrefd{2012c}{Nicolo Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella}.
\codes{\CSG:~\CClu}%
\itemspace

\bsgitem{2012c}{}  A linear time active learning algorithm for link classification.  
In:  F.\ Pereira, C.J.C.\ Burges, L.\ Bottou, and K.Q.\ Weinberger, eds., \emph{Advances in Neural Information Processing Systems 25 (NIPS 2012)} (Lake Tahoe, Nev., 2012), 9 pp., electronic.  
URL (9/2015) \url{http://papers.nips.cc/paper/4598-a-linear-time-active-learning-algorithm-for-link-classification}\\
\arXiv{1301.4767} (full version).  
\annot{} 
Less short version of \xrefd{2012b}{Nicolo Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella} and \xrefd{2012d}{Nicolo Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella}.
\annotctd{} See \arXiv{1301.4767} for the full version (16 pp.).
\codes{\CSG:~\CClu}%
\itemspace

\bsgitem{2012d}{}  A fast active learning algorithm for link classification.  In:  \emph{13th Italian Conference on Theoretical Computer Science} (ICTCS 2012, Varese, Italy, 2012), paper 38, 4 pp., electronic.
URL (9/2015) \url{http://ictcs.di.unimi.it/papers/paper\_38.pdf}
\annot{} Shortest version of \xrefd{2012c}{Nicolo Cesa-Bianchi, Claudio Gentile, Fabio Vitale, \& Giovanni Zappella}.
\codes{\CSG:~\CClu}%
\itemspace

\author{Seth Chaiken} 						%CHAIKEN
\authorspace
\bsgitem{1982a}{} A combinatorial proof of the all minors matrix tree theorem.  \emph{SIAM J.\ Algebraic Discrete Methods} 3 (1982), 319--329.
\MR{0666857} (83h:05062).  \Zbl{495.05018}.					
 \annot{} \S4: ``Extension to signed graphs''.  Generalizing \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, an all-minors matrix-tree theorem for weighted signed digraphs and a corollary for weighted signed graphs.  
Given: a signed graph on vertex set $[n]$.  For a Laplace-type $n \times n$ matrix $\Lap$ ($A$ in the paper), $\Lap(\bar U, \bar W)$ is $\Lap$ with the rows indexed by $U$ and the columns indexed by $W$ deleted.  Take $U, W \subseteq V$ with $\#U = \#W = k \leq n$.  Then $\det \Lap(\bar U, \bar W)$ is a sum of terms, one for each independent set $F$ of rank $n-k$ in $\bfF(\Sigma)$ in which each tree component contains just one vertex from $U$ and one from $W$.  Each term has a sign depending partly on the number of negative paths by which $F$ links $U$ to $W$ and partly on the linking pattern, and with magnitude $4^c\cdot$(weight product of $F$), where $c =$ \# of circles in $F$.  
[The credit to Zaslavsky is overly generous: only the case $U = W = \eset$ is his; the others are new.]  
The digraph version replaces 4 by 2 and imposes conditions on arc directions in the tree and nontree components of $F$.
 \annotctd{} A brief remark describes a gain-graphic (``voltage-graphic'') generalization.
\codes{\CSD,~\CSG,~\CGG:~\CLap,~\CInc,~\Cmatrd}%
\itemspace

\bsgitem{1996a}{}  Oriented matroid pairs, theory and an electrical application.  
In:  Joseph E.\ Bonin, James G.\ Oxley, and Brigitte Servatius, eds., \emph{Matroid Theory} (Proc., Seattle, 1995), pp.\ 313--331.  
Contemp.\ Math., Vol.\ 197.  Amer.\ Math.\ Soc., Providence, R.I., 1996.  
\MR{1411693} (97e:05058).  Zbl{866.05016}.
\annot{} Connects a problem on common covectors of two subspaces of $\bbR^m$, and more generally of a pair of oriented matroids, to the problem of sign-solvability of a matrix and the even-cycle problem for signed digraphs.
\par
\codes{\CSol,~\Csd:~\CPar,~\CAlgor}%
\itemspace

\bsgitem{1996b}{}  Open problem 5.  A problem about common covectors and bases in oriented matroid pairs.  In:  Joseph E.\ Bonin, James G.\ Oxley, and Brigitte Servatius, eds., \emph{Matroid Theory} (Proc., Seattle, 1995), pp.\ 415--417.  Contemp.\ Math., Vol.\ 197.  Amer.\ Math.\ Soc., Providence, R.I., 1996.
\annot{} Possible generalizations to oriented matroids of sign-nonsingularity of a matrix.
\codes{\CSol,~\CSD:~\CPar}%
\itemspace

\author{Seth Chaiken, Christopher R.H.\ Hanusa, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2010a}{}  Nonattacking queens in a rectangular strip.  \emph{Ann.\ Combin.}\ 14 (2010), 419--441.
\MR{2776757} (2012d:05034).  \Zbl{1233.05022}.  \arXiv{1105.5087}.			
\annot{} Affinographic hyperplanes and rooted integral gain graphs, from \xref{Forge--Zaslavsky}{2007a}{David Forge \& Thomas Zaslavsky}, imply the structure of formulas counting nonattacking arrangements of identical chess pieces in an $m\times n$ strip, as a function of $n$.
\codes{\CGG:~\CGeom,~\CInv}%
\itemspace

\bsgitem{2019a}{}  A $q$-queens problem.  VI.  The bishops' period.  \emph{Ars Math.\ Contemp.}\ 16 (2019), no.\ 2, 549--561.
\MR{3963222}.  \Zbl{1416.05131}.  \arXiv{1405.3001}.  				
\annot{} Hyperplane representation of sign-colored graphs with a new associated matroid, assisted by a related signed graph and its frame matroid.
\dated{4 Apr 2011}
\codes{\CSGc:~\CGeom,~\CMatrd}\codesctd{\CSG:~\Cmatrd}%
\itemspace

\author{Bikas K.\ Chakrabarti}						%CHAKRABARTI
\authorspace
\authref  See also \hyperlink{Sei Suzuki}{S.~Suzuki}.
\authrefspace

\author{Bikas K.\ Chakrabarti, Amit Dutta, \& Parongama Sen}
\authorspace
\bsgitem{1996a}{}  \emph{Quantum Ising Phases and Transitions in Transverse Ising Models}.  Lect.\ Notes in Phys., New Ser.\ m: Monographs, Vol.\ 41.  Springer, Berlin, 1996.
\Zbl{872.60090}.  
\annot{} Shows aspects of what physicists may ask about signed graphs.  ``Transverse'' means external magnetic field(s), modellable as extra dominating vertices.  
Ch.\ 4, ``ANNI model in transverse field'':  \Cf\ \xref{Liebmann}{1986a}{Rainer Liebmann}.
Ch.\ 6, ``Transverse Ising spin glass and random field systems'':  The typical mountainous energy landscape.  \S6.1, ``Classical Ising spin glasses:  A summary'':  Random signs.  The $\pm J$ model is unweighted signed graphs.  \S6.2, ``Quantum spin glasses'':  Height of energy barriers between valleys may be less important than width due to quantum tunnelling.
\annotctd{} Second edition:  \xref{Suzuki, Inoue, and Chakrabarti}{2013a}{Sei Suzuki, Jun-ichi Inoue, \& Bikas K.\ Chakrabarti}.
\dated{8 Aug 2018} 
\codes{\CPhys:~\CSG,~\Cwg:~\CFr}%
\itemspace

\author{Nilanjan Chakraborty}						%CHAKRABORTY
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Sudip Chakravarty}						%CHAKRAVARTY
\authorspace
\authref  See \hyperlink{Rajiv R.P.\ Singh}{R.R.P.~Singh}.
\itemspace

\authorx{AtMa P.O.\ Chan}						%CHAN APO
\author{AtMa P.O.\ Chan, Jeffrey C.Y.\ Teo, \& Shinsei Ryu}
\authorspace
\bsgitem{2015a}{}  Topological phases on non-orientable surfaces: twisting by parity symmetry.  \emph{New J.\ Phys.}\ 18 (2016), art.\ 035005, 22 pp.
\MR{3484878}.  
\annot{} A physics approach to embedding a signed graph on a surface, via orientable surfaces with parity defects.  Also, in part, the same for gain graphs.
\dated{2 Apr 2016} 
\codes{\Csg:~\CTop}\codesctd{\Cgg:~\CTop}%
\itemspace

\author{Sarah Chand}						%CHAND S
\authorspace
\authref  See \hyperlink{Rashid Farooq}{R.~Farooq}.
\itemspace

\author{Karthekeyan Chandrasekaran}					%CHANDRASEKARAN
\authorspace
\authref  See \hyperlink{Charles Carlson}{C.~Carlson}.
\itemspace

\authorx{Vijaya Chandru}						%CHANDRU
\author{Vijaya Chandru, Collette R.\ Coullard, \& Donald K.\ Wagner}
\authorspace
\bsgitem{1985a}{} On the complexity of recognizing a class of generalized networks.  \emph{Operations Res.\ Lett.}\ 4 (1985), 75--78.  
\MR{0811167} (87a:90144).  \Zbl{565.90078}.						
\annot{} Determining whether a gain graph with real multiplicative gains has a balanced circle, i.e., is not contrabalanced, is NP-hard.  So is determining whether a real matrix is projectively equivalent to the incidence matrix of a contrabalanced real gain graph.
\codes{\CGN,~\CBic:~\CInc,~\CAlgor}%
\itemspace

\authorx{Chung-Chien Chang}						%CHANG CC
\author{Chung-Chien Chang \& Cheng-Ching Yu}	
\authorspace
\bsgitem{1990a}{}  On-line fault diagnosis using the signed directed graph.  \emph{Industrial and Engineering Chem.\ Res.}\ 29 (1990), 1290--1299.  
\annot{} Modifies the method of \xref{Iri, Aoki, O'Shima, and Matsuyama}{1979a}{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama} of constructing the diagnostic signed digraph, e.g.\ by considering transient and steady-state situations.
\codes{\CSD:~\CAppl,~\CRef}%
\itemspace

\author{Gerard J.\ Chang}						%CHANG GJ
\authorspace
\authref  See \hyperlink{Jing-Ho Yan}{J.H.~Yan}.
\itemspace

\author{Hsien-Chih Chang}						%CHANG HC
\authorspace
\authref  See \hyperlink{Charles Carlson}{C.~Carlson}.
\itemspace

\author{Maw-Shang Chang}						%CHANG MS
\authorspace
\authref  See \hyperlink{Li-Hsuan Chen}{L.-H.~Chen}.
\itemspace

\author{Michael D.\ Chang}						%CHANG MD
\authorspace
\authref  See \hyperlink{Michael Engquist}{M.~Engquist}.
\itemspace

\authorx{Ting-Chung Chang}						%CHANG TC
\author{Ting-Chung Chang [Ting-Jung Chang]}
\authorspace
\authref  See \hyperlink{Ting-Jung Chang}{T.J.~Chang}.
\itemspace

\author{Ting-Jung Chang [Ting-Chung Chang]}				%CHANG TJ
\authorx{Ting-Jung Chang}
\authrefspace

\author{Ting-Jung Chang \& Bit-Shun Tam}
\authorspace
\bsgitem{2010a}{}  Graphs with maximal signless Laplacian spectral radius.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 7, 1708--1733. 
\MR{2592913} (2011e:15014).  \Zbl{1231.05166}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ting-Jung Chang (as Ting-Chung Chang) \& Bit-Shun Tam}
\authorspace
\bsgitem{2011a}{}  Connected graphs with maximal $Q$-index: The one-dominating-vertex case.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2451--2461.
\MR{2811129} (2012d:05220).  \Zbl{1222.05029}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ting-Jung Chang, Bit-Shun Tam, \& Shu-Hui Wu}
\authorspace
\bsgitem{2011a}{}  Theorems on partitioned matrices revisited and their applications to graph spectra.  \emph{Linear Algebra Appl.}\ 434 (2011), 559--581.
\MR{2741241} (2012g:05131).  \Zbl{1225.05160}.  
\codes{\Csg:~\CPar:~\CEig}%
\itemspace

\author{Yi Chang}							%CHANG Y
\authorspace
\authref  See \hyperlink{Jiliang Tang}{J.-L.~Tang} and \hyperlink{Shuang Hong Yang}{S.H.~Yang}.
\itemspace

\author{Claudine Chaouiya}						%CHAOUIYA
\authorspace
\authref  See \hyperlink{Gilles Didier}{G.~Didier} and \hyperlink{Aurelien Naldi}{A.~Naldi}.
\itemspace

\author{Guillaume Chapuy}						%CHAPUY
\authorspace
\authref  See \hyperlink{Olivier Bernardi}{O.~Bernardi}.
\itemspace

\author{Pierre Charbit}						%CHARBIT
\authorspace
\authref  See \hyperlink{Pierre Aboulker}{P.~Aboulker}.
\itemspace

\author{Moses Charikar}						%CHARIKAR
\authorspace
\authref  See also \hyperlink{Nir Ailon}{N.~Ailon}.
\authrefspace

\author{Moses Charikar, Neha Gupta, \& Roy Schwartz}
\authorspace
\bsgitem{2017a}{}  Local guarantees in graph cuts and clustering.  In:  Friedrich Eisenbrand \etal, eds., \emph{Integer Programming and Combinatorial Optimization} (19th Int.\ Conf., IPCO 2017, Waterloo, Ont., 2017), pp.\ 136--147.  \LNCS 10328.  Springer, Cham, 2017.
\MR{3678780}.  \Zbl{06767445}.  \arXiv{1704.00355}.				
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Moses Charikar, Venkatesan Guruswami, \& Anthony Wirth}
\authorspace
\bsgitem{2003a}{}  Clustering with qualitative information.  In:  \emph{Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science} (FOCS'03), pp.\ 524--533.  IEEE, 2003.
\annot{} Conference version of \xrefd{2005a}{Moses Charikar, Venkatesan Guruswami, \& Anthony Wirth}.
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\bsgitem{2005a}{}  Clustering with qualitative information.  Learning Theory 2003.  \emph{J.\ Computer System Sci.}\ 71 (2005), no.\ 3, 360--383.  
\MR{2168358} (2006f:68141).  \Zbl{1094.68075}.					
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace
 
\author{Ankit Charls}							%CHARLS
\authorspace
\authref  See \hyperlink{Tushar Sharma}{T.~Sharma}.
\itemspace

\authorx{A.\ Charnes}		 				%CHARNES
\author{A.\ Charnes, M.\ Kirby, \& W.\ Raike}
\authorspace
\bsgitem{1966a}{}  Chance-constrained generalized networks.  \emph{Operations Res.}\ 14 (1966), 1113--1120.  
\Zbl{152.18302} (152, p.\ 183b).  
\codes{\CGN}%
\itemspace

\author{A.\ Charnes \& W.M.\ Raike} 
\authorspace
\bsgitem{1966a}{}  One-pass algorithms for some generalized network problems.  \emph{Operations Res.}\ 14 (1966), 914--924.  
\Zbl{149.38106} (149, p.\ 381f).  
\codes{\CGN:~\CInc}%
\itemspace

\authorx{Cl\'ement Charpentier}						%CHARPENTIER
\authorx{Clement Charpentier}
\author{Cl\'ement Charpentier, Reza Naserasr, \& ?\'Eric Sopena}
\authorx{Clement Charpentier, Reza Naserasr, \& ?\'Eric Sopena}
\authorspace
\bsgitem{2020a}{}  Homomorphisms of sparse signed graphs.  \emph{Electronic J.\ Combin.}\ 27 (2020), no.\ 3, art.\P3.6, 28 pp.
\MR{4245119}.  
\HAL{hal-03035030}.  
\codes{\CSG:~\CHom}%
\itemspace

\author{Gary Chartrand} 						%CHARTRAND
\authorspace
\authref  See also \hyperlink{M.\ Behzad}{M.~Behzad}.
\authrefspace

\bsgitem{1977a}{} \emph{Graphs as Mathematical Models}.  Prindle, Weber and Schmidt, Boston, 1977. 
\MR{0490611} (58 \#9947).  \Zbl{384.05029}.
\annotctd{} [Repr.\ \xrefd{1985a}{Gary Chartrand}.]
\codes{\CSG:~\CBal,~\CClu}%
\itemspace

\bsgitem{1985a}{}  \emph{Introductory Graph Theory}.  Dover Publications, New York, 1985.
\MR{0783826} (86c:05001).  
\annot{} ``Corrected reprint'' of \xrefd{1977a}{Gary Chartrand}.
\codes{\CSG:~\CBal,~\CClu}%
\itemspace

\author{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz}
\authorspace
\bsgitem{1994a}{}  On signed degrees in signed graphs.  \emph{Czechoslovak Math.\ J.}\ 44(119) (1994), 677--690.  
\MR{1295143} (95g:05084).  \Zbl{837.05110}.					
\annot{} Net degree sequences (i.e., $d^\pm := d^+ - d^-$; called ``signed degree sequences'') of signed simple graphs.  A Havel--Hakimi-type reduction formula, but with an indeterminate length parameter; a determinate specialization to complete graphs.  A necessary condition for a sequence to be a net degree sequence.  Examples: paths, stars, double stars.  [Continued in \xref{Yan, Lih, Kuo, and Chang}{1997a}{Jing-Ho Yan, Ko-Wei Lih, David Kuo, \& Gerard J.\ Chang}.  Solved in \xref{Michael}{2002a}{T.S.\ Michael}.]
\annotctd{} [This is a special case of weighted degree sequences of $K_n$ with integer edge weights chosen from a fixed interval of integers.  Here the interval is $[-1, +1]$.  The theory of such degree sequences is due to V.\ Chungphaisan, Conditions for sequences to be $r$-graphic, \emph{Discrete Math.}\ 7 (1974), 31--39.  \MR{0351903} (50 \#4391).  \xref{Michael}{2002a}{T.S.\ Michael} characterizes net degree sequences by noticing this connection.]
\codes{\CSGw:~\CInvar}%
\annotctd{} [One can interpret net degrees as the net indegrees, $d^{\text{\rm in}} - d^{\text{\rm out}}$, of certain bidirected graphs.  Change the positive (negative) edges to extroverted (resp., introverted).  Then we have the net indegree sequence of an oriented $-\Gamma$.  \Problem\ 1.  Generalize to all bidirected (simple, or simply signed) graphs, especially $K_n$'s.  \Problem\ 2.  Find an Erd\H{o}s--Gallai-type characterization of net degree sequences of signed simple graphs.  [Solved by \xref{Michael}{2002a}{T.S.\ Michael}.]  \Problem\ 3.  Characterize the separated signed degree sequences of signed simple graphs, where the separated signed degree is $(d^+(v), d^-(v))$.  \Problem\ 4.  Generalize Problem 3 to edge $k$-colorings of $K_n$.]
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\author{Gary Chartrand, Frank Harary, Hector Hevia, \& Kathleen A.\ McKeon}
\authorspace
\bsgitem{1992a}{}  On signed graphs with prescribed positive and negative graphs.  \emph{Vishwa Int.\ J.\ Graph Theory} 1 (1992), 9--18.  
\MR{1196220} (93m:05095).  
\annot{} What is the smallest order of an edge-disjoint union of two (isomorphism types of) simple graphs, $\Gamma$ and $\Gamma'$?  Bounds, constructions, and special cases.  (The union is called a signed graph with $\Gamma$ and $\Gamma'$ as its positive and negative subgraphs.)  Thm.\ 13:  If $\Gamma'$ is bipartite (i.e., the union is balanced) with color classes $V'_1$ and $V'_2$, the minimum order $= \min(\#V'_1, \#V'_2) + \max(\#V, \#V'_1, \#V'_2)$.
\codes{\Cwg}\codesctd{\CSG:~\CBal}%
\itemspace

\author{Bilal A.\ Chat}						%CHAT
\authorspace
\authref  See \hyperlink{Hilal A.\ Ganie}{H.A.~Ganie}.
\itemspace

\author{Sourav Chatterjee}						%CHATTERJEE S
\authorspace
\bsgitem{2007a}{}  Estimation in spin glasses: a first step.  \emph{Ann.\ Stat.}\ 38 (2007), no.\ 5, 1931--1946.
\MR{2363958} (2009k:62063).  \Zbl{1126.62128}.  \arXiv{math/0604634}.		
\annot{} Advanced statistics are applied to states $s:V \to \signnos$ of weighted signed complete graphs to estimate the probability of $s$.  Thm.\ 1.1 (too complicated to state here) gives estimators for some choices of weights.  \S1.4, ``Application to the Sherrington--Kirkpatrick (S--K) model''.
\dated{31 Aug 2012} 
\codes{\CSG,~\CWG:~\CState,~\CPhys}%
\itemspace

\bsgitem{2008a}{}  Chaos, concentration, and multiple valleys.  Manuscript, 2008.  
\arXiv{0810.4221}.								
\annot{} \S9, ``Example: Generalized SK model of spin glasses'':  A general theory is applied to convex combinations $\sum_{p\ge1} c_p (K_n^{(p)},\sigma,w)$ of signed, weighted complete $p$-uniform hypergraphs, where $c_p \to 0$ slowly as $p \to \infty$.  (Obviously, $n$ also $\to \infty$.)
[Published by inclusion in \xrefd{2014a}{Sourav Chatterjee}.]
\dated{31 Aug 2012} 
\codes{\CSH,~\CWH:~\CState(\Cfr),~\CPhys}%
\itemspace

\bsgitem{2009a}{}  Disorder chaos and multiple valleys in spin glasses.  Manuscript, 2009.  
\arXiv{0907.3381}.								
\annot{} Weighted signed complete graph, $(\Sigma,w)$ where $w:E\to\bbR_{>0}$.  Near-ground states $s:V\to\signnos$ (nearly smallest switched weight, $\sum_E \sigma^s(e)w(e)$) are highly dispersed in $\signnos^V$, suggesting that 
\linebreak[4]
$\signnos^V$ has many near-minimal valleys.  Perturbation of $w$ can drastically change the valley structure.  The same is not true on the square grid.  
[Published by inclusion in \xrefd{2014a}{Sourav Chatterjee}.]
\dated{31 Aug 2012} 
\par
\codes{\CSG,~\CWG:~\CState(\Cfr),~\CPhys}%
\itemspace

\bsgitem{2014a}{}  \emph{Superconcentration and Related Topics}.  Springer Monographs in Math.  Springer, Cham, 2014.
\MR{3157205}.  \Zbl{1288.60001}.  
\annot{} Ch.\ 4: ``Multiple valleys''.
	
\codes{\CPhys:~\Csg:~}%
\itemspace

\author{Guy Chaty}							%CHATY
\authorspace
\bsgitem{1988a}{}  On signed digraphs with all cycles negative.  \emph{Discrete Appl.\ Math.}\ 20 (1988), 83--85.  
\MR{0936899} (89d:05148).  \Zbl{647.05028}.					
\annot{} Clarifies the structure of ``free cyclic'' digraphs and shows they include strong ``upper'' digraphs (see \xref{Harary, Lundgren, and Maybee}{1985a}{Frank Harary, J.\ Richard Lundgren, \& John S.\ Maybee}).
\par
\codes{\CSD:~\CStr}%
\itemspace

\author{Vaggos Chatziafratis}						%CHATZIAFRATIS
\author{Vaggos Chatziafratis, Neha Gupta, \& Euiwoong Lee}
\authorspace
\bsgitem{20xxa}{}   Inapproximability for local correlation clustering and dissimilarity hierarchical clustering.  \emph{}\ 
\arXiv{2010.01459}.  
\codes{\Csg:~\CClu:~\CAlgor}%
\itemspace

\author{M.\ Chaves}							%CHAVES
\authorspace
\authref  See \hyperlink{L.\ Tournier}{L.~Tournier}.
\itemspace

\authorx{P.D.\ Chawathe}		 				%CHAWATHE
\author{P.D.\ Chawathe \& G.R.\ Vijayakumar}
\authorspace
\bsgitem{1990a}{\dag} A characterization of signed graphs represented by root system $D_\infty$.  \emph{European J.\ Combin.}\ 11 (1990), 523--533.  
\MR{1078708} (91k:05071).  \Zbl{764.05090}.					
\annot{} A list of the 49 switching classes of signed simple graphs that are the forbidden induced subgraphs for a signed simple graph to be a reduced line graph of a simply signed graph without loops or half edges.  The graphs have orders 4, 5, and 6.  [See several other works of \hyperlink{G.R.\ Vijayakumar}{Vijayakumar} \etal]
\codes{\CSG:~\Cadj,~\CLG,~\CGeom,~\Cinc}%
\itemspace

\author{Shuchi Chawla}						%CHAWLA
\authorspace
\authref  See \hyperlink{Nikhil Bansal}{N.~Bansal}.
\itemspace

\author{Charalampos Chelmis}						%CHELMIS
\authorspace
\authref  See \hyperlink{Ajitesh Srivastava}{A.~Srivastava}.
\itemspace

\author{Beifang Chen}						%CHEN BF
\bsgitem{2018a}{}  Conformal decomposition of integral tensions and potentials of signed graphs.  \emph{SIAM J.\ Discrete Math.}\ 31 (2018), 2457--2478.
\MR{3719012}.  \Zbl{1373.05078}.  
\annot{} Dual to \xref{Chen--Wang--Zaslavsky}{2017a}{Beifang Chen, Jue Wang, \& Thomas Zaslavsky}.
	
\codes{\CSG:~\CFlows(\CDual)}%
\itemspace

\bsgitem{2021a}{}  Conformal decomposition of integral flows on signed graphs with outer-edges.  \emph{Graphs Combin.}\ 37 (2021), 2207--2225.
\MR{4338725}.  \Zbl{1479.05135}.  
\annot{} Improves upon \xref{Chen--Wang--Zaslavsky}{2017a}{Beifang Chen, Jue Wang, \& Thomas Zaslavsky}.
\codes{\CSG:~\CFlows}%
\itemspace

\author{Beifang Chen \& Shuchao Li}
\authorspace
\bsgitem{2011a}{}  The number of nowhere-zero tensions on graphs and signed graphs.  \emph{Ars Combin.}\ 102 (2011), 47--64. 
\MR{2847958}.  \Zbl{1265.05527}.  
\codes{\CSG}%
\itemspace

\author{Beifang Chen \& Jue Wang}
\authorspace
\bsgitem{2009a}{\dag}  The flow and tension spaces and lattices of signed graphs.  \emph{European J.\ Combin.}\ 30 (2009), 263--279.
\MR{2460231} (2009i:05102).  \Zbl{1198.05085}.					
\annot{} Introduces cuts, and directed circuits and cuts, of a signed graph; and the cycle (or circuit) and cut (or cocycle) spaces of a signed graph over a commutative, unital ring in which $2$ is invertible.
Definitions, basic theory, and graphical proofs.  Orthogonal complementarity between real, or integral, circuit and cut spaces.  Relationships between real and integral spaces.  Interpretations in terms of flows and tensions.
\annotctd{} A cut is an edge set $U := E\langle X,X^c \rangle \cup U_X$ where $X \subseteq V$ and $U_X$ is a minimal balancing set of $E\ind X$.  A minimal cut is a bond, i.e., a cocircuit in $\bfF(\Sigma)$.  A circuit or cut has two possible ``directions''.  A minimal directed cut need not be a directed bond.  The indicator vectors of directed circuits generate the cycle (``circuit'') space; the indicator vectors of directed cuts generate the cocycle (``cut'') space.
\annotctd{} The flow space or lattice is the real or integral null space of the incidence matrix.  The tension space or lattice is the real or integral row space.  The spaces equal lattices equal the real cycle and cut spaces and the lattices are their integral parts.  
Not every integral flow is in the integral span of circuit indicator vectors; but every integral tension is spanned by cut indicator vectors.
\annotctd{} [Based upon and extending parts of \xref{J.\ Wang}{2007a}{Jue Wang}.]
\par
\codes{\CSG:~\CStr,~\COri,~\CInc}%
\itemspace

\bsgitem{2010a}{}  Torsion formulas for signed graphs.  \emph{Discrete Appl.\ Math.}\ 158 (2010), 1148--1157.
\MR{2629892} (2011j:05131).  \Zbl{1255.05090}.					
\annotctd{} [Based upon part of \xref{J.\ Wang}{2007a}{Jue Wang}.]
\codes{\CSG:~\CFlows(\CDual)}%
\itemspace

\bsgitem{2011a}{}  Classification of indecomposable integral flows on signed graphs.  Manuscript, 2011, rev.\ 2013.  
\arXiv{1112.0642}.  
\codes{\CSG:~\CFlows}%
\itemspace

\author{Beifang Chen, Jue Wang, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2017a}{}  Resolution of irreducible integral flows on signed graphs.  \emph{Discrete Math.}\ 340 (2017), no.\ 6, 1271--1286.
\MR{3624612}.  \Zbl{1369.05098}.  \arXiv{1701.04494}.				
\annot{} Irreducible integral flows include circuit flows as well as others of a complicated and unexpected nature.  Resolved by lifting to the signed covering graph.  
[Based upon part of \xref{J.\ Wang}{2007a}{Jue Wang} and also \xref{Chen and Wang}{2011a}{Beifang Chen \& Jue Wang}, with a different method.]
\codes{\CSG:~\CInc,~\CStr}%
\itemspace

\author{Chao-Yang Chen}						%CHEN CY
\authorspace
\authref  See \hyperlink{Bin Hu}{B.~Hu}.
\itemspace

\authorx{Chen Chen}							%CHEN C
\author{Chen Chen, Jin Huang, \& Shuchao Li}
\authorspace
\bsgitem{2018a}{}  On the relation between the $H$-rank of a mixed graph and the matching number of its underlying graph.  \emph {Linear Multilinear Algebra} 66 (2018), no.\ 9, 1853--1869.
\MR{3825702}.  \Zbl{1392.05071}.  
\annot{} For $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$ ($\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges), $-2\xi(\|\Phi\|) \leq \rk A(\Phi) - \mu(\|\Phi\|) \leq \xi(\|\Phi\|)$.  Characterizes equalities.  
[A different matching formula in \xref{Tian--Chen--Chu}{2018a}{Fenglei Tian, Li Chen, \& Rui Chu}.]
\dated{15 Dec 2020} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Chen Chen, Shuchao Li, \& Minjie Zhang}
\authorspace
\bsgitem{2019a}{}  Relation between the $H$-rank of a mixed graph and the rank of its underlying graph.  \emph{Discrete Math.}\  342 (2019), no.\ 5, 1300--1309.
\emph{Discrete Math.}\ 342 (2019), no.\ 5, 1300--1309.
\MR{3905201}.  \Zbl{1407.05144}.  \arXiv{1812.05309}.				
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.  
$|\rk A(\Phi) - \rk A(\|\Phi\|)| \leq 2\xi(\|\Phi\|)$.  Equality is characterized.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Doron Chen}							%CHEN D
\authorspace
\authref  See also \hyperlink{Erik G.\ Boman}{E.G.~Boman}.
\authrefspace

\author{Doron Chen \& Sivan Toledo}
\authorspace
\bsgitem{2005a}{} Combinatorial characterization of the null spaces of symmetric H-matrices.  \emph{Linear Algebra Appl.}\ 392 (2004), 71--90.
\MR{2095908} (2005h:15016).  \Zbl{1061.65028}.					
\annot{} Certain matrices are related to gain graphs and others to signed graphs.
	
\par
\codes{\CGG,~\CSG:~\CInc,~\CMatrd}%
\itemspace

\author{Ge Chen}							%CHEN G
\authorspace
\authref  See \hyperlink{Wenjun Mei}{W.-J.~Mei}.
\itemspace

\authorx{Gina Chen}							%CHEN G
\author{Gina Chen, Vivian Liu, Ellen Robinson, Lucas J.\ Rusnak, \& Kyle Wang}
\authorspace
\bsgitem{2018a}{}  A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients.  \emph{Linear Algebra Appl.}\ 556 (2018), 323--341. 
\MR{3842586}.  \Zbl{1394.05070}.  \arXiv{1704.03599}.				
\codes{\CSH:~\COri:~\CAdj,~\CLap}%
\itemspace

\author{Haiyan Chen}						%CHEN HY
\authorspace
\authref  See \hyperlink{Yanan Zhang}{Y.-N.~Zhang}.
\itemspace

\author{Hanlin Chen}							%CHEN HL
\authorspace
\authref  See \hyperlink{Fayun Cao}{F.-Y.~Cao}.
\itemspace

\author{Jia-Fen Chen}						%CHEN JF
\authorspace
\authref  See also \hyperlink{Bang Ye Wu}{B.Y.~Wu}.
\authrefspace

\author{Jia-Fen Chen \& Bang Ye Wu}
\authorspace
\authorspace
\bsgitem{2013a}{}  \emph{Balancing a Complete Signed Graph by Editing Edges and Deleting Nodes}.  Thesis, National Chung Cheng Univ., Taiwan, 2013.
\annot{} Minimum number of edge and vertex deletions and edge additions to arrive at balance, by two algorithms.  [This thesis is presumably by Chen.]
\dated{5 Jun 2017} 
\codes{\CSG:~\CKG:~\CFr:~\CAlgor,~\CClu}%
\itemspace

\authorx{Jianer Chen}						%CHEN JE
\author{Jianer Chen \& Jonathan L.\ Gross}	
\authorspace
\bsgitem{1995a}{}  Voltage graphs for parallel architecture layouts.  In:  Y.\ Alavi and A.\ Schwenk, eds., \emph{Graph Theory, Combinatorics, Algorithms and Applications} (Proc.\ Seventh Quadren.\ Int.\ Conf.\ Theory Appl.\ Graphs, Kalamazoo, Mich., 1992), Vol. 1, pp.\ 455--466.  Wiley, New York, 1995.
\MR{1405831} (97j:05023).  \Zbl{843.68085}.  
\codes{\CGG:~\CCov,~\CTop}%
\itemspace

\author{Jianer Chen, Jonathan L.\ Gross, \& Robert G.\ Rieper}
\authorspace
\bsgitem{1994a}{}  Overlap matrices and total imbedding distributions.  \emph{Discrete Math.}\ 128 (1994), 73--94.  
\MR{1271857} (95f:05031).  \Zbl{798.05017}.
\codes{\CSG:~\CTop,~\CSw}%
\itemspace

\author{Jianer Chen \& Jie Meng}
\authorspace
\bsgitem{2010a}{}  A $2k$ kernel for the cluster editing problem.  In:  \emph{Computing and Combinatorics} (Proc.\ 16th Ann.\ Int.\ Conf., COCOON 2010, Nha Trang, Vietnam, 2010), pp.\ 459--468.  \LNCS 6196.   Springer, Berlin, 2010.
\MR{2720122} (no rev).  \Zbl{1286.05164}.  
\annot{} See \xrefd{2012a}{Jianer Chen \& Jie Meng}.
\codes{\Csg:~\Ckg:~\CClu:~\CAlgor}%
\itemspace

\bsgitem{2012a}{}  A $2k$ kernel for the cluster editing problem.  \emph{J.\ Computer Syst.\ Sci.}\ 78 (2012), no.\ 1, 211--220.
\MR{2896358}.  \Zbl{1238.68062}.  
\annot{} 	
Equivalent:  Is the clusterability index $l_{\clu}(K_n,\sigma) \leq k$?  
Dictionary:  ``graph'' = positive subgraph of $(K_n,\sigma)$; ``editing'' = edge sign changes.  
[Definition:  $l_{\clu}=$ smallest number of sign changes that give a clusterable (\cf\ \xref{Davis}{1967a}{James A.\ Davis}) signed $K_n$.]
	
\dated{13 Jun, 1 July 2017} 
\par
\codes{\Csg:~\Ckg:~\CClu:~\CAlgor}%
\itemspace

\author{Jianwen Chen}						%CHEN JW
\authorspace
\authref  See \hyperlink{Wenqiang Duan}{W.-Q.~Duan}.
\itemspace

\author{Jie Chen}							%CHEN JE
\authorspace
\authref  See \hyperlink{Ye Jiang}{Y.~Jiang} and \hyperlink{Hongwei Zhang}{H.W.~Zhang}.
\itemspace

\authorx{Jing Chen}							%CHEN J
\author{Jing Chen \& Genghua Fan}
\authorspace
\bsgitem{2018a}{}  Short signed circuit covers of signed graphs.  \emph{Discrete Appl.\ Math.}\ 235 (2018), 51--58.
\MR{3732594}.  \Zbl{1375.05123}.  
\codes{\CSG:~\Cmatrd}%
\itemspace

\bsgitem{2021a}{}  Circuit $k$-covers of signed graphs.  \emph{Discrete Appl.\ Math.}\ 294 (2021), 41--54.
\MR{4214950}.  \Zbl{1459.05111}.  
\codes{\CSG:~\Cmatrd}%
\itemspace

\author{Li Chen}							%CHEN L
\authorspace
\authref  See \hyperlink{Fenglei Tian}{F.-L.~Tian}.
\itemspace

\author{Li-Hsuan Chen}						%CHEN LH
\authorspace
\authref  See also \hyperlink{Bang Ye Wu}{B.Y.~Wu}.
\authrefspace

\author{Li-Hsuan Chen, Maw-Shang Chang, Chun-Chieh Wang, \& Bang Ye Wu}
\authorspace
\bsgitem{2013a}{}  On the min-max 2-cluster editing problem.  In:  \emph{Advances in Intelligent Systems and Applications}, Vol.\ 1, pp.\ 133--142.  Springer, Berlin, 2013.
\annot{} \Cf\  \xrefd{2013b}{Li-Hsuan Chen, Maw-Shang Chang, Chun-Chieh Wang, \& Bang Ye Wu}, \xref{Chen and Wu}{2017a}{Li-Hsuan Chen \& Bang Ye Wu}.
	
\codes{\Csg:~\Ckg:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2013b}{}  On the min-max 2-cluster editing problem.  \emph{J.\ Inform.\ Sci.\ Engineering} 29 (2013), no.\ 6, 1109--1120.
\MR{3137567}.  
\codes{\Csg:~\Ckg:~\CFr:~\CAlgor}%
\itemspace

\author{Li-Hsuan Chen \& Bang Ye Wu}
\authorspace
\bsgitem{2017a}{}  Parameterized algorithms for min--max 2-cluster editing.  \emph{J.\ Combin.\ Optim.}\ 34 (2017), 47--63.
\MR{3661065}.  \Zbl{1378.05197}.  
\annot{} Equivalent to:  Is $l(K_n,\sigma) \leq k$?  Dictionary:  ``graph'' = positive subgraph of $(K_n,\sigma)$; ``editing'' = edge sign changes; ``$2$-clustering'' = bipartition of $V$.  
	
\dated{13 Jun 2017} 
\codes{\Csg:~\Ckg:~\CFr:~\CAlgor}%
\itemspace

\authorx{Ming-Zhu Chen}						%CHEN MZ
\author{Ming-Zhu Chen, A-Ming Liu, \& Xiao-Dong Zhang}
\authorspace
\bsgitem{20xxa}{}  The signless Laplacian spectral radius of graphs without intersecting odd cycles.  \emph{}\ 
\arXiv{2108.03895}.  
\annot{} The matrix is $\Lap(-\Gamma)$ where $\Gamma$ excludes a subgraph of odd circles with one common vertex.
[\Problem.  Generalize to negative circles in a signed graph.  That allows even circles; it should be easy.]
\dated{17 Oct 2022} 
\par
\codes{\Csg:~\CPar:~\CLap:~\CEig}%
\itemspace

\author{Qiyao Chen}							%CHEN QY
\author{Qiyao Chen \& Yichao Chen}
\authorspace
\bsgitem{2022a}{}  Parallel edges in ribbon graphs and interpolating behavior of partial-duality polynomials.  \emph{European J.\ Combin.}\ 102 (2022), art.\ 103492, 12 pp. 
\MR{4350495}.  
\arXiv{2106.00381}.  
\annot{} \Cf\ \xref{Gross, Mansour, and Tucker}{2021a}{Jonathan L.\ Gross, Toufik Mansour, \& Thomas W.\ Tucker}.
\codes{\Csg:~\CTop:~\CDual:~\CInvar}%
\itemspace

\author{Rong Chen}							%CHEN R
\authorspace
\authref  See also \hyperlink{Bo Bao}{B.~Bao}.
\authrefspace

\bsgitem{2017a}{}  The excluded minors for the class of matroids that are graphic or bicircular lift.  \emph{Adv.\ Appl.\ Math.}\ 83 (2017), 97--114.
\MR{3573220}.  \Zbl{1351.05048}.  
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Rong Chen, Matt DeVos, Daryl Funk, \& Irene Pivotto}
\authorspace
\bsgitem{2015a}{}  Graphical representations of graphic frame matroids.  \emph{Graphs Combin.}\ 31 (2015), no.\ 6, 2075--2086.
\MR{3417216}.  \Zbl{1327.05143}.  \arXiv{1403.7733}.				
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Rong Chen \& Zifei Gao}
\authorspace
\bsgitem{2016a}{}  Representations of bicircular lift matroids.  \emph{Electronic J.\ Combin.}\ 23 (2016), no.\ 3, Paper 3.42, 11 pp. 
\MR{3558079}.  \Zbl{1344.05037}.  \arXiv{1510.02643}.				
\par
\codes{\CBic}%
\itemspace

\author{Rong Chen \& Irene Pivotto}
\authorspace
\bsgitem{2018a}{}  Biased graphs with no two vertex-disjoint unbalanced cycles.  \emph{J.\ Combin.\ Theory Ser.\ B} 130 (2018), 207--245.
\MR{3772740}.  \Zbl{1384.05102}.  \arXiv{1403.1919}.				
\codes{\CGG:~\CStr}%
\itemspace

\author{Rong Chen \& Geoff Whittle}
\authorspace
\bsgitem{2018a}{}  On recognizing frame and lifted-graphic matroids.  \emph{J.\ Graph Theory} 86 (2018), no.\ 1, 72--76.
\MR{3729836}.  \Zbl{06843119}.  \arXiv{1601.01791}.				
\codes{\CGG:~\CMatrd:~\CAlgor}%
\itemspace

\author{Rong Chen \& Kai-nan Xiang}
\authorspace
\bsgitem{2012a}{}  Decomposition of 3-connected representable matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 102 (2012), no.\ 3, 647--670.
\MR{2900809}.  \Zbl{1238.05046}.  
\annot{} A ``spike-like'' or ``swirl-like'' matroid is $\bfL(\Gamma,\cB)$ or $\bfF(\Gamma,\cB)$ where $\Gamma$ is a circle with all edges doubled or more.  Thm.\ 1.1:  A 3-connected vector matroid with $\#E\geq9$ decomposes into spike-like, swirl-like, and ``freely-placed-line'' matroids and sequentially 4-connected matroids, assembled in a tree pattern along modular lines.
\dated{18 Apr 2013} 
\par
\codes{\Cgg:~\CMatrd:~\CStr}%
\itemspace

\author{Siyuan Chen}						%CHEN SY
\authorspace
\authref  See \hyperlink{Yufei Huang}{Y.F.~Huang}.
\itemspace

\authorx{Tianran Chen}						%CHEN TR
\author{Tianran Chen, Robert Davis, \& Evgeniia Korchevskaia}
\authorspace
\bsgitem{20xxa}{}  Facets and facet subgraphs of adjacency polytopes.  Submitted.  \emph{}\ 
\arXiv{2107.12315}.  
\annot{} Thm.\ 12 and Rem.\ 13 implicitly state the connection with balanced flats of an even-cycle matroid $\bfF(-\Gamma)$.
\dated{20 Jun 2022} 
\par
\codes{\Csg:~\Cpar:~\CMatrd,~\CGeom}%
\itemspace

\authorx{Vinciane Chen}						%CHEN V
\author{Vinciane Chen, Angeline Rao, Lucas J.\ Rusnak, \& Alex Yang}
\authorspace
\bsgitem{2015a}{}  A characterization of oriented hypergraphic balance via signed weak walks.  \emph{Linear Algebra Appl.}\ 485 (2015), 442--453.
\MR{3394156}.  \Zbl{1322.05087}.  
\par
\codes{\CSH:~\CBal}%
\itemspace

\author{Wei Chen}							%CHEN W
\authorspace
\authref  See also \hyperlink{Yanhua Li}{Y.-H.~Li}.
\authrefspace

\author{Wei Chen, Dan Wang, Ji Liu, Tamer Ba\c{s}ar, \& Li Qiu}
\authorspace
\bsgitem{2017a}{}  On spectral properties of signed Laplacians for undirected graphs.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 1999--2002.  IEEE, 2017.
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Weisheng Chen}						%CHEN WS
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\authorx{William Y.C.\ Chen}						%CHEN WYC
\author{William Y.C.\ Chen, Larry X.W.\ Wang, \& Arthur L.B.\ Yang}
\authorspace
\bsgitem{2011a}{}  Recurrence relations for strongly $q$-log-convex polynomials.  \emph{Canadian Math.\ Bull.}\ 54 (2011), no.\ 2, 217--229.
\MR{2884236} (2012k:05029).  \Zbl{1239.05190}.  
\annot{} \S3.4, ``The Dowling polynomials'': $D_m(n;x)$ and $F_{m,1}(n;x)$ are from \xref{Benoumhani}{1997a}{Moussa Benoumhani}.
	
\dated{28 Jan 2015} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\authorx{Xiaodan Chen}						%CHEN XD
\author{Xiaodan Chen \& Yaoping Hou}
\authorspace
\bsgitem{2016a}{}  Upper bounds for the signless Laplacian spectral radius of graphs on surfaces.  \emph{Filomat} 30 (2016), no.\ 13, 3473--3481. 
\MR{3593047}.  \Zbl{1462.05214}.  
\codes{\Csg:~\Cpar:~\CLap:~\CEig,~\CTop}%
\itemspace

\authorx{Xiaolin Chen}						%CHEN XL
\author{Xiaolin Chen, Xueliang Li, \& Yingying Zhang}
\authorspace
\bsgitem{2016a}{}  3-Regular mixed graphs with optimum Hermitian energy.  \emph{Linear Algebra Appl.}\ 496 (2016), 475--486.
\MR{3464084}.  \Zbl{1331.05139}.  
\annot{} 
In defining the Hermitian adjacency matrix there seems to be a missing line $h_{lk} = +i$ if $l \to k$.
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Xiaoyue Chen}						%CHEN XY
\authorspace
\authref  See \hyperlink{Ligang Jin}{L.-G.~Jin}.
\itemspace

\authorx{Xudong Chen}						%CHEN XD
\author{Xudong Chen, Mohamed-Ali Belabbas, \& Tamer Ba\c{s}ar}
\authorx{Xudong Chen, Mohamed-Ali Belabbas, \& Tamer Basar}
\authorspace
\bsgitem{2017a}{}  Cluster consensus with point group symmetries.  \emph{SIAM J.\ Control Optim.}\ 55 (2017), no.\ 6, 3869--3889.
\MR{3732941}.  \Zbl{1376.05036}.  \arXiv{1601.06346}.  
\annot{} Gain-graph generalization of \xref{Altafini}{2012a}{Claudio Altafini} \etal
\codes{\CGG:~\CClu:~\CDyn}%
\itemspace

\authorx{Y.\ Chen}							%CHEN Y
\author{Y.\ Chen, X.L.\ Wang, B.\ Yuan, \& B.Z.\ Tang}
\authorspace
\bsgitem{2014a}{}  Overlapping community detection in networks with positive and negative links.  \emph{J.\ Stat.\ Mech.}\ 2014 (2014), art.\ P03021, 22 pp.
\arXiv{1310.4023}.  
\codes{\CSG:~\CClu}%
\itemspace

\authorx{Ya-Hong Chen}						%CHEN YH
\author{Ya-Hong Chen, Rong-Ying Pan, \& Xiao-Dong Zhang}
\authorspace
\bsgitem{2011a}{}  Two sharp upper bounds for the signless Laplacian spectral radius of graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ 3 (2011), no.\ 2, 185--191.
\MR{2822283} (2012f:\-05173).  \Zbl{1222.05149}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Yanqing Chen}						%CHEN YQ
\author{Yanqing Chen \& Ligong Wang}
\authorspace
\bsgitem{2010a}{}  Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 5, 908--913.
\MR{2658641} (2011h:05149).  \Zbl{1215.05100}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Yichao Chen}						%CHEN YC
\authorspace
\authref  See also \hyperlink{Qiyao Chen}{Q.-Y.~Chen}.
\authrefspace

\bsgitem{20xxa}{}  Permutation-bipartition pairs.  Submitted.  \emph{}\ 
\arXiv{2204.04359}.  
\codes{\CSG:~\CTop}%
\itemspace

\authorx{Yu Chen}							%CHEN Y
\author{Yu Chen \& Yaoping Hou}
\authorspace
\bsgitem{2021a}{}  Eigenvalue multiplicity in cubic signed graphs.  \emph{Linear Algebra Appl.}\ 630 (2021), 95--111.
\MR{4299098}.  \Zbl{1473.05167}.  
\annot{} Multiplicity of eigenvalue $\mu$ of $A(\Sigma)$ is $\leq\frac12n$ ($=$ iff $\mu=\pm\sqrt3$), except $\leq\frac12n+1$ when $\mu=\pm1$ ($=$ iff $\Sigma \simeq(K_4,-\mu)$) or $\mu=0$ ($=$ iff $\Sigma \simeq +K_{33}$).
\dated{1 Oct 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Yu Qing Chen}						%CHEN YQ
\author{Yu Qing Chen, Anthony B.\ Evans, Xiaoyu Liu, Daniel C.\ Slilaty, \& Xiangqian Zhou}
\authorspace
\bsgitem{20xxa}{}  Representations of signed graphs.  Submitted. \emph{}\ 
\codes{\CSG,~\CGG:~\CGeom}%
\itemspace

\authorx{Zhibin Chen}						%CHEN ZB
\author{Zhibin Chen \& Wenan Zang}
\authorspace
\bsgitem{2009a}{}  Odd-$K_{4}$'s in stability critical graphs.  \emph{Discrete Math.}\  309 (2009), no.\ 20, 5982--5985.
\MR{2552630} (2010j:05291).  \Zbl{1229.05134}.  
\codes{\Csg:~\Cpar:~\CStr}%
\itemspace

\authorx{Zhi-Hong Chen}						%CHEN ZH
\author{Zhi-Hong Chen, Ying-Qiang Kuang, \& Hong-Jian Lai}
\authorspace
\bsgitem{1999a}{}  Connectivity of cycle matroids and bicircular matroids.  \emph{Ars Combin.}\ 52 (1999), 239--250.
\MR{1705651} (2001d:05032).  \Zbl{977.05027}.						
\annot{} The relationship between graph structure and the Tutte, verticial, and cyclic connectivities of the bicircular matroid.
\codes{\CBic:~\CStr}%
\itemspace

\authorx{Bo Cheng}							%CHENG B
\author{Bo Cheng \& Bolian Liu}
\authorspace
\bsgitem{2008a}{}  The base sets of primitive zero-symmetric sign pattern matrices.  \emph{Linear Algebra Appl.}\ 428 (2008), 715--73.
\MR{2382083} (2009c:15028).  \Zbl{1135.15014}.					
\annot{} The \xref{Abelson--Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} algebra is employed, with symbols $0,1,-1,\#$ for $o,p,n,a$.  ``Generalized sign pattern matrix'': $\#$ entries are allowed.   ``Generalized signed digraph'': $\#$-arcs are allowed.  
\codes{\CQM:~\CSD}%
\itemspace

\bsgitem{2010a}{}  Primitive zero-symmetric sign pattern matrices with the maximum base.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 2, 365--379.
\MR{2645090} (2011e:15058).  \Zbl{1193.15029}.					
\codes{\CQM:~\CSD}%
\itemspace

\authorx{Feng Cheng}						%CHENG F
\author{Feng Cheng \& Li Hua You}
\authorspace
\bsgitem{2012a}{}  The base set of primitive anti-symmetric signed digraphs with no loops.  (In Chinese.)  \emph{J.\ South China Normal Univ.\ Natural Sci.\ Ed.}\  44 (2012), no.\ 4, 13--19.
\MR{3052990} (no rev).  \Zbl{1289.05188} (\emph{q.v.}).  
\codes{\CSD:~\Cqm:~\CDyn}%
\itemspace

\author{Jian Cheng}							%CHENG JA
\authorspace
\bsgitem{2017a}{}  \emph{Integer Flows and Circuit Covers of Graphs and Signed Graphs}.  Doctoral dissertation, West Virginia University, 2017. 
\MR{3732040} (no rev).  
\par
\codes{\CSG:~\CFlows,~\Cflows}%
\itemspace

\author{Jian Cheng, You Lu, Rong Luo, \& Cun-Quan Zhang}
\authorspace
\bsgitem{2019a}{}  Shortest circuit covers of signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B}\ 134 (2019), 164--178.
\MR{3906634}.  \Zbl{1402.05095}.  \arXiv{1510.05717}.				
\codes{\CSG:~\Cflows}%
\itemspace

\author{Ying Cheng} 							%CHENG Y
\authorspace
\bsgitem{1986a}{}  Switching classes of directed graphs and $H$-equivalent matrices.  \emph{Discrete Math.}\ 61 (1986), 27--40.  
\MR{0850927} (88a:05075).  \Zbl{609.05039}.					
\annot{} This article studies what are described as $\bbZ_4$-gain graphs $\Phi$ with underlying simple graph $\Gamma$.  [However, see below.]  They are regarded as digraphs $D$, the gains being determined by $D$ as follows: $\phi(u,v) = 1$ or $2$ if $(u,v)$ is an arc, $2$ or $3$ if $(v,u)$ is an arc.  [N.B.  $\Gamma$ is not uniquely determined by $D$.]  Cheng's ``switching'' is gain-graph switching but only by switching functions $\eta : V \to \{0,2\}$; I will call this ``semiswitching''.  His ``isomorphisms'' are vertex permutations that are automorphisms of $\Gamma$; I will call them ``$\Gamma$-isomorphisms''.  The objects of study are equivalence classes under semiswitching (semiswitching classes) or semiswitching and $\Gamma$-isomorphism (semiswitching $\Gamma$-isomorphism classes).
Prop.\ 3.1 concerns adjacency of vertex orbits of a $\Gamma$-isomorphism that preserves a semiswitching class (call it a $\Gamma$-automorphism of the class).
Thm.\ 4.3 gives the number of semiswitching $\Gamma$-isomorphism classes.
Thm.\ 5.2 characterizes those $\Gamma$-automorphisms of a semiswitching class that fix an element of the class; Thm.\ 5.3 characterizes the $\Gamma$-isomorphisms $g$ that fix an element of every $g$-invariant semiswitching class.
\annotctd{} [Likely the right viewpoint, as is hinted in \S6, is that the edge labels are not $\bbZ_4$-gains but weights from the set $\{\pm1, \pm2, \ldots, \pm k\}$ with $k = 2$.  Then semiswitching is ordinary signed switching, and so forth.  However, I forbear to reinterpret everything here.]
\annotctd{} In \S6, $\bbZ_4$ is replaced by $\bbZ_{2k}$ [but this should be $\{\pm1, \pm2, \ldots, \pm k\}$]; semiswitching functions take values $0, k$ only.  Generalizations of \S\S3, 4 are sketched and are applied to find the number of $H$-equivalent matrices of given size with entries $\pm1, \pm1, \ldots, \pm k$.  ($H$- [or Hadamard] equivalence means permuting rows and columns and scaling them by $-1$.)
\par
\codes{\Csg,~\Cwg,~\CGG:~\CSw,~\CAut,~\CEnum}%
\itemspace

\author{Ying Cheng \& Albert L.\ Wells, Jr.}
\authorspace
\bsgitem{1984a}{}  Automorphisms of two-digraphs.  (Summary.)  
Proc.\ Fifteenth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing (Baton Rouge, 1984), [Vol.\ 3].  \emph{Congressus Numer.}\ 45 (1984), 335--336.  
\MR{0777706} (86c:05004c) (volume).  
\annot{} A two-digraph is a switching class of $\bbZ_3$-gain graphs based on $K_n$.
\par
\codes{\Cgg,~\CSD:~\CSw,~\CAut}%
\itemspace

\bsgitem{1986a}{\dag}  Switching classes of directed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 40 (1986), 169--186.  
\MR{0838217} (87g:05104).  \Zbl{565.05034}, (\Zbl{579.05027}). 			
\annot{} This exceptionally interesting paper treats a digraph as a ternary gain graph $\Phi$ (i.e., with gains in $\GF(3)^+$) based on $K_n$.  
A theory of switching classes and triple covering graphs, analogous to that of signed complete graphs (and of two-graphs) is developed.  The approach, analogous to that in \xref{Cameron}{1977b}{Peter J.\ Cameron}, employs cohomology.  The basic results are those of general gain-graph theory specialized to the ternary gain group and graph $K_n$.  
\annotctd{} The main results concern a switching class $[\Phi]$ of digraphs and an automorphism group $\fA$ of $[\Phi]$.
\S3, ``The first invariant'':  Thm.\ 3.2 characterizes, by a cohomological obstruction $\gamma$, the pairs $([\Phi], \fA)$ such that some digraph in $[\Phi]$ is fixed.  Thm.\ 3.5 is an [interestingly] more detailed result for cyclic $\fA$.
\S4: ``Triple covers and the second invariant''.  Digraph triple covers of the complete digraph are considered.  Those that correspond to gain covering graphs of ternary gain graphs $\Phi$ are characterized (``cyclic triple covers'', pp.\ 178--180).  Automorphisms of $\Phi$ and its triple covering $\tilde\Phi$ are compared.  Given $([\Phi], \fA)$, Thm.\ 4.4 finds the cohomological obstruction $\beta$ to lifting $\fA$ to $\tilde\Phi$.  Thm.\ 4.7 establishes an equivalence between $\gamma$ and $\beta$ in the case of cyclic $\fA$.
\annotctd{} \S5: ``Enumeration''.  Thm.\ 5.1 gives the number of isomorphism types of switching classes on $n$ vertices, based on the method of \xref{Wells}{1984a}{Albert L.\ Wells, Jnr.} for signed graphs.
\S6: ``The fixed signing property''.  Thm.\ 6.1 characterizes the permutations of $V(K_n)$ that fix a gain graph in every invariant switching class, based on the method of \xref{Wells}{1984a}{Albert L.\ Wells, Jnr.}
\annotctd{} Dictionary: ``Alternating function'' on $X \times X$ = $\GF(3)^+$-valued gain function on $K_X$.
\annotctd{} [See \xref{Babai and Cameron}{2000a}{L.\ Babai \& P.J.\ Cameron} for a treatment of tournaments as nowhere-zero ternary gain graphs based on $K_n$.]
\par
\codes{\Cgg:~\CSw,~\CAut,~\CEnum,~\CCov}%
\itemspace

\authorx{Zhiyun Cheng}						%CHENG ZY
\author{Cheng Zhiyun \& Gao Hongzhu}
\authorspace
\bsgitem{2012a}{}  Some applications of planar graph in knot theory.  \emph{Acta Math.\ Sci.\ Ser.\ B, Eng.\ Ed.}\ 32B (2012), no.\ 2, 663--671.
\MR{2921907}.  \Zbl{1265.57002}.  
\annot{} Which planar sign-colored graphs from link diagrams correspond to knots.
\dated{26 Jul 2013} 
\codes{\CSGc:~\CKnot}%
\itemspace

\author{Zhiyun Cheng, Ziyi Lei, Yitian Wang, \& Yanguo Zhang}
\authorspace
\bsgitem{2022a}{}  A categorification for the signed chromatic polynomial.   \emph{Electronic J.\ Combin.}\ 29 (2022), no.\ 2, art.\ P2.49.
\MR{4441085}.  
\arXiv{2101.01661}.  
\annot{} Cohomology theories for the chromatic polynomial and the balanced chromatic polynomial of $\Sigma$ using a chain complex for each.  Cohomology properties coordinate nicely with some polynomial properties.  There are no categories.  [\Question.  Is their cohomology related to that of (the complement of) the associated complex hyperplane arrangements of the signed graph?]  
Dictionary:  ``signed chromatic polynomial'' = chromatic quasipolynomial.  [Their history of signed graph coloring is somewhat misleading.]
\dated{6 Jan 2021, 17 Jun 2022} 
\par
\codes{\CSG:~\CCol:~\CAlgeb,~\CInvar}%
\itemspace

\authorx{Gi-Sang Cheon}						%CHEON GS
\author{Gi-Sang Cheon \& Ji-Hwan Jung}
\authorspace
\bsgitem{2012a}{}  $r$-Whitney numbers of Dowling lattices.  \emph{Discrete Math.}\ 312 (2012), no.\ 15, 2337--2348.
\MR{2926106}.  \Zbl{1246.05009}.  
\annot{} \Cf\ \xref{Mez\H{o}}{2010a}{Istvan Mezo}.  [Cheon and Jung seem to have made an independent discovery.]
\dated{8 Apr 2016} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\authorx{Danila Cherkashin}					%CHERKASHIN
\author{Danila Cherkashin \& Pavel Prozorov}
\authorspace
\bsgitem{2022a}{}  On the minimal sum of edges in a signed edge-dominated graph.  \emph{Electronic J.\ Combin.}\ 29 (2022), no.\ 3, art.\ P3.38, 18 pp.
\MR{4467140}.  
\arXiv{2012.09956}.  
\annot{} Improves \xref{Akbari, Bolouki, \etal\!}{2009a}{S.\ Akbari, S.\ Bolouki, P.\ Hatami, and M.\ Siami}, i.e.,  $s \geq -n^2/25$, and $s=-(1+o(1)) n^2/8(1+\sqrt2)^2$ is attainable.
\dated{15 Aug 2022} 
\codes{\CSGw}%
\itemspace

\author{T.C.\ Chern}							%CHERN TC
\authorspace
\authref  See \xref{Kuo--Chern--Shih}{1988a}{Y.S.\ Kuo, T.C.\ Chern, \& Wei-kuan Shih}.
\itemspace

\author{Yonah Cherniavsky}						%CHERNIAVSKY
\authorspace
\authref  See also \hyperlink{Vadim Bugaenko}{V.~Bugaenko}.
\authrefspace

\author{Yonah Cherniavsky, Avraham Goldstein, \& Vadim E.\ Levit}
\authorspace
\bsgitem{2013a}{}  On the structure of the group of balanced labelings on graphs.  
In:  Jaroslav Ne\v{s}et\v{r}il and Marco Pellegrini, eds., \emph{The Seventh European Conference on Combinatorics, Graph Theory and Applications} (EuroComb 2013, Pisa), pp.\ 117--122.  CRM Ser., Vol.\ 16.  Edizioni della Normale, Scuola Normale Superiore Pisa, Pisa, Italy, 2013.
\MR{3184274} (no rev).  \Zbl{1293.05157}.  
\par
\codes{\CGG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2014a}{}  Groups of balanced labelings on graphs.  \emph{Discrete Math.}\ 320 (2014), 15--25.
\MR{3147203}.  \Zbl{1281.05117}.  \arXiv{1301.4206}.				
\codes{\CGG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2017a}{}  Balanced Abelian group-valued functions on directed graphs.  \emph{Ars Math.\ Contemp.}\ 13 (2017), no.\ 2, 307--315. 
\MR{3720533}.  \Zbl{1380.05093}.  
\annot{} Abelian gains on a graph (``flexible case'') and a digraph (``rigid case'').
	
\codes{\CGG,~\CGD:~\CBal}%
\itemspace

\author{Yonah Cherniavsky, Avraham Goldstein, Vadim E. Levit, \& Robert Shwartz}
\authorspace
\bsgitem{2016a}{}  Enumeration of balanced finite group valued functions on directed graphs.  \emph{Inform.\ Processing Lett.}\ 116 (2016), no.\ 7, 484--488. 
\MR{3479183}.  \Zbl{1353.05064}.  \arXiv{1405.3686}.				
\codes{\CGG(\CGen):~\CBal(\CGen)}%
\itemspace

\author{Yonah Cherniavsky \& Robert Shwartz}
\authorspace
\bsgitem{2020a}{}  Quotients of Coxeter groups associated to signed line graphs.  \emph{Adv.\ Appl.\ Discrete Math.}\ 25 (2020), no.\ 2, 213--261.
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{William K.\ Cheung}						%CHEUNG
\authorspace
\authref  See \hyperlink{Bo Yang}{B.~Yang}.
\itemspace

\author{Ming Chi}							%CHI M
\authorspace
\authref  See \hyperlink{Bin Hu}{B.~Hu}.
\itemspace

\author{Kai-Yang Chiang}						%CHIANG KY
\authorspace
\authref  See also \hyperlink{Cho-Jui Hsieh}{C.-J.~Hsieh}.
\authrefspace

\author{Kai-Yang Chiang, Cho-Jui Hsieh, Nagarajan Natarajan, Ambuj Tewari, \& Inderjit S.\ Dhillon}
\authorspace
\bsgitem{2014a}{}  Prediction and clustering in signed networks: A local to global perspective.  \emph{J.\ Machine Learning Res.}\ 15 (2014), 1177--1213.
\MR{3195342}.  \Zbl{1319.91134}.  \arXiv{1302.5145}.				
\codes{\CSG:~\CClu,~\CPred}%
\itemspace

\author{Kai-Yang Chiang, Nagarajan Natarajan, Ambuj Tewari, \& Inderjit S.\ Dhillon}
\authorspace
\bsgitem{2011a}{}  Exploiting longer cycles for link prediction in signed networks.  In:  \emph{Proceedings of the 20th ACM Conference on Information and Knowledge Management} (CIKM '11, Glasgow, 2011), pp.\ 1157--1162.  ACM, New York, 2011.
\par
\codes{\CSG,~\CSD:~\CPred:~\CFr:~\CAlgor,~\CPsS}%
\itemspace

\author{Kai-Yang Chiang, Joyce Jiyoung Whang, \& Inderjit S.\ Dhillon}
\authorspace
\bsgitem{2012a}{}  Scalable clustering of signed networks using balance normalized cut.  In:  \emph{Proceedings of the 21st ACM Conference on Information and Knowledge Management} (CIKM'12, Maui, 2012), pp.\ 615--624.  ACM, New York, 2012.
\par
\codes{\CSG:~\CBal,~\CClu:~\CAlgor}%
\itemspace

\authorx{Shuya Chiba}						%CHIBA
\author{Shuya Chiba, Shinya Fujita, Ken-ichi Kawarabayashi, \& Tadashi Sakuma}
\authorspace
\bsgitem{2009a}{}  Disjoint even cycles packing.  European Conf.\ Combin.\ Graph Theory Appl.\ (EuroComb 2009).  \emph{Electronic Notes Discrete Math.}\ 34 (2009), 113--119.
\MR{2591427} (no rev).  \Zbl{1273.05169}.  
\codes{\CPar:~\CCircles}%
\itemspace

\bsgitem{2014a}{}  Minimum degree conditions for vertex-disjoint even cycles in large graphs.  \emph{Adv.\ Appl.\ Math.}\ 54 (2014), 105--120.
\MR{3199776}.  \Zbl{1284.05209}.  
\codes{\CPar:~\CCircles}%
\itemspace

\author{K.P.\ Chithra}							%CHITRA
\authorspace
\authref  See \hyperlink{Naduvath K.\ Sudev}{N.K.~Sudev}.
\itemspace

\author{Sergei Chmutov}						%CHMUTOV
\authorspace
\bsgitem{2009a}{}  Generalized duality for graphs on surfaces and the signed Bollob\'as--Riordan polynomial.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), no.\ 3, 617--638.
\MR{2507944} (2010f:05046).  \Zbl{1172.05015}.  
\annot{} Sign-colored graphs embedded in a surface (\xref{Chmutov and Pak}{2007a}{Sergei Chmutov \& Igor Pak}).  Duality with respect to an edge subset, applied to a sign-colored Bollob\'as--Riordan polynomial, gives a polynomial duality.  [Further developments in \xref{Vignes-Tourneret}{2009a}{Fabien Vignes-Tourneret} and \xref{Krushkal}{2011a}{Vyacheslav Krushkal}.]
\par
\codes{\CSGc:~\CTop,~\CInvar}%
\itemspace

\author{Sergei Chmutov \& Igor Pak}
\authorspace
\bsgitem{2007a}{}  The Kauffman bracket of virtual links and the Bollob\'as-Riordan polynomial.  \emph{Moscow Math.\ J.}\ 7 (2007), no.\ 3, 409--418.
\MR{2343139} (2008h:57006).  \Zbl{1155.57004}.
\annot{} Sign-colored graphs embedded in a surface (orientable or not, independently of the edge signs.  [The orientation properties of the ribbons make a signed graph, independent of the sign-colors.]  
\codes{\CSGc,~\Csg:~\CTop,~\CInvar}%
\itemspace

\author{Kwang-Hyun Cho}						%CHO KH
\authorspace
\authref  See \hyperlink{Jeong-Rae Kim}{J.-R.~Kim} and \hyperlink{Yung-Keun Kwon}{Y.-K.~Kwon}.
\itemspace

\authorx{Hyeong-ah Choi}				 		%CHOI HA
\author{Hyeong-ah Choi, Kazuo Nakajima, \& Chong S.\ Rim}
\authorspace
\bsgitem{1989a}{}  Graph bipartization and via minimization.  \emph{SIAM J.\ Discrete Math.}\ 2 (1989), 38--47.  
\MR{0976786} (89m:90132).  \Zbl{677.68036}.					
\annot{} Vertex biparticity [i.e., vertex frustration number $l_0(-\Gamma)$] is compared to edge biparticity [frustration index $l(-\Gamma)$] (for cubic graphs) and studied algorithmically.  Proved: $l_0(-\Gamma)=l(-\Gamma)$ for cubic graphs; thus, cubic ``$l_0\leq k$'' is NP-complete because ``$l\leq k$'' is and $l_0=l$.  [Equality is generalized in \xref{Sivaraman and Zaslavsky}{20xxa}{Vaidy Sivaraman \& Thomas Zaslavsky}.]
\codes{\Cpar:~\CFr:~\CAlgor}%
\itemspace

\author{Timothy Y.\ Chow}						%CHOW
\authorspace
\bsgitem{2003a}{}  Symplectic matroids, independent sets, and signed graphs.  \emph{Discrete Math.}\ 263 (2003), 35--45.
\MR{1955713} (2004a:05033).  \Zbl{1014.05017}.					
\annot{} \S4, ``From graphs to symplectic matroids'':  The matroid union of $\bfF(\Gamma,\sigma)$ over all signatures of a fixed graph yields a symplectic matroid.
\par
\codes{\CSG:~\CMatrd}%
\itemspace

\author{Debashish Chowdhury}						%CHOWDHURY
\authorspace
\bsgitem{1986a}{}  \emph{Spin Glasses and Other Frustrated Systems}.  Princeton Univ.\ Press, Princeton, and World Scientific, Singapore, 1986.
\annot{} Includes brief survey of how physicists look upon frustration.  See esp.\ \S1.3, ``An elementary introduction to frustration'', where the signed square lattice graph illustrates balance vs.\ imbalance; Ch.\ 20, ``Frustration, gauge invariance, defects and SG [spin glasses]'', discussing planar duality (see e.g.\ \xref{Barahona}{1982a}{Francisco Barahona}, ``gauge theories'', where gains are in the orthogonal or unitary group (switching is called ``gauge transformation'' by physicists), and functions of interest to physicists; Addendum to Ch.\ 20, pp.\ 378--379, mentioning results on when the proportion of negative bonds is fixed, and on gauge theories. 
\par
\codes{\CPhys:~\CSG,~\CGG,~\CVS,~\CFr:~\CExp,~\CRef}%
\itemspace

\author{Nicholas A.\ Christakis}						%CHRISTAKIS
\authorspace
\authref  See \hyperlink{Derek Feng}{D.~Feng}.
\itemspace

\author{Dianhui Chu}							%CHU DH
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Rui Chu}							%CHU RI
\authorspace
\authref  See \hyperlink{Fenglei Tian}{F.-L.~Tian}.
\itemspace

\author{San Yan Chu}						%CHU SY
\authorspace
\authref  See \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\author{Ying Chu}							%CHU Y
\authorspace
\authref  See \hyperlink{Ruimei Gao}{R.M.~Gao}.
\itemspace

\author{Maria Chudnovsky}						%CHUDNOVSKY
\authorspace
\bsgitem{2005a}{}  Even hole free graphs.  \emph{Graph Theory Notes N.Y.}\ 49 (2005), 22--24.
\MR{2202297} (2006h:05185).  
\codes{\CSG:~\CCircles:~\CAlgor}%
\itemspace

\author{Maria Chudnovsky, William H.\ Cunningham, \& Jim Geelen}
\authorspace
\bsgitem{2008a}{}  An algorithm for packing non-zero $A$-paths in group-labelled graphs.  \emph{Combinatorica} 28 (2008), no.\ 2, 145--161.
\MR{2399016} (2009a:05103).  \Zbl{1164.05029}.					
\annot{} See \xref{Chudnovsky, Geelen, \etal{}}{2006a}{Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis Goddyn, Michael Lohman, \& Paul Seymour}.  Structure theorem for optimal $A$-paths in terms of switching only vertices in $A^c$; algorithm for finding such.  
Lemma 3.1 generalizes the basis result of \xref{Chudnovsky, Geelen, \etal{}}{2006a}{Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis Goddyn, Michael Lohman, \& Paul Seymour}.
[\Question.  $B(\Pi)$ is a subset of $V \times \fG$.  How is this related to the covering graph?  Can one simplify their proofs?  A ``non-zero'' path is like a level-changing path in $\tilde \Phi$ (covering graph).  This suggests modelling their picture by $\Phi' = \Phi \cup 1K_n$, i.e., with distinguished identity-gain complete subgraph.  Or, by $\Omega \subseteq M\cdot\Delta =$ a biased expansion, with a distinguished maximal balanced subgraph.]
\codes{\CGG:~\CPaths:~\CStr,~\CAlgor}%
\itemspace

\author{Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis Goddyn, Michael Lohman, \& Paul Seymour}
\authorspace
\bsgitem{2006a}{}  Packing non-zero $A$-paths in group-labelled graphs.  \emph{Combinatorica} 26 (2006), no.\ 5, 521--532.
\MR{2279668} (2007j:05184).  \Zbl{1127.05050}.					
\annot{} In a gain graph $\Phi$, find the maximum number of vertex-disjoint paths with non-identity gain and with endpoints in $A \subseteq V$ (non-zero $A$-paths).  Thm.:  If max $< k$, there is a set $X$ of up to $2k-2$ vertices such that every $A$-path in $\Phi \setminus X$ has identity gain.  This is not best possible.
\annotctd{} They prove: $\{ B(\Pi) : \Pi \in \cP^*(G,A) \}$ is the set of bases of a matroid.
\annotctd{} Dictionary:  ``Group-labelled graph'' = gain graph; $\Gamma$-labelled graph = $\Gamma$-gain graph (for a group $\Gamma$); ``weight'' = gain.  ``Shifting'' = switching; ``$A$-equivalent'' = $A^c$-switching equivalent, i.e., obtained by switching vertices not in $A$.
\codes{\CGG:~\CStr,~\CPaths}%
\itemspace

\author{Maria Chudnovsky, Ken-ichi Kawarabayashi, \& Paul Seymour}
\authorspace
\bsgitem{2005a}{}  Detecting even holes.  \emph{J.\ Graph Theory} 48 (2005), no.\ 2, 85--111.
\MR{2110580} (2006k:05197).  \Zbl{1062.05135}.						
\annot{} Algorithm to detect positive holes (induced circles) in a signed graph.  A polynomially equivalent problem is to decide whether a graph is negative-hole signable, i.e., has a signature in which every hole is negative.
\par
\codes{\CSG:~\CCircles:~\CAlgor}%
\itemspace

\author{S.T.\ Chui}							%CHUI ST
\authorspace
\authref  See also \hyperlink{B.W.\ Southern}{B.W.~Southern}.
\authrefspace

\author{S.T.\ Chui, G.\ Forgacs, \& D.M.\ Hatch}
\authorspace
\bsgitem{1982a}{}  Ground states and the nature of a phase transition in a simple cubic fully frustrated Ising model.  \emph{Phys.\ Rev.\ B} 25 (1982), no.\ 11, 6952--6958.
\annot{} Physics of ``fully frustrated'' $3$-dimensional cubic lattice, i.e., every square (``plaquette'') is negative.  Each square has one negative edge.  This is the unique fully frustrated signature up to switching [short proof:  the squares generate the cycle space], but there are many nonisomorphic ground states ($\zeta: V \to \signnos$ such that $\min_\zeta \#(E^\zeta)^-$); they are said to form 12 mutually unreachable classes.  App.\ A characterizes the ground states [and implies $l(\Sigma) = \frac{1}{4} \#V$ since each cube has one negative edge in each direction, neglecting boundary effects---or assuming toroidality].  The signed lattice is at times assumed to have a $2\times2$ fundamental domain; under that assumption there are 8 translational symmetry types of vertex, each forming a double-sized sublattice.  Approximate clustering is discussed.
\dated{18 Jun 2012} 
\par
\codes{\CPhys,~\CSG:~\CState(\Cfr),~\Csw,~\CClu}%
\itemspace

\authorx{Deborah Chun}						%CHUN D
\author{Deborah Chun, Tyler Moss, Daniel Slilaty, \& Xiangqian Zhou}
\authorspace
\bsgitem{2015a}{}  Unavoidable minors of large 4-connected bicircular matroids.  \emph{Ann.\ Combin.}\ 19 (2015), no.\ 1, 95--105. 
\MR{3319862}.  \Zbl{1310.05043}.  
\codes{\CBic:~\CStr}%
\itemspace

\bsgitem{2016a}{}  Bicircular matroids representable over $GF$ (4) or $GF$ (5).  \emph{Discrete Math.}\ 339 (2016), 2239--2248.
\MR{3512338}. \Zbl{1338.05038}.  
\annot{} Implies characterization of $\Gamma$ with antivoltages in $\bbZ_3$. $\bbZ_4$.  \Cf\ \xref{Zaslavsky}{2007a}{Thomas Zaslavsky}.
\dated{11 Jun 2019} 
\codes{\CBic:~\CGeom}%
\itemspace

\author{Yang Chun}							%CHUN Y
\authorspace
\authref  See \hyperlink{Bao Jiao}{B.~Jiao}.
\itemspace

\authorx{F.R.K.\ Chung}						%CHUNG FRK
\authorx{Fan Chung}
\author{F.R.K.\ Chung, Wayne Goddard, \& Daniel J.\ Kleitman}
\authorspace
\bsgitem{1994a}{}  Even cycles in directed graphs.  \emph{SIAM J.\ Discrete Math.}\  7 (1994), 474--483.
\MR{1285584} (95e:05050).  \Zbl{809.05062}.  
\annot{} A strongly connected digraph with $\#E \geq \lfloor (n+1)^2/4 \rfloor$ has an even cycle.  This is best possible.  
[This equals \xref{Petersdorf's}{1966a}{M.\ Petersdorf} bound for $l(K_{n+2},\sigma)$.  \Question.  Are they related?]
\dated{12 Jun 2012} 
\par
\codes{\Csd:~\CPar:~\CBal}%
\itemspace

\author{Fan Chung \& Mark Kempton}
\authorspace
\bsgitem{2013a}{}  A local clustering algorithm for connection graphs.  In:  Anthony Bonato \etal, eds., \emph{Algorithms and Models for the Web Graph} (Proc.\ 10th Int.\ Workshop, WAW 2013, Cambridge, Mass., 2013), pp.\ 26--43.  \LNCS 8305.  Springer, Cham, 2013.
\MR{3163708}.  \Zbl{1342.05158}.  
\codes{\CGG,~\CWG:~\CBal,~\CLap:~\CAlgor}%
\itemspace

\bsgitem{2015a}{}  A local clustering algorithm for connection graphs.  \emph{Internet Math.}\ 11 (2015), no.\ 4-5, 333--351.
\MR{3373768}.  
\annot{} Connection graph:  A real-weighted $GL(\bbR,d)$-gain graph.	
\par
\codes{\CGG,~\CWG:~\CBal,~\CLap:~\CAlgor}%
\itemspace

\author{Fan Chung, Wenbo Zhao, \& Mark Kempton}
\authorspace
\bsgitem{2014a}{}  Ranking and sparsifying a connection graph.  \emph{Internet Math.}\ 10 (2014), no.\ 1-2, 87--115.
\MR{3274541}.  \Zbl{1342.05182}.  
\codes{\CGG,~\CWG:~\CBal:~\CAlgor}%
\itemspace

\author{Fu-Lai Chung}						%CHUNG FL
\authorspace
\authref  See \hyperlink{Xiao Shen}{X.~Shen}.
\itemspace

\authorx{Taeyoung Chung}						%CHUNG TY
\author{Taeyoung Chung, Jack Koolen, Yoshio Sano, \& Tetsuji Taniguchi}
\authorspace
\bsgitem{2011a}{}  The non-bipartite integral graphs with spectral radius three.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2544--2559.
\MR{2811137} (2012d:05224).  \Zbl{1222.05151}.					
\annot{} \S2.2, ``Generalized line graphs and generalized signless Laplace matrices'':  The generalized signless Laplace matrix of $(\Gamma,f)$, where $f: V \to \bbZ_{\geq0}$, is $\Lap(-\Gamma) + 2D(f)$.  The incidence matrix of $(\Gamma,f)$ is $\Eta(\Sigma)$ where $\Sigma$ consists of $-\Gamma$ with $f(x)$ negative digons adjoined to $x \in V$.  [See \xref{Zaslavsky}{1984c}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky} for this construction, which is not stated here.]
\dated{20 Dec 2011} 
\codes{\Csg:~\CPar:~\CEig,~\CInc,~\CLG}%
\itemspace

\author{V.\ Chv\'atal} 						%CHVATAL 
\authorx{V.\ Chvatal} 						%CHVATAL 
\authorspace
\authref  See \hyperlink{Jin Akiyama}{J.~Akiyama}.
\itemspace

\author{Adriana Ciampella}						%CIAMPELLA
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo} and \hyperlink{Maurizio Brunetti}{M.~Brunetti}.
\itemspace

\authorx{Olivier Cinquin}						%CINQUIN
\author{Olivier Cinquin \& Jacques Demongeot}
\authorspace
\bsgitem{2002a}{}  Roles of positive and negative feedback in biological systems.  \emph{C.\ R.\ Biologies} 325 (2002), 1085--1095.
\annot{} Stability of systems of nonlinear differential equations.  Some mathematical treatment.
\codes{\CSD:~\CSta,~\CAppl}%
\itemspace

\bsgitem{2002b}{}  Positive and negative feedback: Striking a balance between necessary antagonists.  \emph{J.\ Theor.\ Biol.}\ 216 (2002), 229--241.
\MR{1941484} (no rev).  
\codes{\CSD:~\CExp}%
\itemspace

\author{S.M.\ Cioab\u{a}}						%CIOABA
\authorx{S.M.\ Cioaba}
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Francesco Belardo}{F.~Belardo}, and \hyperlink{Michael S.\ Cavers}{M.~Cavers}.
\itemspace

\authorx{Valerio Ciotti}						%CIOTTI
\author{Valerio Ciotti, Ginestra Bianconi, Andrea Capocci, Francesca Colaiori, \& Pietro Panzarasa}
\authorspace
\bsgitem{2015a}{}  Degree correlations in signed social networks.  \emph{Physica A}\ 422, (2015), 25--39.
\arXiv{1412.1024}.  
\codes{\CPsS:~\CSG:~\CAppl:~\CBal,~\CClu}%
\itemspace

\author{Pedro Cisneros-Velarde}					%CISNEROS
\authorspace
\authref  See also \hyperlink{Wenjun Mei}{W.-J.~Mei}.
\authrefspace

\author{Pedro Cisneros-Velarde \& Francesco Bullo}
\authorspace
\bsgitem{2020a}{}  Signed network formation games and clustering balance.  \emph{Dyn. Games Appl.}\ 10 (2020), no.\ 4, 783--797. 
\MR{4181812}.  \Zbl{1457.91105}.  
\annot{} Players change edge signs to converge on a clustered signed graph.
\dated{3 Jun 2022} 
\codes{\CSG:~\CDyn:~\CClu}%
\itemspace

\author{Lane Clark}							%CLARK L
\authorspace
\bsgitem{2004a}{}  Limit theorems for associated Whitney numbers of Dowling lattices.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 50 (2004), 105--113.
\MR{2075859} (2005b:06007).  \Zbl{1053.06003}.  
\annot{} Limit theorems and asymptotics for modified Whitney numbers (first kind) introduced by \xref{Benoumhani}{1997a}{Moussa Benoumhani}.
\dated{12 Jul 2016} 
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\authorx{F.W.\ Clarke}			 			%CLARKE FW
\author{F.W.\ Clarke, A.D.\ Thomas, \& D.A.\ Waller}
\authorspace
\bsgitem{1980a}{}  Embeddings of covering projections of graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 28 (1980), 10--17.  
\MR{0565507} (81f:05066).  \Zbl{351.05126}, (\Zbl{416.05069}).
\codes{\Cgg:~\CTop}%
\itemspace

\authorx{Nancy E.\ Clarke}						%CLARKE NE
\author{Nancy E.\ Clarke, Samuel Fiorini, Gwena\"el Joret, \& Dirk Oliver Theis}
\authorspace
\bsgitem{2014a}{}  A note on the cops and robber game on graphs embedded in non-orientable surfaces.  \emph{Graphs Combin.}\ 30 (2014), no.\ 1, 19--124.
\MR{3143863}.  \Zbl{1295.05153}.  
\annot{} [Thm.\ 1 actually proves Thm.\ 1$'$:  The cop number $c(\Gamma)$ satisfies $c(|\Sigma|)\leq c(\tilde\Sigma)$, the cop number of the double cover of $\Sigma$.]  Thm.\ 1 is stated and proved for $\Sigma$ that is orientation embedded in a surface [but the proof uses only the double covering graph].  Lem.\ 1 includes Thm.\ 1$'$.
\dated{30 Nov 2014} 
\codes{\Csg:~\Ccov:~\CTop}%
\itemspace

\author{Nolan J.\ Coble}						%COBLE
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo}.
\itemspace

\author{A.M.\ Cohen}							%COHEN AM
\authorspace
\authref  See \hyperlink{Andries E.\ Brouwer}{A.E.~Brouwer}.
\itemspace

\author{Bernard P.\ Cohen} 						%COHEN BP
\authorspace
\authref  See \hyperlink{Joseph Berger}{J.~Berger}.
\itemspace

\authorx{Edith Cohen}						%COHEN E
\author{Edith Cohen \& Nimrod Megiddo}
\authorspace
\bsgitem{1989a}{}  Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs.  In: \emph{Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing} (Seattle, 1989), pp.\ 523--534.
\annot{} Partial version of \xrefd{1993a}{Edith Cohen \& Nimrod Megiddo}.
\codes{GD:~\CBal:~\CAlgor}%
\itemspace

\bsgitem{1991a}{}  Recognizing properties of periodic graphs.  
In: Peter Gritzmann and Bernd Sturmfels, eds., \emph{Applied geometry and Discrete Mathematics: The Victor Klee Festschrift},  pp.\ 135--146.  DIMACS Ser.\ Discrete Math.\ Theor.\ Computer Sci., Vol.\ 4.  Amer.\ Math.\ Soc., Providence, R.I., and Assoc.\ Computing Mach., 1991.  
\MR{1116344} (92g:05166).  \Zbl{753.05047}.					
\annot{} Given: a gain graph $\Phi$ with gains in $\bbZ^d$ (a ``static graph'').  Found: algorithms for (1) connected components and (2) bipartiteness of the covering graph $\tilde\Phi$ (the ``periodic graph'') and, (3) given costs on the edges of $\Phi$, for a minimum-average-cost spanning tree in the covering graph.  Many references to related work.
\codes{\CGG:~\CCov:~\CAlgor,~\CRef}%
\itemspace

\bsgitem{1992a}{}  New algorithms for generalized network flows.  
In: D.~Dolev, Z.~Galil, and M.~Rodeh, eds., \emph{Theory of Computing and Systems} (Proc., Haifa, 1992), pp.\ 103--114.  \LNCS 601.  Springer-Verlag, Berlin, 1992.  
\MR{1233831} (no rev).  
\annot{} Preliminary version of \xrefd{1994a}{Edith Cohen \& Nimrod Megiddo}, differing only slightly.
\par
\codes{\CGN:~\CAlgor}\codesctd{\Csg:~\COri:~\CAlgor}%
\itemspace

\bsgitem{1993a}{}  Strongly polynomial-time and NC algorithms for detecting cycles in periodic graphs.  \emph{J.\ Assoc.\ Comput.\ Mach.}\ 40 (1993), 791--830.  
\MR{1369189} (96h:05182).  \Zbl{782.68053}.					
\annot{} Looking for a closed walk (``cycle'') with gain $0$ in a gain digraph with (additive) gains in $\bbQ^d$.  [\Cf\ \xref{Kodialam and Orlin}{1991a}{Muralidharan Kodialam \& James B.\ Orlin}.]
\par
\codes{\CGD:~\CBal:~\CAlgor}%
\itemspace

\bsgitem{1994a}{}  New algorithms for generalized network flows.  \emph{Math.\ Programming} 64 (1994), 325--336.  
\MR{1286453} (95k:90111).  \Zbl{816.90057}.					
\annot{} Maximize the fraction of demand satisfied by a flow on a network with gains.  Positive real gains in \S3.  Bidirected networks with positive gains in \S4; these are more general than networks with arbitrary non-zero real gains.
\codes{\CGN:~\CAlgor}\codesctd{\Csg:~\COri:~\CAlgor}%
\itemspace

\bsgitem{1994b}{}  Improved algorithms for linear inequalities with two variables per inequality.  \emph{SIAM J.\ Comput.}\ 23 (1994), 1313--1347.  
\MR{1303338} (95i:90040).  \Zbl{833.90094}.					
\codes{\CGN:~\CInc:~\CDual:~\CAlgor}%
\itemspace

\author{Olivier Cohen}						%COHEN O
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena} and \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{Francesca Colaiori}						%COLAIORI
\authorspace
\authref  See \hyperlink{Valerio Ciotti}{V.~Ciotti}.
\itemspace

\authorx{Charles J.\ Colbourn}			 			%COLBOURN
\author{Charles J.\ Colbourn \& Derek G.\ Corneil}
\authorspace
\bsgitem{1980a}{} On deciding switching equivalence of graphs.  \emph{Discrete Appl.\ Math.}\ 2 (1980), 181--184.  
\MR{0588697} (81k:05090).  \Zbl{438.05054}.					
\annot{} Deciding switching isomorphism of graphs is polynomial-time equivalent to graph isomorphism.  
\codes{\CTG:~\CAlgor}%
\itemspace

\authorx{Tom Coleman}						%COLEMAN
\author{Tom Coleman, James Saunderson, \& Anthony Wirth}
\authorspace
\bsgitem{2008a}{}  A local-search 2-approximation for 2-correlation-clustering.  
In:  D.\ Halperin and K.\ Mehlhorn, eds., \emph{Algorithms -- ESA 2008} (16th Ann.\ Europ.\ Symp.\ Algorithms, Karlsruhe, 2008), pp.\ 308--319.  \LNCS 5193.  Springer, Berlin, 2008.
\Zbl{1158.68549}.  
\codes{\CSG:~\CClu}%
\itemspace

\author{D.A.\ Coley}							%COLEY
\authorspace
\authref  See \hyperlink{T.\ Wanschura}{T.~Wanschura}.
\itemspace

\author{L.\ Collatz}							%COLLATZ*
\authorspace
\bsgitem{1978a}{}  Spektren periodischer Graphen.  \emph{Resultate Math.}\ 1 (1978), 42--53.  
\MR{0510149} (80b:05042).  \Zbl{402.05054}.					
\annot{} Introducing periodic graphs: these are connected canonical covering graphs $\Gamma = \tilde\Phi$ of finite $\bbZ^d$-gain graphs $\Phi$.  The ``spectrum'' of $\Gamma$ is the set of all eigenvalues of $A(\|\Phi\|)$ for all possible $\Phi$.  The spectrum, while infinite, is contained in the interval $[-r, r]$ where $r$ is the largest eigenvalue of each $A(\|\Phi\|)$ [the ``index'' of \xref{von Below}{1994a}{Joachim von Below}].  The inspiration is tilings.
\codes{\CGG:~\CCov:~\CEig}%
\itemspace

\authorx{Barry E.\ Collins}			 			%COLLINS
\author{Barry E.\ Collins \& Bertram H.\ Raven}
\authorspace
\bsgitem{1968a}{} Group structure:  attraction, coalitions, communication, and power.  
In:  Gardner Lindzey and Elliot Aronson, eds., \emph{The Handbook of Social Psychology}, second ed., Vol.\ 4, Ch.\ 30, pp.\ 102--204.  Addison-Wesley, Reading, Mass., 1968.  
\annot{} ``Graph theory and structural balance,'' pp.\ 106--109.
\par
\codes{\CPsS:~\CSG:~\CExp,~\CRef}%
\itemspace

\author{Luke Collins}							%COLLINS
\authorspace
\authref  See \hyperlink{Irene Sciriha}{I.~Sciriha}.
\itemspace

\authorx{Barbara Coluzzi}						%COLUZZI
\author{Barbara Coluzzi, Enzo Marinari, Giorgio Parisi, \& Heiko Rieger}
\authorspace
\bsgitem{2000a}{}  On the energy minima of the Sherrington-Kirkpatrick model.  \emph{J.\ Phys.\ A} 33 (2000), no.\ 21, 3851--3862.
\MR{1769547} (no rev).  \Zbl{945.82004}.  \arXiv{cond-mat/0003287}.			
\codes{\CPhys:~\CSG}%
\itemspace

\authorx{Ph.\ Combe}						%COMBE
\author{Ph.\ Combe \& H.\ Nencka}
\authorspace
\bsgitem{1995a}{}  Non-frustrated signed graphs.  
In:  J.\ Bertrand \etal, eds., \emph{Modern Group Theoretical Methods in Physics} (Proc.\ Conf.\ in Honour of Guy Rideau, Paris, 1995), pp.\ 105--113.  
Math.\  Phys.\ Stud., Vol.\ 18.  Kluwer, Dordrecht, 1995.  
\MR{1361440} (96j:05105).  \Zbl{905.05071}.
\annot{} $\Sigma$ is balanced iff a fundamental system of circles is balanced [as is well known; see \emph{i.a.}\ \xref{Popescu}{1979a}{Dragos Popescu}, \xref{Zaslavsky}{1981b}{Thomas Zaslavsky}].  An algorithm [incredibly complicated, compared to the obvious method of tracing a spanning tree] to determine all vertex signings of $\Sigma$ that switch it to all positive.  Has several physics references.
\codes{\CSG:~\CBal,~\CFr,~\CAlgor,~\CRef}%
\itemspace

\bsgitem{1997a}{}  Cooperative networks and frustration on graphs.  \emph{Methods Funct.\ Anal.\ Topology} 3 (1997), 40--50.
\MR{1770677} (2001e:91135).  \Zbl{933.92005}.						
\annot{} A signed-graphic model $\Sigma$ of a neuron network.  Obs.:  A network is cooperative iff $\Sigma$ has a non-frustrated state $s : V \to \signnos$, i.e., the Hamiltonian (``energy'') $H(s) := -\frac12 \sum_{uv \in E} \sigma(uv) s(u) s(v) = -\#E$.  [Should be $-\frac12\#E$.]  $H$ [i.e., $\Sigma$] is non-frustrated if some state is.  Assertion:  $H$ is non-frustrated iff $\Sigma$ is balanced.  A proof idea (not a proof) is by setting up (real-valued) linear equations of positivity of generating circles; carried out for $K_n$.  [See \xrefd{1997b}{Ph.\ Combe \& H.\ Nencka}.]
[Easy proof:  $H(s) = -\frac12\#E+\#E^-(\Sigma^s)$, hence $H$ is non-frustrated iff $\Sigma^s$ is all positive for some $s$ iff $\Sigma$ is balanced.  See e.g.\ \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Cor.\ 3.3.]
\dated{17 Jun, 17 Aug 2012} 
\codes{\CSG:~\CBal,~\Csw,~\CFr,~\CBiol}%
\itemspace

\bsgitem{1997b}{}  Frustration and overblocking on graphs.  \emph{Math.\ Computer Modelling}\ 26 (1997), no.\ 8-10, 307--309.
\MR{1492513} (no rev).  \Zbl{1185.05147}.  
\annot{} No proofs.  Prop.\ 1:  The signatures of $\Gamma$ are a ``$\GF(2)$-vector space''.  [Meaning:  They are the points in $\{\pm1\}^{\#E}\subset\bbR^{\#E}$.]  Prop.\ 2:  Nonfrustration corresponds to a large family of [real] linear systems.  
``Minimal'' circles generalize plaquettes (girth circles) to arbitrary graphs.  [``Minimal'' =(?) minimum length, assuming such circles generate the cycle space.  In general, choice of generating circles remains a good question.]  ``Fully frustrated'':  all minimal circles are negative.  Prop.\ 3:  Full frustration corresponds to another family of [real] linear systems.  
[``Overblocking'':  Fully frustrated and some nonminimal circles are negative.]  Prop.\ 4:  Linear system for overblocking in a fully frustrated signature.  Cor.\ 5:  $K_5$ is overblocking.  $K_{3,2}$ cannot be fully frustrated.  
\dated{17 Jun 2012} 
\codes{\CSG:~\CBal,~\Csw,~\CFr,~\CPhys}%
\itemspace

\author{Jean-Paul Comet}						%COMET
\authorspace
\authref  See also \hyperlink{Adrien Richard}{A.~Richard}.
\authrefspace

\author{Jean-Paul Comet, Mathilde Noual, Adrien Richard, Julio Aracena, Laurence Calzone, Jacques Demongeot, Marcelle Kaufman, Aur\'elien Naldi, El Houssine Snoussi, \& Denis Thieffry}
\authorx{Jean-Paul Comet, Mathilde Noual, Adrien Richard, Julio Aracena, Laurence Calzone, Jacques Demongeot, Marcelle Kaufman, Aurelien Naldi, El Houssine Snoussi, \& Denis Thieffry}
\authorspace
\bsgitem{2013a}{}  On circuit functionality in Boolean networks.  \emph{Bull.\ Math.\ Biol.}\ 75 (2013), 906--919.
\MR{3070304}.  \Zbl{1272.92016}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{F.G.\ Commoner} 						%COMMONER
\authorspace
\bsgitem{1973a}{}  A sufficient condition for a matrix to be totally unimodular.  \emph{Networks} 3 (1973), 351--365.  
\MR{0335550} (49 \#331).  \Zbl{352.05012}.
\codes{\CSD:~\CBal}%
\itemspace

\author{Michele Conforti}						%CONFORTI
\authorspace
\authref  See also \hyperlink{Francisco Barahona}{F.~Barahona}.
\authrefspace

\author{Michele Conforti \& G\'erard Cornu\'ejols}
\authorx{Michele Conforti \& Gerard Cornuejols}
\authorspace
\bsgitem{1995a}{}  Balanced $0$, $\pm1$-matrices, bicoloring and total dual integrality.  \emph{Math.\ Programming} 71 (1995), 249--258.
\MR{1378792} (97a:90103).  \Zbl{0849.90095}.  
\annot{} Bicolorability means every square submatrix contains the incidence matrix of a balanced signed graph.  
\codes{\CSGw,~\Csg:~\CBal(\CGen)}%
\itemspace

\author{Michele Conforti, G\'erard Cornu\'ejols, Ajai Kapoor, \& Kristina Vu\v{s}kovii\'c}
\authorx{Michele Conforti, Gerard Cornuejols, Ajai Kapoor, \& Kristina Vuskoviic}
\authorspace
\bsgitem{1994a}{}  Recognizing balanced $0, \pm1$ matrices.  In:  \emph{Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms} (Arlington, Va., 1994), pp.\ 103--111.  Assoc.\ Computing Mach., New York, 1994.
\MR{1285156} (95e:05022).  \Zbl{867.05014}.					
\codes{\CSGw,~\Csg:~\CBal(\CGen)}%
\itemspace

\bsgitem{1995a}{}  A mickey-mouse decomposition theorem.  
In: Egon Balas and Jens Clausen, eds., \emph{Integer Programming and Combinatorial Optimization} (4th Int.\ IPCO Conf., Copenhagen, 1995, Proc.),  pp.\ 321--328.  \LNCS 920.  Springer, Berlin, 1995.  
\MR{1367991} (96i:05139).  \Zbl{875.90002} (book). 
\annot{} The structure of graphs that are signable to be ``without odd holes'': that is, so that each triangle is negative and each chordless circle of length greater than 3 is positive.  Proof based on \xref{Truemper}{1982a}{Klaus Truemper}.
\par
\codes{\CSG:~\CBal(\CGen),~\CStr}%
\itemspace

\bsgitem{1997a}{}  Universally signable graphs.  \emph{Combinatorica} 17 (1997), 67--77.
\MR{1466576} (98g:05134).  \Zbl{980.00112}.					
\annot{} $\Gamma$ is ``universally signable'' if it can be signed so as to make every triangle negative and the holes independently positive or negative at will.  Such graphs are characterized by a decomposition theorem which leads to a polynomial-time recognition algorithm.
\codes{\CSG:~\CBal,~\CStr}%
\itemspace

\bsgitem{1999a}{}  Even and odd holes in cap-free graphs.  \emph{J.\ Graph Theory} 30 (1999), 289--308. 
\MR{1669460} (99m:05155).  \Zbl{920.05028}.					
\annot{} Recognition of graphs that are ``strongly even-signable'' (signable so triangles are $-$ and longer circles with $\leq1$ chord are $+$) and ``strongly odd-signable'' (signable so quadrilaterals with a unique chord are $+$ and all other circles with $\leq1$ chord are $-$).  
[Description adapted from \xref{Trotignon and Vu\v{s}kovi\'c}{2010a}{Nicolas Trotignon \& Kristina Vuskovic}.]
\dated{19 Jan 2015} 
\codes{\Csg:~\CBal(\CGen):~\CAlgor}%
\itemspace

\bsgitem{2000a}{}  Triangle-free graphs that are signable without even holes.  \emph{J.\ Graph Theory} 34 (2000), 204--220.
\MR{1762021} (2001b:05188).  \Zbl{953.05061}.						
\annot{} ``Even hole'' means a chordless circle, bigger than a triangle, that is positive in a given signing of the graph.  The graphs of the title are characterized in several ways.  Most of them have significant wheels.	
	
\par
\codes{\CSG:~\CBal,~\CStr,~\CAlgor}%
\itemspace

\bsgitem{2001a}{}  Balanced $0$, $\pm1$ matrices.  I.  Decomposition.  \emph{J.\ Combin.\ Theory Ser.\ B} 81 (2001), no.\ 2, 243--274.
\MR{1814907} (2002c:05041).  \Zbl{1026.05016}.  
\par
\codes{\CSGw,~\Csg:~\CBal(\CGen)}%
\itemspace

\bsgitem{2001b}{}  Balanced $0$, $\pm1$ matrices.  II.  Recognition algorithm.  \emph{J.\ Combin.\ Theory Ser.\ B} 81 (2001), no.\ 2, 275--306.
\MR{1814908} (2002c:05042).  \Zbl{1026.05017}.  
\par
\codes{\CSGw,~\Csg:~\CBal(\CGen)}%
\itemspace

\bsgitem{2001c}{}  Perfect, ideal and balanced matrices.  \emph{European J.\ Oper.\ Res.}\ 133 (2001), 455--461.
\MR{1842697} (2002e:05062).  \Zbl{1053.15014}.  
\annot{} \S5, ``Balanced matrices'':  Expounds part of \xref{Conforti and Cornu\'ejols}{1995a}{Michele Conforti \& Gerard Cornuejols}.
\dated{23 Aug 2014} 
\codes{\CSGw,~\Csg:~\CBal(\CGen):~\CExp}%
\itemspace

\bsgitem{2002a}{}  Even-hole-free graphs.  I.\  Decomposition theorem.  \emph{J.\ Graph Theory} 39 (2002), 6--49.
\MR{1871344} (2003c:05189).  \Zbl{1005.05034}.					
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{2002b}{}  Even-hole-free graphs.  II.\  Recognition algorithm.  \emph{J.\ Graph Theory} 40 (2002), 238--266.
\MR{1913849} (2004e:05182).  \Zbl{1003.05095}.					
\codes{\CSG:~\CBal}%
\itemspace

\author{Michele Conforti, G\'erard Cornu\'ejols, \& M.R.\ Rao}
\authorx{Michele Conforti, Gerard Cornuejols, \& M.R.\ Rao}
\authorspace
\bsgitem{1999a}{}  Decomposition of balanced matrices.  \emph{J.\ Combin.\ Theory Ser.\ B} 77 (1999), 292--406.
\MR{1719340} (2001d:05126).  \Zbl{1023.05025}.  
\codes{\CSGw,~\Csg:~\CBal(\CGen)}%
\itemspace

\author{Michele Conforti, G\'erard Cornu\'ejols, \& Klaus Truemper}
\authorx{Michele Conforti, Gerard Cornuejols, \& Klaus Truemper}
\authorspace
\bsgitem{1994a}{}  From totally unimodular to balanced $0$, $\pm1$ matrices: A family of integer polytopes.  \emph{Math.\ Oper.\ Res.}\ 19 (1994), no.\ 1, 21--23.
\MR{1290007} (96e:15023).  \Zbl{799.15010}.  
\annot{} The forbidden matrix type $A'$ in Rem.\ 3, par.\ 2 is the transpose of a signed-graph incidence matrix.
\dated{23 Aug 2014} 
\par
\codes{\Csgw,~\Csg:~\Cbal(\Cgen)}%
\itemspace

\author{Michele Conforti, G\'erard Cornu\'ejols, \& Kristina Vu\v{s}kovii\'c}
\authorx{Michele Conforti, Gerard Cornuejols, \& Kristina Vuskoviic}
\authorspace
\bsgitem{1999a}{}  Balanced cycles and holes in bipartite graphs.  \emph{Discrete Math.}\ 199 (1999), 27--33. 
\MR{1675908} (99j:05119).  \Zbl{939.05050}.					
\codes{\CSGw,~\Cgg,~\Csg:~\CBal}%
\itemspace

\bsgitem{2006a}{}  Balanced matrices.  \emph{Discrete Math.}\ 306 (2006), 2411--2437.
\MR{2261909} \linebreak (2007g:05131).  \Zbl{1102.05013}.  
\annot{} Bipartite $\Gamma$ is ``balanceable'' if it can be signed so each hole (chordless circle) is positive iff it is evenly even.  [\xref{Truemper}{1982a}{Klaus Truemper} implies a characterization by forbidden induced signed subgraphs.]  ``Strongly balancedable'': also no circle has a unique chord.
\dated{19 Jan 2015} 
\codes{\CSGw,~\Csg:~\CBal(\CGen):~\CExp}%
\itemspace

\author{Michele Conforti, Samuel Fiorini, Tony Huynh, Gwena\"el Joret, \& Stefan Weltge}
\authorspace
\bsgitem{2020a}{}  The stable set problem in graphs with bounded genus and bounded odd cycle packing number.  In:  Shuchi Chawla, ed., \emph{ Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms} (SODA 2020, Salt Lake City), pp.\ 2896--2915.  Soc.\ Industrial Appl.\ Math.\ (SIAM), Philadelphia, 2020.
\MR{4141358}.  
\arXiv{1908.06300}.  
\annot{} Orientably embedded [i.e., positive] odd circles in a nonorientable surface satisfy the Erd\H{o}s--Posa property.  [\Problem.  Generalize to negative circles in signed graphs.]
\dated{25 Mar 2022} 
\codes{\Csg:~\CPar:~\CCircles:~\CAlgor}%
\itemspace

\author{Michele Conforti, Samuel Fiorini, Tony Huynh, \& Stefan Weltge}
\authorspace
\bsgitem{2020a}{}  Extended formulations for stable set polytopes of graphs without two disjoint odd cycles.  In:  Daniel Bienstock and Giacomo Zambelli, eds., \emph{Integer Programming and Combinatorial Optimization} (Proc.\ 21st Int.\ Conf., IPCO 2020, London), pp.\ 104--116.  \LNCS 12125.  Springer, Cham, 2020.
\MR{4139563}.  
\annot{} See \xrefd{2022a}{Michele Conforti, Samuel Fiorini, Tony Huynh, \& Stefan Weltge}.
\codes{\Csg:~\Cpar:~\CFr:~\CCircles,~\CAlgor}%
\itemspace

\bsgitem{2022a}{}  Extended formulations for stable set polytopes of graphs without two disjoint odd cycles.  \emph{Math.\ Programming} 192 (2022), no.\ 1-2, 547--566.
\MR{4391781}.  \Zbl{1489.90151}.  \arXiv{1911.12179}.  
\annot{} 
[\Problem.  Generalize to negative circles in signed graphs.]
\dated{25 Mar 2022} 
\codes{\Csg:~\Cpar:~\CFr:~\CCircles,~\CAlgor}%
\itemspace

\author{Michele Conforti \& Bert Gerards}
\authorspace
\bsgitem{2007a}{}  Packing odd circuits.  \emph{SIAM J.\ Discrete Math.}\ 21 (2007), no.\ 2, 273--302.
\MR{2318666} (2008g:05162).  \Zbl{1139.05323}.  
\annot{} The problem is to find the most vertex-disjoint negative circles in a signed graph (thus, odd-length circles in an ordinary graph).  It is NP-hard but it can be solved in polynomial time for the signed graphs that exclude the switching classes $[-K_5]$, $[K_{3,3}^{1,1}]$, $[K_{3,3}^{1,2}]$, $[K_{3,3}^2]$, which are defined as:
$K_{3,3}^{1,1}$ = $+K_{3,3}$ with edge $u_1v_1$ made negative and the additional negative edge $-u_2v_2$, 
$K_{3,3}^{1,2}$ = $+K_{3,3}$ with $u_1v_1$ made negative and added edges $-u_1u_2$ and $-u_1u_3$, and 
$K_{3,3}^2$ = $+K_{3,3}$ with edges $u_1v_1$ and $u_2v_2$ made negative. 
\codes{\CSG:~\CCircles:~\CStr}%
\itemspace

\author{Michele Conforti, Bert Gerards, \& Ajai Kapoor}
\authorspace
\bsgitem{2000a}{}  A theorem of Truemper.  \emph{Combinatorica} 20 (2000), no.\ 1, 15--26.
\MR{1770518} (2001h:05085).  \Zbl{949.05071}.					
\annot{} Full version of \xref{Conforti and Kapoor}{1998a}{Michele Conforti \& Ajai Kapoor}.
\codes{\CSG:~\CBal}%
\itemspace

\author{Michele Conforti \& Ajai Kapoor}
\authorspace
\bsgitem{1998a}{}  A theorem of Truemper.
In: Robert E.\ Bixby, E.\ Andrew Boyd, and Roger Z.\ R\'{\i}os-Mercado, eds., \emph{Integer Programming and Combinatorial Optimization} (6th Int.\ IPCO Conf., Houston, 1998, Proc.), pp.\ 53--68.  \LNCS 1412.  Springer, Berlin, 1998.
\MR{1726335} (2000h:05184).  \Zbl{907.90269}.					
\annot{} A new proof of \xref{Truemper's}{1982a}{Klaus Truemper} theorem on prescribed hole signs.  Discussion of applications.
\codes{\CSG:~\CBal}%
\itemspace

\author{Antonio Coniglio}						%CONIGLIO
\authorspace
\bsgitem{1994a}{}  Frustrated percolation, spin glasses and glasses.  \emph{Nuovo Cimento D} 16 (1994), no.\ 8, 1027--1037. 
\annot{} The ``site-frustrated percolation model'':  On a signed graph a vertex (``site'') may be occupied or not, but a frustrated circle cannot be fully occupied.  Unoccupied vertices are considered ``defects or holes''.  Each possible configuration has a weight; the statistical properties are examined.  
[The unoccupied vertices constitute a balancing vertex set (i.e., its deletion leaves a balanced subgraph).  Maximally occupied configurations correspond to minimum balancing vertex sets.  \Question.  What does the physics mean for such sets, and vice versa?]
	
\dated{22 Aug 2014} 
\codes{\Csg,~\CPhys:~\CFr,~\CState(\CGen)}%
\itemspace

\bsgitem{1998a}{}  Spin glasses, glasses and granular materials.  \emph{Philosophical Mag.\ B} 77 (1998), no.\ 2, 213-- 219.
\annot{} \S2, ``Frustrated lattice gas'':  a signed-graph ($\pm J$ Ising) model ``diluted with lattice gas variables''.  
\S3, ``Percolation in phase space'':  see xrefd{1994a}{Antonio Coniglio}.
	
\codes{\Csg,~\CPhys:~\CFr}%
\itemspace

\bsgitem{1999a}{}  Frustrated percolation.  \emph{Physica A} 266 (1999), 379--389.
\annot{} \S2, ``Clusters in the Ising spin glass model''.  \S4, ``Site-frustrated percolation'':  see \xrefd{1994a}{Antonio Coniglio}.  \S6, ``Hamiltonian formalism for FP'':  Eq.\ (24) shows a Potts--Ising model, corresponding to a signed graph with a variable vertex coloring as well as the spin state (i.e., vertex signs).
	
\dated{22 Aug 2014} 
\codes{\Csg,~\CPhys:~\CState}%
\itemspace

\bsgitem{2002a}{}  Clusters in frustrated systems.  \emph{Physica A} 306 (2002), 76--89.
\annot{} See esp.\ \S3, ``Clusters in the Ising spin-glass model''.
	
\codes{\Csg,~\CPhys:~\CState}%
\itemspace

\author{A.\ Coniglio, F.\ di Liberto, G.\ Monroy, \& F.\ Peruggi}
\authorspace
\bsgitem{1991a}{}  Cluster approach to spin glasses and the frustrated-percolation problem.  \emph{Phys.\ Rev.\ B} 44 (1991), no.\ 22, 12605--12608.
\annot{} A cluster approach to the Ising spin glass model, i.e., vertex signs (``spins'') on a signed graph.  
Dictionary:  ``NN'' = ``nearest-neighbor'' = interactions only between adjacent vertices.
	
\dated{22 Aug 2014} 
\par
\codes{\Csg,~\CPhys:~\CState}%
\itemspace

\author{Joseph G.\ Conlon}						%CONLON
\authorspace
\bsgitem{2004a}{}  Even cycles in graphs.  \emph{J.\ Graph Theory} 45 (2004), no.\ 3, 163--223.
\MR{2037758} (2004m:05145).  \Zbl{1033.05062}.					
\annot{} Main theorem:  For 3-connected $G \neq K_4$, there is an even circle, deletion of whose vertices or edges leaves a 2-connected graph. 
[\Problem.  Generalize to signed graphs.  And see \xref{Voss}{1991a}{Heinz-Jurgen Voss}.]
\codes{\Cpar}%
\itemspace

\author{S.\ Contreras}						%CONTRERAS
\authorspace
\authref  See \hyperlink{A.J.\ Ramirez-Pastor}{A.J.~Ram{\'\i}rez-Pastor}, \hyperlink{M.C.\ Salas-Solis}{M.C.~Salas-Sol{\'\i}s}, and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\authorx{George Converse}						%CONVERSE
\author{George Converse \& M.\ Katz}
\authorspace
\bsgitem{1975a}{}  Symmetric matrices with given row sums.  \emph{J.\ Combin.\ Theory Ser.\ A} 18 (1975), 171--176.
\MR{0363945} (51 \#200).  \Zbl{297.05024}.					
\annot{} An equivalent of Thm.\ 8.2.1 in \xref{Brualdi}{2006a}{Richard A.\ Brualdi}.
\dated{13 Oct 2012} 
\par
\codes{\Csg:~\Cpar:~\CAdj}%
\itemspace

\author{S.N.\ Coppersmith}						%COPPERSMITH
\authorspace
\authref  See \hyperlink{J.W.\ Landry}{J.W.~Landry}.
\itemspace

\authorx{Oscar Coppola}						%COPPOLA
\author{Oscar Coppola, Jake Huryn, \& Michael Reilly}
\authorspace
\bsgitem{20xxa}{}  An extension of Stanley's symmetric acyclicity theorem to signed graphs.  Submitted.  
\arXiv{2302.05992}.  
\codes{\CSG:~\CInvar:~\CCol,~\COri}%
\itemspace

\author{Raul Cordovil} 						%CORDOVIL
\authorspace
\authref  See \hyperlink{Pascal Berthome}{P.~Berthom\'e}.
\itemspace

\author{Denis Cornaz}						%CORNAZ
\authorspace
\bsgitem{2006a}{}  On  co-bicliques.  \emph{RAIRO Oper.\ Res.}\ 41 (2006), 295--304.
\MR{2348004} (2009f:\-90053).  \Zbl{1227.90043}.  
\codes{\CSG}%
\itemspace

\author{Denis Cornaz \& A.\ Ridha Mahjoub}
\authorspace
\bsgitem{2007a}{}  The maximum induced bipartite subgraph problem with edge weights.  \emph{SIAM J.\ Discrete Math.}\ 21 (2007), no.\ 3, 662--675.
\MR{2353996} (2008j:05331).  \Zbl{1141.05076}.  
\codes{\CSD}%
\itemspace

\author{Derek G.\ Corneil} 						%CORNEIL
\authorspace
\authref  See \hyperlink{Charles J.\ Colbourn}{C.J.~Colbourn} and \xref{Seidel}{1991a}{J.J.\ Seidel}.
\itemspace

\author{G\'erard Cornu\'ejols}						%CORNUEJOLS
\authorx{Gerard Cornuejols}
\authorspace
\authref  See also \hyperlink{Manoel Campelo}{M.~Campelo} and \hyperlink{Michele Conforti}{M.~Conforti}.
\authrefspace

\bsgitem{2001a}{}  \emph{Combinatorial Optimization: Packing and Covering}.  
CBMS-NSF Reg.\ Conf.\ Ser.\ Appl.\ Math., Vol.\ 74.  Soc.\ Indust.\ Appl.\ Math., Philadelphia, 2001.
\MR{1828452} (2002e:90004).  \Zbl{972.90059}.					
\annot{} The topic is linear optimization over a clutter, esp.\ a ``binary clutter'', which is the class of negative circuits of a signed binary matroid.  The class $\cC^-(\Sigma)$ is an important example (see \xref{Seymour}{1977a}{P.D.\ Seymour}), as is its blocker $b\big(\cC^-(\Sigma)\big)$ [which is the class of minimal balancing edge sets; hence the frustration index $l(\Sigma) =$ minimum size of a member of the blocker].
\annotctd{} Ch.~5: ``Graphs without odd-$K_5$ minors'', i.e., signed graphs without $-K_5$ as a minor.  Some esp.\ interesting results:  Thm.\ 5.0.7 (special case of \xref{Seymour}{1977a}{P.D.\ Seymour}, Main Thm.):  The clutter of negative circles of $\Sigma$ has the ``Max-Flow Min-Cut Property'' (Seymour's ``Mengerian'' property) iff $\Sigma$ has no $-K_4$ minor.  Conjecture 5.1.11 is \xref{Seymour's}{1981a}{P.D.\ Seymour} beautiful conjecture (his ``weak MFMC'' is here called ``ideal'').  \S5.2 reports the partial result of \xref{Guenin}{2001a}{Bertrand Guenin}.  (See also \S8.4.)
\annotctd{} Def.\ 6.2.6 defines a signed graph ``$G(A)$'' of a $0, \pm1$-matrix $A$, whose transposed incidence matrix is a submatrix of $A$.  \S6.3.3: ``Perfect $0,\pm1$-matrices, bidirected graphs and conjectures of \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg}, associates a bidirected graph with a system of 2-variable pseudoboolean inequalities; reports on \xref{Sewell}{1996a}{E.C.\ Sewell} (\qv).
\annotctd{} \S8.4: ``On ideal binary clutters'', reports on \xref{Cornu\'ejols and Guenin}{2002a}{Gerard Cornuejols \& Bertrand Guenin}, \xref{Guenin}{1998a}{Bertrand Guenin}, and \xref{Novick and Seb\"o}{1995a}{Beth Novick \& Andras Sebo} (\emph{qq.v.}).
\par
\codes{\CS(\CMatrd),~\CSG:~\CMatrd,~\CGeom,~\CInc(\CGen),~\COri:~\CExp,~\CRef,~\CExr}%
\itemspace

\author{G\'erard Cornu\'ejols \& Bertrand Guenin}
\authorx{Gerard Cornuejols \& Bertrand Guenin}
\authorspace
\bsgitem{2002a}{}  Ideal binary clutters, connectivity, and a conjecture of Seymour.  \emph{SIAM J.\ Discrete Math.}\ 15 (2002), no.\ 3, 329--352.
\MR{1921026} (2003h:05057).  \Zbl{1035.90045}.					
\annot{} A partial proof of \xref{Seymour's}{1981a}{P.D.\ Seymour} conjecture.  Main Thm.:  A binary clutter is ideal if it has as a minor none of the circuit clutter of $F_7$, $\cC^-(-K_5)$ or its blocker, or $\cC^-(-K_4)$ or its blocker.  Important are the lift and extended lift matroids, $\bfL(M,\sigma)$ and $\bfL_0(M,\sigma)$, defined as in signed graph theory.  [See \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}, \S8.4.]
	
\par
\codes{\CS(\CMatrd),~\CSG:~\CMatrd,~\CGeom}%
\itemspace

\authorx{Sylvie Corteel}						%CORTEEL
\author{Sylvie Corteel, David Forge, \& V\'eronique Ventos}
\authorx{Sylvie Corteel, David Forge, \& Veronique Ventos}
\authorspace
\bsgitem{2015a}{}  Bijections between affine hyperplane arrangements and valued graphs.  \emph{European J.\ Combin.}\ 50 (2015), 30--37.
\MR{3361409}.  \Zbl{1323.52012}.  \arXiv{1403.2573}.  \HAL{hal-01365957}.		
\codes{\Cgg:~\CGeom,~\CMatrd}%
\itemspace

\author{S.\ Cosares}							%COSARES
\authorspace
\authref  See \hyperlink{L.\ Adler}{L.~Adler}.
\itemspace

\author{Collette R.\ Coullard} 						%COULLARD
\authorspace
\authref  See also \hyperlink{Vijaya Chandru}{V.~Chandru}.
\authrefspace

\author{Collette R.\ Coullard, John G.\ del Greco, \& Donald K.\ Wagner} 
\authorspace
\bsgitem{1991a}{\dag\dag}  Representations of bicircular matroids.  \emph{Discrete Appl.\ Math.}\ 32 (1991), 223--240.  
\MR{1120878} (92i:05072).  \Zbl{755.05025}.					
\annot{} \S4:  \S4.1 describes 4 fairly simple types of ``legitimate'' graph operation that preserve the bicircular matroid.  Thm.\ 4.11 is a converse: if $\Gamma_1$ and $\Gamma_2$ have the same connected bicircular matroid, then either they are related by a sequence of legitimate operations, or they belong to a small class of exceptions, all having order $\leq 4$, whose bicircular matroid isomorphisms are also described.  This completes the isomorphism theorem of \xref{Wagner}{1985a}{Donald K.\ Wagner}.  
\S5:  If finitely many graphs are related by a sequence of legitimate operations (so their bicircular matroids are isomorphic), then they have contrabalanced real gains whose incidence matrices are row equivalent.  These results are also found by a different approach in \xref{Shull, Orlin, \etal\!}{1989a}{Randy Shull, James B.\ Orlin, Alan Shuchat, \& Marianne L.\ Gardner}, \xref{Shull, Shuchat, \etal\!}{1993a}{Randy Shull, Alan Shuchat, James B.\ Orlin, \& Marianne Lepp}, \xrefd{1997a}{Randy Shull, Alan Shuchat, James B.\ Orlin, \& Marianne Lepp}.
	
\par
\codes{\CBic:~\CStr,~\CInc}%
\itemspace

\bsgitem{1993a}{}  Recognizing a class of bicircular matroids.  \emph{Discrete Appl.\ Math.}\ 43 (1993), 197--215.  
\MR{1223421} (94i:05021).  \Zbl{777.05036}.					
\codes{\CBic:~\CAlgor}%
\itemspace

\bsgitem{1993b}{}  Uncovering generalized-network structure in matrices.  \emph{Discrete Appl.\ Math.}\ 46 (1993), 191--220.  
\MR{1243724} (95c:68179).  \Zbl{784.05044}.					
\codes{\CGN:~\CBic:~\CInc,~\CAlgor}%
\itemspace

\authorx{G.\ Coutinho}						%COUTINHO
\author{G.\ Coutinho, C.\ Godsil, H.\ Shirazi, \& H.\ Zhan}
\authorspace
\bsgitem{2016a}{}  Equiangular lines and covers of the complete graph.  \emph{Linear Algebra Appl.}\ 488 (2016), 264--283.
\MR{3419786}.  \Zbl{1409.05220}.  \arXiv{1504.00085}.				
\par
\codes{\Csg:~\CLap,~\CCov}%
\itemspace

\author{Gheorghe Craciun}						%CRACIUN
\authorspace
\authref  See also \hyperlink{Murad Banaji}{M.~Banaji} and \hyperlink{Maya Mincheva}{M.~Mincheva}.
\authrefspace

\author{Gheorghe Craciun \& Martin Feinberg}
\authorspace
\bsgitem{2005a}{}  Multiple equilibria in complex chemical reaction networks:  I.  The injectivity property.  \emph{SIAM J.\ Appl.\ Math.}\ 65 (2005), 1526--1546.
\MR{2177713} (2006g:\-92075).  \Zbl{1094.80005}.					
\codes{\CSG,~\CChem}%
\itemspace

\bsgitem{2006a}{}  Multiple equilibria in complex chemical reaction networks:  II.  The species-reaction graph.  \emph{SIAM J.\ Appl.\ Math.}\ 66, no.\ 4, 1321--1338.
\MR{2246058} (2007e:92027).  \Zbl{1136.80306}.					
\codes{\CSG,~\CChem}%
\itemspace

\bsgitem{2006b}{}  Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models.  \emph{IEEE Proc.\ Systems Biol.}\ 153 (2006), no.\ 4, 179--186.
\par
\codes{\CSG,~\CChem}%
\itemspace

\author{Gheorghe Craciun, Casian Pantea, \& Eduardo D.\ Sontag}
\authorspace
\bsgitem{2011a}{}  Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks.  In:  Heinz Koeppl, Douglas Densmore, Gianluca Setti, and Mario di Bernardo, eds., \emph{Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology}, pp.\ 63--72.  Springer, New York, 2011.
\codes{\CSG,~\CChem:~\CExp}%
\itemspace

\author{Yves Crama} 						%CRAMA
\authorspace
\authref  See also \hyperlink{Endre Boros}{E.~Boros}.
\authrefspace

\bsgitem{1989a}{}  Recognition problems for special classes of polynomials in 0--1 variables.  \emph{Math.\ Programming} A44 (1989), 139--155.  
\MR{1003557} (90f:90091).  \Zbl{674.90069}.					
\annot{} Balance and switching are used to study pseudo-Boolean functions.  (\S\S2.2 and 4.)
\codes{\CSG:~\CBal,~\CSw}%
\itemspace

\author{Yves Crama \& Peter L.\ Hammer} 
\authorspace
\bsgitem{1989a}{}  Recognition of quadratic graphs and adjoints of bidirected graphs.  
Combinatorial Math.:  Proc.\ Third Int.\ Conf.  \emph{Ann.\ New York Acad.\ Sci.}\ 555 (1989), 140--149.  
\MR{1018617} (91d:05044).  \Zbl{744.05060}.					
\annot{} ``Adjoint'' = unoriented positive part of the line graph of a bidirected graph.  ``Quadratic graph'' = graph that is an adjoint.  Recognition of adjoints of bidirected simple graphs is NP-complete. 
\codes{\Csg:~\COri:~\CLG:~\CAlgor}%
\itemspace

\author{Yves Crama, Peter L.\ Hammer, \& Toshihide Ibaraki} 
\authorspace
\bsgitem{1986a}{}  Strong unimodularity for matrices and hypergraphs.  Appl.\ Combin.\ Methods Math.\ Programming (Gainesville, Fla., 1985).  \emph{Discrete Appl.\ Math.}\ 15 (1986), 221--239.  
\MR{0865003} (88a:05105).  \Zbl{647.05042}.					
\annot{} \S7:  Signed hypergraphs, with a surprising generalization of balance.
\par
\codes{\CSH:~\CBal}%
\itemspace

\author{Y.\ Crama, M.\ Loebl, \& S.\ Poljak}
\authorspace
\bsgitem{1992a}{}  A decomposition of strongly unimodular matrices into incidence matrices of digraphs.  \emph{Discrete Math.}\ 102 (1992), 143--147.
\MR{1170457} (93g:05097).  \Zbl{776.05071}.					
\par
\codes{\CSG}%
\itemspace

\authorx{R.\ Crowston}						%CROWSTON
\author{R.\ Crowston, G.\ Gutin, M.\ Jones, \& G.\ Muciaccia}
\authorspace
\bsgitem{2013a}{}  Maximum balanced subgraph problem parameterized above lower bound.  \emph{Theoretical Computer Sci.}\ 513 (2013), 53--64.
\MR{3128945}.  \Zbl{1396.68054}.  \arXiv{1212.6848}.				
\annot{} An algorithm for $l(\Sigma) \leq \frac12 \#E - \frac14 (n-1) - \frac14 k$ with time linear in $n$ and exponential in $k$, assuming $\Sigma$ is simple.  (The upper bound $l \leq \frac12 \#E - \frac14 (n-1)$ is from \xref{Poljak and Turz\'{\i}k}{1982a}{Svatopluk Poljak \& Daniel Turzik}, \xrefd{1986a}{Svatopluk Poljak \& Daniel Turzik}.)
\dated{2 Mar 2014} 
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\authorx{Anne Crumi\`ere}						%CRUMIERE
\authorx{Anne Crumiere}
\author{Anne Crumi\`ere \& Paul Ruet}
\authorx{Anne Crumiere \& Paul Ruet}
\authorspace
\bsgitem{2008a}{}  Spatial differentiation and positive circuits in a discrete framework.  Proc.\ Workshop (DCM 2007, Wroc{\l}aw).  \emph{Electronic Notes Theor.\ Computer Sci.}\ 192 (2008), 85--100.
\Zbl{1277.92008}.  \HAL{hal-00692089}.  
\annot{} Regulatory graph: a signed digraph.
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Anne Crumi\`ere \& Mathieu Sablik}
\authorx{Anne Crumiere \& Mathieu Sablik}
\authorspace
\bsgitem{2008a}{}  Positive circuits and $d$-dimensional spatial differentiation: Application to the formation of sense organs in \emph{Drosophila}.  \emph{BioSystems} 94 (2008), 102--108.
\annot{} Regulatory graph: a signed digraph.
\codes{\CBiol:~\CSD:~\CDyn}%
\itemspace

\authorx{Lin Cui}							%CUI L
\author{Lin Cui \& Yi-Zheng Fan}
\authorspace
\bsgitem{2010a}{}  The signless Laplacian spectral radius of graphs with given number of cut vertices.  \emph{Discuss.\ Math.\ Graph Theory} 30 (2010), no.\ 1, 85--93.
\MR{2676064} (2011j:05196).  \Zbl{1215.05101}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Qing Cui}							%CUI Q
\authorspace
\authref  See \hyperlink{Wenzhong Liu}{W.-Z.~Liu}.
\itemspace

\authorx{Shu-Yu Cui}							%CUI SY
\author{Shu-Yu Cui \& Gui-Xian Tian}
\authorspace
\bsgitem{2012a}{}  The signless Laplacian spectrum of the (edge) corona of two graphs.  \emph{Utilitas Math.}\ 88 (2012), 287--297.
\MR{2975841}.  \Zbl{1256.05137}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{G.J.\ Culos}							%CULOS
\author{G.J.\ Culos, D.D.\ Olesky, \& P. van den Driessche}
\authorspace
\bsgitem{2016a}{}  Using sign patterns to detect the possibility of periodicity in biological systems.  \emph{J.\ Math.\ Biol.}\ 72 (2016), 1281--1300.
\MR{3464203}.  \Zbl{1337.92015}.  
\codes{\CQM:~\CSD}%
\itemspace

\author{William H.\ Cunningham} 					%CUNNINGHAM
\authorspace
\authref  See \hyperlink{Julian Araoz}{J.~Ar\'aoz} and \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}.
\itemspace

\author{Drago\v{s} M.\ Cvetkovi\'c} 					%CVETKOVIC
\authorx{Dragos M.\ Cvetkovic}
\authorspace
\authref  See also \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}, \hyperlink{F.C.\ Bussemaker}{F.C.~Bussemaker}, \hyperlink{Domingos M.\ Cardoso}{D.M.~Cardoso}, and \hyperlink{Michael Doob}{M.~Doob}.
\authrefspace

\bsgitem{1978a}{}  The main part of the spectrum, divisors and switching of graphs.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 23 (37) (1978), 31--38. 
\MR{0508125} (80h:05045).  \Zbl{423.05028}.
\itemspace

\bsgitem{1995a}{}  Star partitions and the graph isomorphism problem.  \emph{Linear Algebra Appl.}\ 39 (1995), 109--132.  
\MR{1374474} (97b:05105).  \Zbl{831.05043}.
\annot{} Pp.\ 128--130 discuss switching-equivalent graphs.  Some of the theory is invariant, hence applicable to two-graphs.  [\Question.  How can this be generalized to signed graphs and their switching classes?]
\codes{\Ctg:~\CAdj}%
\itemspace

\bsgitem{2005a}{}  Signless Laplacians and line graphs.  \emph{Bull.\ Cl.\ Sci.\ Math.\ Nat.\ Sci.\ Math.}\ No.\ 30 (2005), 86--92.
\MR{2213761} (2006m:05152).  \Zbl{1119.05066}.					
\par
\codes{\Cpar:~\CLap,~\CLG:~\CEig}%
\itemspace

\bsgitem{2008a}{}  New theorems for signless Laplacian eigenvalues.  \emph{Bull.\ Cl.\ Sci.\ Math.\ Nat.\ Sci.\ Math.}\ No.\ 33 (2008), 131--146.
\MR{2609604} (2011b:05145).  \Zbl{1199.05212}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  Spectral theory of graphs based on the signless laplacian (A quick outline).  Res.\ report, 2010.  
URL \url{http://www.mi.sanu.ac.rs/projects/signless\_L\_reportJan28.pdf}
\annot{} Surveys the spectral theory of $\Lap(-\Gamma)$.
\dated{23 Nov 2014} 
\par
\codes{\Cpar:~\CLap:~\CEig:~\CExp}%
\itemspace

\bsgitem{2010b}{\dag}  Bibliography on the signless laplacian eigenvalues: First one hundred references.  Manuscript, 2010.  \url{http://www.mi.sanu.ac.rs/projects/signless_100.pdf}
\annot{} Eigenvalues of $\Lap(-\Gamma)$, with a brief history.  $\Lap(-\Gamma)$ is important because $\Spec \Lap(-\Gamma)$ ``dominates'' $\Spec \Lap(\Phi)$ for any signed or complex unit gain graph; \cf\ \xref{Reff}{2012a}{Nathan Reff}.
\dated{12 Dec 2020} 
\codes{\Csg:~\CPar:~\CLap:~\CEig}%
\itemspace

\author{Drago\v{s} M.\ Cvetkovi\'c \& Michael Doob}
\authorx{Dragos M.\ Cvetkovic \& Michael Doob}
\authorspace
\bsgitem{1984a}{}  Root systems, forbidden subgraphs, and spectral characterizations of line graphs.  In:  \emph{Graph Theory} (4th Yugosl.\ Sem., Novi Sad, 1983), pp.\ 69--99.  Univ.\ Novi Sad, Novi Sad, 1984.
\MR{0751442} (86a:05088).  \Zbl{533.05041}.					
\par
\codes{\Csg:~\Cpar:~\CGeom,~\CLG}%
\itemspace

\author{Dragos M.\ Cvetkovi\'c, Michael Doob, Ivan Gutman, \& Aleksandar Torga\v{s}ev}
\authorx{Dragos M.\ Cvetkovic, Michael Doob, Ivan Gutman, \& Aleksandar Torgasev}
\authorspace
\bsgitem{1988a}{}  \emph{Recent Results in the Theory of Graph Spectra}.  Ann.\ Discrete Math., 36.  North-Holland, Amsterdam, 1988.  
\MR{0926481} (89d:05130).  \Zbl{634.05034}.						
\annot{} Signed graphs mentioned:  P.\ 40 cites \xref{Zaslavsky}{1981a}{Thomas Zaslavsky}.  
Pp.\ 44--45 (with unusual terminology) describe \xref{B.D.\ Acharya}{1980a}{B.\ Devadas Acharya} and \xref{M.K.\ Gill}{1981b}{Mukhtiar Kaur Gill}.  
P.\ 100 cites \xref{B.D.\ Acharya}{1979b}{B.\ Devadas Acharya}.  
All-negative signatures are implicated in the infinite-graph eigenvalue theorem of \xref{Torga\v{s}ev}{1982a}{Aleksandar Torgasev}, Thm.\ 6.29 of this book.  
M\"obius molecules (with signed molecular graphs) mentioned on p.\ 149.
\codes{\CSG,~\Cpar:~\CEig:~\CExp,~\CAppl,~\CRef}%
\itemspace

\author{Drago\v{s} M.\ Cvetkovi\'c, Michael Doob, \& Horst Sachs} 
\authorx{Dragos M.\ Cvetkovic, Michael Doob, \& Horst Sachs} 
\authorspace
\bsgitem{1980a}{}  \emph{Spectra of Graphs: Theory and Application}.  VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.  Copublished as:  Pure and Appl.\ Math., Vol.\ 87.  Academic Press, New York-London, 1980.  
\MR{0572262} (81i:05054).  \Zbl{458.05042}.					
\annot{} \S4.6:  Signed digraphs with multiple edges are employed to analyze the characteristic polynomial of a digraph.  (Signed) switching, too.  
Pp.\ 187--188:  Exercises involving Seidel switching and the Seidel adjacency matrix.  Thm.\ 6.11 (\xref{Doob}{1973a}{Michael Doob}):  The even-cycle matroid determines the eigenvaluicity of $-2$.  \S7.3: ``Equiangular lines and two-graphs.''
\dated{$\leq$ 2000, rev 20 Sept 2010}
\annotctd{} [Russian ed.:  \xref{Tsvetkovich, Dub, and Zakhs}{1984a}{D.\ Tsvetkovich, M.\ Dub, \& Kh.\ Zakhs}.]
\par
\codes{\CSD,~\Cpar,~\CTG:~\CSw,~\CAdj,~\CEig,~\CGeom:~\CExp,~\CExr,~\CRef}%
\itemspace

\bsgitem{1982a}{}  \emph{Spectra of Graphs: Theory and Application}.  VEB Deutscher Verlag der Wissenschaften, Berlin, 1982.
\MR{0690768} (84a:05046).  
\annot{} Update of \xrefd{1980a}{Dragos M.\ Cvetkovic, Michael Doob, \& Horst Sachs}.
\par
\codes{\CSD,~\Cpar,~\CTG:~\CSw,~\CAdj,~\CEig,~\CGeom:~\CExp,~\CExr,~\CRef}%
\itemspace

\bsgitem{1995a}{}  \emph{Spectra of Graphs: Theory and Applications}.  Third ed.  Johann Ambrosius Barth, Heidelberg, 1995.  
\MR{1324340} (96b:05108).  \Zbl{824.05046}.
\annot{} Appendices update \xrefd{1982a}{Dragos M.\ Cvetkovic, Michael Doob, \& Horst Sachs}, beyond the updating in \xref{Cvetkovi\'c, Doob, Gutman, and Torga\v{s}ev}{1988a}{Dragos M.\ Cvetkovic, Michael Doob, Ivan Gutman, \& Aleksandar Torgasev}.  App.\ B.3, p.\ 381 mentions work of \hyperlink{G.R.\ Vijayakumar}{Vijayakumar} (\qv).  
P.\ 422: Pseudo-inverse graphs (when $A(\Gamma)^{-1} = A(\Sigma)$ for some balanced $\Sigma$, $|\Sigma|$ is the ``pseudo-inverse'' of $\Gamma$). 
\par
\codes{\CSD,~\Cpar,~\CTG:~\CAdj,~\CLap,~\CEig,~\CSw,~\CGeom,~\CBal:~\CExp,~\CExr,~\CRef}%
\itemspace

\author{Drago\v{s} Cvetkovi\'c, Michael Doob, \& Slobodan Simi\'c} 
\authorx{Dragos Cvetkovic, Michael Doob, \& Slobodan Simic} 
\authorspace
\bsgitem{1980a}{} Some results on generalized line graphs.  \emph{C.\ R.\ Math.\ Rep.\ Acad.\ Sci.\ Canada} 2 (1980), 147--150.  
\MR{0576993} (81f:05136).  \Zbl{434.05057}.
\annot{} Abstract of \xrefd{1981a}{Dragos Cvetkovic, Michael Doob, \& Slobodan Simic}.  
\codes{\Csg:~\CLG,~\CEig(\CLG),~\CAut(\CLG)}%
\itemspace

\bsgitem{1981a}{}  Generalized line graphs.  \emph{J.\ Graph Theory} 5 (1981), 385--399.  
\MR{0635701} (82k:05091).  \Zbl{475.05061}.
\codes{\Csg:~\CLG,~\CEig(\CLG),~\CAut(\CLG)}%
\itemspace

\author{Drago\v{s} Cvetkovi\'c, Peter Rowlinson, \& Slobodan K.\ Simi\'c}
\authorx{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2004a}{}  \emph{Spectral Generalizations of Line Graphs:  On Graphs with Least Eigenvalue $-2$.}\  
London Math.\ Soc.\ Lect.\ Note Ser., 314.  Cambridge Univ.\ Press, Cambridge, Eng., 2004.
\MR{2120511} (2005m:05003).  \Zbl{1061.05057}.					
\annot{} Generalized line graphs are the fundamental example.  Pp.\ 190--191 mention signed graphs representable in root systems as in papers of \hyperlink{G.R.\ Vijayakumar}{G.R.\ Vijayakumar} (\qv) [but not mentioning line graphs of signed graphs].
\dated{13 Oct 2010}
\codes{\CLG:~\CGen,~\CGeom,~\CEig}\codesctd{\CSG:~\CGeom:~\CExp}%
\itemspace

\bsgitem{2007a}{}  Signless Laplacians of finite graphs.  \emph{Linear Algebra Appl.}\ 423 (2007), no.\ 1, 155--171.
\MR{2312332} (2008c:05105).  \Zbl{1113.05061}.					
\annot{} ``Signless Laplacian'' $Q(\Gamma) :=$ Laplacian matrix $\Lap(-\Gamma) = D(\Gamma) + A(\Gamma)$.  Spectral properties; bounds for graph invariants; combinatorics of coefficients of characteristic polynomial of $\Lap(-\Gamma)$.  	
[\Problem.  Find all articles on ``signless Laplacians'', herein called $\Lap(-\Gamma)$.  Generalize to signed graphs, with nonbipartite graphs generalizing to unbalanced graphs.]
\dated{14 Sept 2010} 
\codes{\Csg:~\CPar:\CLap:~\CEig}%
\itemspace

\bsgitem{2007b}{}  Eigenvalue bounds for the signless Laplacian.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 81(95) (2007), 11--27.
\MR{2401311} (2009e:05181).  \Zbl{1164.05038}.					
\annot{} See \xrefd{2007a}{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}.  Thm.:  For connected $\Gamma$ with $\#V=n$ and $\#E=m$, $\lambda_1(\Lap(-\Gamma))$ is maximized when $\Gamma$ is a nested split graph.  Also, many computer-generated conjectures (\cf\ \xref{Aouchiche and Hansen}{2010a}{Mustapha Aouchiche \& Pierre Hansen}); some are proved (here or elsewhere) or disproved; some are difficult.
\dated{4 Sept 2010, 22 Jan 2012} 
\codes{\CPar:~\CLap:~\CEig,~\CLG}%
\itemspace

\bsgitem{2010a}{}  \emph{An Introduction to the Theory of Graph Spectra}.  London Math.\ Soc.\ Student Texts, 75.  Cambridge Univ.\ Press, Cambridge, Eng., 2010.
\MR{2571608} (2011g:\-05004).  \Zbl{1211.05002}.					
\annot{} Graph switching in \S1.1.  Reduced line graphs of simply signed graphs are implicit in the construction of generalized line graphs in \S1.2.
\dated{14 Sept 2010}
\codes{\Ctg:~\CSw:~\CExp}\codesctd{\Csg:~\CLG:~\CExp}%
\annotctd{} \S7.8, ``The signless Laplacian''.
\codes{\CPar,~\CLG:~\CLap:~\CEig:~\CExp}%
\itemspace

\author{Drago\v{s} M.\ Cvetkovi\'c \& Slobodan K.\ Simi\'c} 
\authorx{Dragos M.\ Cvetkovic \& Slobodan K.\ Simic} 
\authorspace
\bsgitem{1978a}{} Graphs which are switching equivalent to their line graphs.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 23 (37) (1978), 39--51.  
\MR{0508126} (80c:05108).  \Zbl{423.05035}.
\par
\codes{\Csw:~\CLG}%
\itemspace

\bsgitem{2009a}{}  Towards a spectral theory of graphs based on the signless Laplacian.  I.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 85(99) (2009), 19--33.
\MR{2536686} (2010i:05203).  \Zbl{224.05293}.					
\annot{} See \xref{Cvetkovi\'c, Rowlinson, and Simi\'c}{2007a}{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic}.	
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  Towards a spectral theory of graphs based on the signless Laplacian.  II.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 9, 2257--2272.
\MR{2599858} (2011d:05217).  \Zbl{1218.05089}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010b}{}  Towards a spectral theory of graphs based on the signless Laplacian.  III.  \emph{Appl.\ Anal.\ Discrete Math.}\ 4 (2010), no.\ 1, 156--166.
\MR{2654936} (2011m:05169).  \Zbl{1265.05360}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011a}{}  Graph spectra in Computer Science.  \emph{Linear Algebra Appl.}\ 434 (2011), no.\ 6, 1545--1562. 
\MR{2775765} (2011m:05170).  \Zbl{1207.68230}.					
\codes{\CPar:~\CEig:~\CExp}%
\itemspace

\author{D.\ Cvetkovi\'c, S.K.\ Simi\'c, \& Z.\ Stani\'c}
\authorx{D.\ Cvetkovic, S.K.\ Simic, \& Z.\ Stanic}
\authorspace
\bsgitem{2010a}{}  Spectral determination of graphs whose components are paths and cycles.  \emph{Computers Math.\ Appl.}\ 59 (2010), 3849--3857.
\MR{2651858} (2011j:05197) (\qv).  \Zbl{1198.05110}.				
\annot{} $\Spec \Lap(-\Gamma)$ performs well to determine the graph from its spectrum.
\dated{29 Apr 2022} 
\codes{\Csg:~\CPar:~\CEig}%
\itemspace

\authorx{Marek Cygan}						%CYGAN
\author{Marek Cygan, Marcin Pilipczuk, Micha\l\ Pilipczuk, \& Jakub Onufry Wojtaszczyk}
\authorx{Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, \& Jakub Onufry Wojtaszczyk}
\authorspace
\bsgitem{2012a}{}  Sitting closer to friends than enemies, revisited.  In:  Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, eds., \emph{Mathematical Foundations of Computer Science 2012} (37th MFCS, Bratislava, 2012), pp.\ 296--307.  \LNCS 7464.  Springer, Heidelberg, 2012.
\MR{3030440}.  \Zbl{1365.68351}.  \arXiv{1201.1869}.				
\annot{} Same as \xrefd{2015a}{Marek Cygan, Marcin Pilipczuk, Micha\l\ Pilipczuk, \& Jakub Onufry Wojtaszczyk}.
\codes{\CSG:~\CKG:~\CBal,~\CAlgor}%
\itemspace

\bsgitem{2015a}{}  Sitting closer to friends than enemies, revisited.  \emph{Theory Computing Systems} 56 (2015), no.\ 2, 394--405.
\MR{3311468}.  \Zbl{1312.05067}.  \arXiv{1201.1869}.				
\annot{} Sequel to \xref{Kermarrec and Thraves}{2011a}{Anne-Marie Kermarrec \& Christopher Thraves}.
\dated{26 Apr 2012} 
\par
\codes{\CSG:~\CKG:~\CBal,~\CAlgor}%
\itemspace

\authorx{V.\ Jude Annie Cynthia}					%CYNTHIA
\author{V.\ Jude Annie Cynthia \& E.\ Padmavathy}
\authorspace
\bsgitem{2018a}{}  Signed product cordiality of circulant network.  \emph{Int.\ J.\ Pure Appl.\ Math.}\ 118 (2018), no.\ 23, 353--361.
\annot{} More as in \xref{Baskar Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\par
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\bsgitem{2021a}{}  Signed cordial labeling and signed product cordial labeling of some interconnection networks.  \emph{Adv.\ Appl.\ Discrete Math.}\ 26 (2021), no.\ 1, 35--51.
\Zbl{1499.05547}.  
\annot{} \Cf\ \xref{Delphy and Devaraj }{2011a}{Prem Delphy P \& Devaraj J} and \xref{Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan} for cordiality and product cordiality, resp.  Examples.
\dated{26 Sept 2022} 
\codes{\CLab:~\CSG,~\CVS}%
\itemspace

\bsgitem{2021b}{}  On signed magic cordial labeling.  \emph{Adv.\ Appl.\ Discrete Math.}\ 27 (2021), no.\ 1, 15--30.
\Zbl{1499.05548}.  
\codes{\CLab:~\CSG,~\CVS}%
\itemspace


\label{LETTER D} \hypertarget{LETTER D}{}	% LETTER  D


\author{[Ilda P.F.\ da Silva]}						%DA SILVA
\authorspace
\authref  See \hyperlink{Ilda P.F.\ da Silva}{I.P.F.~da~Silva} (under `S').
\itemspace

\author{Payal Dabas}						%DABAS
\authorspace
\authref  See \hyperlink{Sangita Kansal}{S.~Kansal}.
\itemspace

\author{A.\ Daemi}							%DAEMI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{F.\ Scott Dahlgren}						%DAHLGREN
\authorspace
\authref  See \hyperlink{Marina Arav}{M.~Arav}.
\itemspace

\authorx{C.\ Dalf'o}							%DALFO
\author{C.\ Dalf'o, M.A.\ Fiol, M.\ Miller, \& J.\ Ryan [Joe Ryan]}
\authorspace
\bsgitem{2017a}{}  From expanded digraphs to lifts of voltage digraphs and line digraphs.  \emph{Australasian J.\ Combin.}\ 69 (2017), 323--333. 
\MR{3714196}.  \Zbl{1375.05113}.  \arXiv{1608.06233}.				
\codes{\CGG:~\CCov}%
\itemspace

\bsgitem{2017b}{}  On quotient digraphs and voltage digraphs.  \emph{Australasian J.\ Combin.}\ 
\MR{3714200}.  \Zbl{1375.05114}.  \arXiv{1612.08855}.				
\codes{\CGG:~\CCov}%
\itemspace

\author{Mina Dalirrooyfard}						%DALIRROOYFARD
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Soudabeh Dalvandi}						%DALVANDI
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}.
\authrefspace

\author{S.\ Dalvandi, F.\ Heydari, \& M.\ Maghasedi}
\authorspace
\bsgitem{2021a}{}  Signed complete graphs with negative paths.  \emph{J.\ Math.\ Extension} 15 (2021), no.\ 1, 127--136.
\Zbl{1473.05115}.  
\annot{} Finds many eigenvalues when $\Sigma^-$ consists of disjoint paths.
\dated{26 Dec 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Signed complete graphs with exactly $m$ non-negative eigenvalues.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 45 (2022), 2107--2122.
\MR{4489553}.  
\annot{} Adjacency eigenvalues of $K_n(-\Delta)$ with non-spanning $\Delta$.  Thm.\ 7:  $c(\Delta) \leq m-1$.  If $= m-1$, each component is complete bipartite.  [I.e., bipartitionally induced in $K_n$.  \Question.  Does that generalize to $\Gamma(-\Delta)$, for 2-connected $\Gamma$?]  
Thm.\ 11:  $m = 3$ implies $c(\Delta) = 2$.
Characterized:  $m=3$ with disconnected or (Thm.\ 15) tree $\Delta$.  
Thm.\ 14:  If connected $\Delta$ has exactly 2 nonnegative eigenvalues, then $K_n(-\Delta)$ has exactly 3.
Lemmas:  Spectra of examples such as $K_{2m}(-M_m)$, $M_m$ an $m$-edge matching.  
\annotctd{} Thm.\ 8:  A signed graph with exactly one positive eigenvalue is balanced.  
Thm.\ 10:  For any $\Delta$, if $m = 2$, $\Delta$ cannot be spanning.  
\dated{17 Oct, 26 Dec 2022, 13 Jan 2023} 
\codes{\CSG:~\CKG:~\CStr,~\CAdj:~\CEig}%
\itemspace

\author{K.V.\ Dalvi}							%DALVI
\authorspace
\authref  See \hyperlink{Ganesh Mundhe}{G.~Mundhe}.
\itemspace

\author{Edwin R.\ van Dam}						%DAM	%VAN DAM
\authorspace
\authref  See also \hyperlink{Pepijn Wissing}{P.~Wissing}.
\authrefspace

\author{E.R.\ van Dam \& M.A.\ Fiol}
\authorspace
\bsgitem{2012a}{}  A short proof of the odd-girth theorem.  \emph{Electronic J.\ Combin.}\ 19 (2012), no.\ 3, art.\ P12, 5 pp.
\MR{2967217}.  \Zbl{1253.05098}.  \arXiv{1205.0153}.				
\annot{} Odd Girth Thm.:  A graph with $d+1$ eigenvalues and odd girth $\geq 2d+1$ is a generalized odd graph.
[\Problem:  Generalize to signed graphs, odd girth becoming negative girth and distance-regular and generalized odd graphs becoming one wants to know what.  Knowing what would indicate what a distance-regular or generalized-odd signed graph should be.]
\dated{17 Dec 2014} 
\codes{\Cpar:~\CEig:~\CStr,~\CRef}%
\itemspace

\author{Edwin R.\ van Dam \& Willem H.\ Haemers}
\authorspace
\bsgitem{2003a}{}  Which graphs are determined by their spectrum?  
Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002).  \emph{Linear Algebra Appl.}\ 373 (2003), 241--272. 
\MR{2022290} (2005a:05135).  \Zbl{1026.05079}.					
\par
\codes{\Cpar:~\CLap:~\CEig}\codesctd{\Csg:~\Cpar:~\CLap:~\CEig:~\CExp}%
\itemspace

\bsgitem{2009a}{}  Developments on spectral characterizations of graphs.  Int.\ Workshop Design Theory, Graph Theory,  Comput.\ Methods -- IPM Combinatorics II.  \emph{Discrete Math.}\ 309 (2009), no.\ 3, 576--586.
\MR{2499010} (2010h:05178).  \Zbl{1205.05156}.					
\annot{} New and old results on $\Lap(-\Gamma)$, the ``signless Laplacian'' of $\Gamma$.
\dated{20 Dec 2011} 
\codes{\Cpar:~\CLap:~\CEig}\codesctd{\CPar:~\CEig:~\CExp}%
\itemspace

\author{Susan S.\ D'Amato}						%D'AMATO
\authorspace
\bsgitem{1979a}{}  Eigenvalues of graphs with twofold symmetry.  \emph{Molecular Phys.}\ 37 (1979), 1363--1369.
\MR{0535191} (80c:05098).  
\annot{} Spectrum of signed covering graph.  [See \xref{Butler}{2010a}{Steve Butler}.]
\dated{9 Mar 2011} 
\codes{\Csg:~\Ccov:~\CEig}%
\itemspace

\bsgitem{1979b}{}  Eigenvalues of graphs with threefold symmetry.  \emph{Theor.\ Chim.\ Acta} 53 (1979), 319--326.
\annot{} Ternary gain graphs: spectrum of covering graph, as with signed graphs in \xrefd{1979a}{Susan S.\ D'Amato}.
\dated{9 Mar 2011} 
\codes{\Cgg:~\Ccov:~\CEig}%
\itemspace

\authorx{Jeffrey M.\ Dambacher}					%DAMBACHER
\author{Jeffrey M.\ Dambacher, Richard Levins, \& Philippe A.\ Rossignol}
\authorspace
\bsgitem{2005a}{}  Life expectancy change in perturbed communities: Derivation and qualitative analysis.  \emph{Math.\ Biosciences} 197 (2005), no.\ 1, 1--14.
\MR{2167483} (2006d:92058).  \Zbl{1074.92037}.					
\codes{\CSD:~\CQM:~\CSta:~\CCycles,~\CRef}%
\itemspace

\author{Jeffrey M.\ Dambacher, Hiram W.\ Li, \& Philippe A.\ Rossignol}
\authorspace
\bsgitem{2003a}{}  Qualitative predictions in model ecosystems.  \emph{Ecological Modelling} 161 (2003), no.\ 1, 79--93.
\annot{} Feedback predictions from signed digraph $(D,\sigma)$ via ``weighted predictions'' $W_{ij} := |C_{ij}(-A(D,\sigma))|/P_{ij}(A(D))$, where $C_{ij}$ is the cofactor and $P_{ij}$ is the permanental cofactor.  $W_{ij} = 1$ means perfect predictability, $= 0$ means no predictability.  Numerical tests.
Dictionary:  ``Community matrix'' = $A(D,\sigma)$.
\dated{9 Sept 2010} 
\par
\codes{\CSD:~\CQM:~\CSta:~\CCycles,~\CRef}%
\itemspace

\authorx{E.\ Damiani}						%DAMIANI
\author{E.\ Damiani, O.\ D'Antona, \& F.\ Regonati}	
\authorspace
\bsgitem{1994a}{}  Whitney numbers of some geometric lattices.  \emph{J.\ Combin.\ Theory Ser.\ A} 65 (1994), 11--25.  
\MR{1255260} (95e:06019).   \Zbl{793.05037}.						
\annot{} E.g., log concavity of Whitney numbers of the second kind of Dowling lattices.
[\Cf\ \xref{Stonesifer}{1975a}{J.\ Randolph Stonesifer} and \xref{Benoumhani}{1999a}{Moussa Benoumhani}.]
\dated{rev.\ 30 Apr 2012}
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{A.\ Danielian}						%DANIELIAN
\authorspace
\bsgitem{1961a}{}  Ground state of an Ising face-centered cubic lattice.  \emph{Phys.\ Rev.\ Lett.}\ 6 (1961), 670--671.
\annot{} ``Ground states'', i.e.\ $\zeta: V \to \signnos$ with smallest $\#(E^\zeta)^-$, of the all-negative (antiferromagnetic) $R \times R \times R$ face-centered cubic lattice graph [assumed toroidal to avoid boundary effects?].  Frustration index $l =2\#V$; the number (``degeneracy'') of ground states is $2^{A\sqrt[3]{\#V}}$ where $A>0$; each ground state has 4-regular $E^-$.  See \xrefd{1964a}{A.\ Danielian} for more structure.  
[\Problem.  Determine the exact number and precise shape of all ground states $\zeta$ in terms of the graph.  Is there something interesting about $(\Sigma^\zeta)^-$, e.g., in its circle decomposition, symmetries, or transformations from one to another?]
\dated{21 Jun 2012} 
\par
\codes{\CSG,~\CPhys:~\CPar:~\CFr,~\CState(\Cfr)}%
\itemspace

\bsgitem{1964a}{}  Low-temperature behavior of a face-centered cubic antiferromagnet.  \emph{Phys.\ Rev.}\ 133 (1964), no.\ 5A, A1344--A1349.
\annot{} \S~II, ``The ground state'', continues \xrefd{1961a}{A.\ Danielian} with more details on the structure of ground states $\zeta$.  The number of them is small compared to the all-negative triangular lattice [\Question:  and other all-negative, highly symmetric graphs?].  $\zeta$ on each $x$-, $y$-, or $z$-layer has a form described in the paper.  Low-weight distance-2 edges will fix the ground state (p.\ A1346).  
\S~III, ``The partition function'', studies the effect of moving out of ground states.  
App.~A derives a formula for the energy change from switching a cluster of vertices, in terms of frustrated and satisfied edges within and without the cluster.  App.~B estimates the effect of switching additional vertices.  [\Problem.  Find rigorous treatments of such switchings; this means studying the energy landscape of state space $\{\zeta\} = \signnos^{\signnos^V}$.]
Dictionary:  ``bond'' = edge, ``even/odd bond'' = frustrated/satisfied edge = switches to $+$ or $-$.  
\dated{21 Jun 2012} 
\par
\codes{\CPhys,~\CSG:~\CPar:~\CFr,~\CState(\Cfr)}%
\itemspace

\author{Daniele D'Angeli}						%D'ANGELI
\authorspace
\authref  See \hyperlink{Matteo Cavaleri}{M.~Cavaleri}.
\itemspace

\author{O.\ D'Antona}						%D'ANTONA
\authorspace
\authref  See \hyperlink{E.\ Damiani}{E.~Damiani}.
\itemspace

\author{George B.\ Dantzig} 						%DANTZIG
\authorspace
\bsgitem{1963a}{} \emph{Linear Programming and Extensions}.  Princeton Univ.\ Press, Princeton, N.J., 1963.  
\MR{0201189} (34 \#1073).  \Zbl{108.33103} (108, p.\ 331c).
\annot{} Chapter 21:  ``The weighted distribution problem.'' 21-2: ``Linear graph structure of the basis.'' 
\codes{\CGN:~\CMatrd(\CBases)}%
\itemspace

\bsgitem{1966a}{} \emph{Linear Programming and Extensions}.  (In Russian.)  Transl.\ G.N.\ Andrianov, L.I.\ Gor'kov, A.A.\ Korbut, and A.N.\ Ljapunov.  ``Progress'' Publishers, Moscow, 1966. 
\Zbl{997.90504}.  
\codes{\CGN:~\CMatrd(\CBases)}%
\itemspace

\bsgitem{1998a}{} \emph{Linear Programming and Extensions}.  Repr.\ of 1968 corr.\ ed.  Princeton Landmarks in Math.  Princeton Univ.\ Press, Princeton, N.J., 1998.  
\MR{1658673} (99g:90004) (no rev).  \Zbl{997.90504}.
\codes{\CGN:~\CMatrd(\CBases)}%
\itemspace

\author{F.A.\ Dar}							%DAR
\authorspace
\authref  See \hyperlink{S.\ Pirzada}{S.~Pirzada}.
\itemspace

\author{Richard D'Ari}						%D'ARI
\authorspace
\authref  See \hyperlink{Rene Thomas}{R.~Thomas}.
\itemspace

\authorx{Avinandan Das}						%DAS A
\author{Avinandan Das, Lawqueen Kanesh, Jayakrishnan Madathil, \& Saket Saurabh}
\authorspace
\bsgitem{2021a}{}  Odd cycle transversal in mixed graphs.  In:  {\L}ukasz Kowalik \etal, eds., \emph{Graph-Theoretic Concepts in Computer Science} (47th Int.\ Workshop, WG 2021, Warsaw), pp.\ 130--142.  \LNCS 12911.  Springer, Cham, 2021.
\MR{4364278}.  
\annot{} ``Odd cycle transversal'' = cycle-balancing set of all-negative signed mixed graph.  Problem:  $\exists$ such set of size $\leq k$?
\dated{25 Mar 2022} 
\par
\codes{\CSD:~\CFr:~\CAlgor}%
\itemspace

\author{Kinkar Ch.\ Das [Kinkar Chandra Das]}				%DAS KC
\authorx{Kinkar Ch.\ Das}
\authorx{Kinkar Chandra Das}
\authorspace
\authref  See also \hyperlink{Jalal Askari}{J.~Askari} and \hyperlink{Santanu Mandal}{S.~Mandal}.
\authrefspace

\bsgitem{2010a}{}  On conjectures involving second largest signless Laplacian eigenvalue of graphs.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 11, 3018--3029.
\MR{2639266} (2011h:05151).  \Zbl{1195.05040}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011a}{}  Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2420--2424.
\MR{2811126} (2012m:05204).  \Zbl{1223.05171}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs.  \emph{Discrete Math.}\ 312 (2012), 992--998.
\MR{2872940}.  \Zbl{1237.05124}.  
\annot{} Assume $n\geq4$; $\lambda_1 =$ max eigenvalue.  Thm.\ 3.2:  $\lambda_1(-\Gamma) + \lambda_n(-\Gamma) \leq 3n-2-2\alpha(\Gamma)$, where $\alpha :=$ independence number; $=$ iff $\Gamma = K_{n-\alpha} \vee \bar K_\alpha$.
Thm.\ 3.3:  $\lambda_1(-\Gamma) - \lambda_n(-\Gamma) \geq 2 + 2\cos(/pi/n)$, with equality iff $\Gamma$ is a path or odd circle.
\dated{21 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Prabir Das}		 					%DAS P
\author{Prabir Das \& S.B.\ Rao}
\authorspace
\bsgitem{1983a}{}  Alternating eulerian trails with prescribed degrees in two edge-colored complete graphs.  \emph{Discrete Math.}\ 43 (1983), 9--20.  
\MR{0680299} (84k:05069).  \Zbl{494.05020}.					
\annot{} Given an all-negative bidirected $K_n$ and a positive integer $f_i = 2g_i$ for each vertex $v_i$.  There is a connected subgraph having in-degree and out-degree $= g_i$ at $v_i$ iff there is a $g$-factor of introverted and one of extroverted edges and the degrees satisfy a complicated degree condition.   Generalizes Thm.\ 1 of \xref{B\'ankfalvi and B\'ankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi}.  
[See \xref{Bang-Jensen and Gutin}{1997a}{Jorgen Bang-Jensen \& Gregory Gutin} for how to convert an edge 2-coloring to an orientation of an all-negative graph and for further developments on alternating walks.]
\codes{\Cpar:~\Cori}%
\itemspace

\author{Sandip Das}							%DAS S
\authorspace
\authref  See also \hyperlink{Julien Bensmail}{J.~Bensmail}.
\authrefspace

\authorx{Sandip Das, Prantar Ghosh, Swathy Prabhu, \& Sagnik Sen}
\author{Sandip Das, Prantar Ghosh, Swathy Prabhu [Swathyprabhu Mj], \& Sagnik Sen}
\authorspace
\bsgitem{2016a}{}  Relative clique number of planar signed graphs.  In:  Sathish Govindarajan and Anil Maheshwari, eds., \emph{Algorithms and Discrete Applied Mathematics} (Proc.\ 2nd Int.\ Conf., CALDAM 2016, Thiruvananthapuram, India, 2016), pp.\ 326--336.  \LNCS 9602, Springer, Cham, 2016.
\MR{3509769}.  \Zbl{1437.05090}.  
\codes{\CSG:~\CInvar}%
�, and �. Sopena. Homomorphisms of signed graphs. Journal of Graph Theory, 2014]. Thus, together with a result from [P. Ochem, A. Pinlou, and S. Sen. Homomorphisms of signed planar graphs. arXiv preprint , 2014.], the lower bound of 8 and upper bound of 40 has already been proved for the signed relative clique number of the family of planar graphs. Here we improve the upper bound to 15. Furthermore, we determine the exact values of signed relative clique number of the families of outerplanar graphs and triangle-free planar graphs.  For the entire collection see [Zbl 1330.68024].
\itemspace

\bsgitem{2020a}{}  Relative clique number of planar signed graphs.  \emph{Discrete Appl.\ Math.}\ 280 (2020), 86--92.
\MR{4096165}.  \Zbl{1439.05103}.  
\codes{\CSG}%
\itemspace

\author{Sandip Das, Soumen Nandi, Soumyajit Paul, \& Sagnik Sen}
\authorspace
\bsgitem{2016a}{}  Chromatic number of signed graphs with bounded maximum degree.  Manuscript, 2016.
\arXiv{1603.09557}.  
\annot{} \Cf\ \xref{Duffy, Jacques, Montassier, and Pinlou}{2009a}{Christopher Duffy, Fabien Jacques, Mickael Montassier, \& Alexandre Pinlou} and \xref{Jacques and Pinlou}{2022a}{Fabien Jacques \& Alexandre Pinlou}.
\codes{\CSG:~\CCol}%
\itemspace

\author{Sandip Das, Soumen Nandi, Sagnik Sen, \& Ritesh Seth}
\authorspace
\bsgitem{2019a}{}  The relative signed clique number of planar graphs is 8.  In:  S.P.\ Pal and A.\ Vijayakumar, eds., \emph{Algorithms and Discrete Applied Mathematics} (5th Int.\ Conf., CALDAM 2019, Kharagpur, India), pp. 245--253.  \LNCS 11394.  Springer, Cham, 2019. 
\MR{3911531}.  \Zbl{1444.05063}.  
\codes{\CSG}%
\itemspace

\authorx{Bhaskar DasGupta}						%DASGUPTA
\author{Bhaskar DasGupta, German Andres Enciso, Eduardo Sontag, \& Yi Zhang}
\authorspace
\bsgitem{2006a}{}  Algorithmic and complexity results for decompositions of biological networks into monotone subsystems.  
In: Carme \`Alvarez and Mar{\'\i}a Serna, eds., \emph{Experimental Algorithms} (5th Int.\ Workshop, WEA 2006, Cala Galdana, Menorca, 2006), pp.\ 253--264.  \LNCS 4007.  Springer, Berlin, 2006. 
\Zbl{196.92016}.  
\annot{} Extended abstract of \xrefd{2007a}{Bhaskar DasGupta, German Andres Enciso, Eduardo Sontag, \& Yi Zhang}.
\codes{\CSD(\Csg):~\CDyn,~\CAlgor,~\CBiol}%
\itemspace

\bsgitem{2007a}{}  Algorithmic and complexity results for decompositions of biological networks into monotone subsystems.  \emph{BioSystems} 90 (2007), no.\ 1, 161--178.
\annot{} ``\dots our problem amounts to finding ground states'' $V \to \signs$ of a signed graph.  
Lemma 3:  A dynamical system is monotone iff the associated signed graph is balanced.   
An algorithm to find $\#E-l(\Sigma)$ to within 7/8.
Dictionary:  ``sign-consistency'' = balance, ``consistent edge'' = satisfied edge (in a state).
\dated{1 Jan 2012} 
\codes{\CSD(\Csg):~\CDyn,~\CAlgor,~\CBiol}%
\itemspace

\author{Brian Davis}							%DAVIS B
\authorspace
\bsgitem{2019a}{}  Unlabeled signed graph coloring.  \emph{Rocky Mountain J.\ Math.} 49(4) (2019), 1111--1122.
\MR{3998912}.  \Zbl{1419.05097}.  \arXiv{1511.07730}.				
\par
\codes{\CSG:~\CCol:~\CInvar,~\Cmatrd,~\CGeom,~\COri}%
\itemspace

\author{James A.\ Davis} 						%DAVIS JA
\authorspace
\bsgitem{1963a}{} Structural balance, mechanical solidarity, and interpersonal relations.  \emph{Amer.\ J.\ Sociology} 68 (1963), 444--463.
Repr.\ with minor changes in:  Joseph Berger, Morris Zelditch, Jr., and Bo Anderson, eds., \emph{Sociological Theories in Progress, Vol.\ One}, Ch.\ 4, pp.\ 74--101.  Houghton Mifflin, Boston, 1966.  
Also reprinted in:  Samuel Leinhardt, ed., \emph{Social Networks: A Developing Paradigm}, pp.\ 199--217.  Academic Press, New York, 1977.  
\codes{\CPsS:~\CSG,~\CWG:~\CExp}%
\itemspace

\bsgitem{1967a}{} Clustering and structural balance in graphs.  \emph{Human Relations} 20 (1967), 181--187.  
Repr.\ in:  Samuel Leinhardt, ed., \emph{Social Networks: A Developing Paradigm}, pp.\ 27--33.  Academic Press, New York, 1977.
\annot{} $\Sigma$ is ``clusterable'' if its vertices can be partioned so that each positive edge is within a part and each negative edge joins different parts.  Thm.:  $\Sigma$ is clusterable $\iff$ no circle has exactly one negative edge.
	
[\Cf\ \xref{Cartwright and Harary}{1968a}{Dorwin Cartwright \& Frank Harary}.]  
[Further developed in \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}.]  
[Complete-graph clustering begins (?)\ in \xref{Zahn}{1964a}{C.T.\ Zahn, Jr.} and \xref{Moon}{1966a}{J.W.\ Moon}, now called ``cluster editing'' and focussed on algorithms; \cf, e.g., \xref{B\"ocker and Baumbach}{2013a}{Sebastian Bocker \& Jan Baumbach}.]
\dated{rev.\ 18 Nov 2017, 11 Jan 2019}
\codes{\CSG:~\CClu}%
\itemspace

\bsgitem{1979a}{}  The Davis/Holland/Leinhardt studies: An overview.  
In:  Paul W.\ Holland and Samuel Leinhardt, eds., \emph{Perspectives on Social Network Research} (Proc.\ Sympos., Dartmouth Coll.,  Hanover, N.H., 1975), 
Academic Press, New York, 1979.  
\annot{} Survey of triad analysis in signed complete digraphs; \cf\ e.g.\ \xref{Davis and Leinhardt}{1972a}{James A.\ Davis \& Samuel Leinhardt}, \xref{Wasserman and Faust}{1994a}{Stanley Wasserman \& Katherine Faust, eds.}.
	
\dated{28 Apr 2009}
\codes{\CPsS,~\CSD:~\CClu(\CGen):~\CExp}%
\itemspace

\author{James A.\ Davis \& Samuel Leinhardt} 
\authorspace
\bsgitem{1972a}{}  The structure of positive interpersonal relations in small groups.  
In:  Joseph Berger, Morris Zelditch, Jr., and Bo Anderson, eds., \emph{Sociological Theories in Progress, Vol.\ Two}, Ch.\ 10, pp.\ 218--251.  
Houghton Mifflin, Boston, 1972.  
\annot{} In ``ranked clusterability'' the vertices of a signed complete, symmetric digraph are divided into levels.  The set of levels is totally ordered.  A symmetric pair, $\{+vw,+wv\}$ or $\{-vw,-wv\}$, should be within a level.  For an asymmetric pair, $\{+vw,-wv\}$, $w$ should be at a higher level than $v$.  Analysis in relation to both randomly generated and observational data.  
	
\dated{28 Apr 2009}
\codes{\CPsS,~\CSD:~\CClu(\CGen)}%
\itemspace

\author{Robert Davis}						%DAVIS R
\authorspace
\authref  See \hyperlink{Tianran Chen}{T.-R.~Chen}.
\itemspace

\authorx{A.C.\ Day}							%DAY
\author{A.C.\ Day, R.B.\ Mallion, \& M.J.\ Rigby}
\authorspace
\bsgitem{1983a}{}  On the use of Riemannian surfaces in the graph-theoretical representation of M\"obius systems.  
In: R.B.\ King, ed., \emph{Chemical Applications of Topology and Graph Theory} (Proc.\ Sympos., Athens, Ga., 1983),  pp.\ 272--284.  Stud.\ Phys.\ Theor.\ Chem., Vol.\ 28.  Elsevier, Amsterdam, 1983.
\MR{0761923} (85h:05039).  
\annot{} A clumsy but intriguing way of representing some signed (or more generally, $\bbZ_n$-weighted) graphs: via $2$-page (or, $n$-page) looseleaf book embedding (all vertices are on the spine and each edge is in a single page), with an edge in page $k$ weighted by the ``sheet parity index'' $\alpha_k = (-1)^k$ (or, $e^{2\pi ik/n}$).  Described in the [unnecessary] terminology of an $n$-sheeted Riemann surface.  
[A $\bbZ_n$-weighted) graph has such a representation iff the subgraph of edges with each weight is outerplanar.]
\annotctd{} A variation to get switching classes of signed circles: replace $\alpha_k$ by the ``connectivity parity index'' $\alpha_k^{\sigma_k}$ where $\sigma_k =$ number of edges in page $k$.  [The variation is valid only for circles.]  
[\Questions\ vaguely suggested by these procedures:  Which signed graphs can be switched so that the edges of each sign form an outerplanar graph?  Also, the same for gain graphs.  And there are many similar questions: for instance, the same ones with ``outerplanar'' replaced by ``planar.'']
\par
\codes{\CSG:~\Csw,~\CAdj,~\CTop,~\CChem:~\CExp,~\CRef}\codesctd{\CWG:~\CAdj,~\CTop:~\CExp,~\CRef}%
\itemspace

\author{Rajat K.\ De}							%DE RK
\authorspace
\authref  See \hyperlink{Anindya Bhattacharya}{A.~Bhattacharya}.
\itemspace

\author{[Nair Maria Maia de Abreu]}					%DE ABREU
\authorspace
\authref  See \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~Abreu} (under `A').
\itemspace

\author{[Mathias Hudoba de Badyn]}					%DE BADYN
\authorspace
\authref  See \hyperlink{Mathias Hudoba de Badyn}{M.H.~de~Badyn} (under `B').
\itemspace

\author{Marisa Debowsky}						%DEBOWSKY
\authorspace
\authref  See \hyperlink{Dan Archdeacon}{D.~Archdeacon}.
\itemspace

\author{Ernesto Ded\`o}						%DEDO
\authorx{Ernesto Dedo}
\authorspace
\bsgitem{1981a}{}  Sulla ricostruibilit\`a del polinomio caratteristico del commutato di un grafo.  [The reconstructibility of the characteristic polynomial of the line-graph of a graph.]  \emph{ Boll.\ Un.\ Mat.\ Ital.\ A} (5) 18 (1981), no.\ 3, 423--429.
\MR{0633676} (82k:05078).  \Zbl{481.05050}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{C.\ De Dominicis}						%DE DOMINICIS
\authorspace
\authref  See \hyperlink{G.J.\ Rodgers}{G.J.~Rodgers}.
\itemspace

\author{Matthias Dehmer}						%DEHMER
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.H.~Yu}.
\itemspace

\authorx{Jos\'e F.\ De Jes\'us}						%DE JESUS	%JESUS
\authorx{Jose F.\ De Jesus}
\author{Jos\'e F.\ De Jes\'us \& Alexander Kelmans}
\authorx{Jose F.\ De Jesus \& Alexander Kelmans}
\authorspace
\bsgitem{2017a}{}   On graphs uniquely defined by their $K$-circular matroids.  \emph{ Discrete Appl.\ Math.}\ 217 (2017), part 3, 474--487.
\MR{3579926}.  \Zbl{1358.05047}.  \arXiv{1508.07627}.				
\codes{\CBic:~\CGen}%
\itemspace

\bsgitem{2017b}{}  $k$-circular matroids of graphs.  \emph{Discrete Appl.\ Math.}\ 225 (2017), 33--950.
\MR{3647486}.  \Zbl{1361.05024}.  \arXiv{1508.05364}.				
\codes{\CBic:~\CGen}%
\itemspace

\author{Hidde de Jong}						%DE JONG	%JONG
\authorspace
\bsgitem{2002a}{}  Modeling and simulation of genetic regulatory systems: A literature review.  \emph{J.\ Comput.\ Biol.}\ 9 (2002), no.\ 1, 67--103.
\MR{3931957}.  
\codes{\CSD:~\CDyn,~\CBiol:~\CExp}%
\itemspace

\authorx{Italo J.\ Dejter}						%DEJTER
\author{I.J.\ Dejter \& V.\ Neumann-Lara}
\authorspace
\bsgitem{1988a}{}  Unboundedness for generalized odd cyclic transversality.  In:  A.\ Hajnal, L.\ Lov\'asz and V.T.\ S\'os, eds., \emph{Combinatorics} (Seventh Hungarian Colloq., Eger, 1987), pp.\ 195--203.  Colloq.\ Math.\ Soc.\ J\'anos Bolyai, Vol.\ 52.  North-Holland, Amsterdam, 1988.
\MR{1221557} (94b:05117).  \Zbl{705.05045}.  
\annot{} Thm.\ 1:  Frustration number $l_0(-\Gamma)$ is unbounded for graphs with no disjoint odd circles.  Their examples are projective-planar antibalanced signed graphs.
Generalized to circles of length $L \mod N$ for many $L,N$.  
\dated{1 May 2017} 
\codes{\Csg:~\CPar:~\CFr,~\CCircles,~\CStr}%
\itemspace

\bsgitem{1991a}{}  Voltage graphs and Hamilton cycles.  In:  V.R.\ Kulli, ed., \emph{Advances in Graph Theory}, pp.\ 141--153.  Vishwa International Publications, Gulbarga, 1991.
\MR{1218635} (94a:05135).  \Zbl{817.05002} (book).  
\codes{\CGG:~\CAut}%
\itemspace

\author{[Pierre de la Harpe]}						%DE LA HARPE
\authorspace
\authref  See \hyperlink{Pierre de la Harpe}{P.~de~la~Harpe} (under `H').
\itemspace

\author{Anne Delandtsheer}						%DELANDTSHEER
\authorspace
\bsgitem{1995a}{}  Dimensional linear spaces.  
In: F.\ Buekenhout, ed., \emph{Handbook of Incidence Geometry: Buildings and Foundations}, Ch.\ 6,  pp.\ 193--294.  North-Holland, Amsterdam, 1995.  
\MR{1360721} (96k:51012).  \Zbl{950.23458}.
\annot{} ``Dimensional linear space'' (DLS) = simple matroid.  \S2.7: ``Dowling lattices,'' from \xref{Dowling}{1973b}{Thomas A.\ Dowling}.  \S6.7: ``Subgeometry-closed and hereditary classes of DLS's,'' from \xref{Kahn and Kung}{1982a}{Jeff Kahn \& Joseph P.S.\ Kung}.  In \S2.6, the ``Enough modular hyperplanes theorem'' from \xref{Kahn and Kung}{1986a}{Jeff Kahn \& Joseph P.S.\ Kung}.
\par
\codes{GG:~\CMatrd:~\CExp}%
\itemspace

\author{Patrick De Leenheer}						%DE LEENHEER
\authorspace
\authref  See also \hyperlink{David Angeli}{D.~Angeli} and \hyperlink{Vincent Antonio Traag}{V.A.~Traag}.
\authrefspace

\author{Patrick De Leenheer, David Angeli, \& Eduardo D.\ Sontag}
\authorspace
\bsgitem{2007a}{}  Monotone chemical reaction networks.  \emph{J.\ Math.\ Chem.}\ 41 (2007), no.\ 3, 295--314.
\MR{2343862} (2009c:92041).  \Zbl{1117.80309}.  
\codes{\CSD,~\CSG:~\CBal,~\CChem}%
\itemspace

\bsgitem{2009a}{}  Monotone chemical reaction networks.  (In Hungarian.)  \emph{Alkalmaz.\ Mat.\ Lapok} 26 (2009), no.\ 2, 381--402.
\MR{2583205} (no rev).  
\annot{} Transl.\ of \xrefd{2007a}{Patrick De Leenheer, David Angeli, \& Eduardo D.\ Sontag} by Judit V\'ardai.
\codes{\CSD,~\CSG:~\CBal,~\CChem}%
\itemspace

\author{[Gloria de Leon-Calio]}						%DE LEON-CALIO
\authorspace
\authref  See \hyperlink{Gloria de Leon-Calio}{G.~de~Leon-Calio} (under `L').
\itemspace

\author{John G.\ del Greco}						%DEL GRECO
\authorspace
\authref  See also \hyperlink{Collette R.\ Coullard}{C.R.\ Coullard}.
\authrefspace

\bsgitem{1992a}{}  Characterizing bias matroids.  \emph{Discrete Math.}\ 103 (1992), 153--159.  
\MR{1171312} (93m:05050).  \Zbl{753.05021}.					
\annot{} How to decide, given a matroid $M$ and a biased graph $\Omega$, whether $M = \bfF(\Omega)$.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Leonardo Silva de Lima}					%DE LIMA
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Aroldo Oliveira}{A.~Oliveira}, and \hyperlink{Carla Silva Oliveira}{C.S.~Oliveira}.
\authrefspace

\author{Leonardo de Lima, Vladimir Nikiforov, \& Carla Oliveira}
\authorspace
\bsgitem{2016a}{}  The clique number and the smallest $Q$-eigenvalue of graphs.  \emph{Discrete Math.}\ 339 (2016), 1744--1752.
\MR{3477106}.  \Zbl{1333.05192}.  \arXiv{1508.01784}.				
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Leonardo Silva de Lima, Carla Silva Oliveira, Nair Maria Maia de Abreu, \& Vladimir Nikiforov}
\authorspace
\bsgitem{2011a}{}  The smallest eigenvalue of the signless Laplacian.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2570--2584.
\MR{2811139} (2012g:05140).  \Zbl{1222.05180}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Prem Delphy P}						%DELPHY
\author{Prem Delphy P \& Devaraj J}
\authorspace
\bsgitem{2011a}{}  On signed cordial graph.  \emph{Int.\ J.\ Math.\ Sci.\ Appl.}\ 1 (2011), no.\ 3, 1159--1167.
\MR{2844560} (n rev).  \Zbl{1266.05049}.  
\annot{} $\Gamma$ is ``signed cordial'' if $\exists$ edge signature $\sigma$ such that $\#E^+ \approx \#E^-$ and $\mu_\sigma\inv(+1) \approx \mu_\sigma\inv(-1)$.  Examples.  
[More in \xref{Cynthia and Padmavathy}{2021a}{V.\ Jude Annie Cynthia \& E.\ Padmavathy}.]
\annotctd{} [\xref{Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan} define ``signed product cordiality'' based on vertex signs.]
\dated{26 Sept 2022} 
\codes{\CLab:~\CSG}%
\itemspace

\authorx{Alberto Del Pia}						%DEL PIA
\author{Alberto Del Pia \& Giacomo Zambelli}
\authorspace
\bsgitem{2009a}{}  Half-integral vertex covers on bipartite bidirected graphs: total dual integrality and cut-rank.  \emph{SIAM J.\ Discrete Math.}\ 23 (2009), no.\ 3, 1281--1296.
\MR{2538651} (2011b:05200).  \Zbl{1227.05209}.  
\annot{} 
Dictionary:  ``Bipartite'' = balanced.
\codes{\Csg:~\COri:~\CInc,~\CAlgor}%
\itemspace

\author{Alberto Del Pia, Antoine Musitelli, \& Giacomo Zambelli}
\authorspace
\bsgitem{2018a}{}  On matrices with the Edmonds--Johnson property arising from bidirected graphs.  \emph{J.\ Combin.\ Theory Ser. B} 130 (2018), 49--91.
\MR{3772734}.  \Zbl{1384.05107}.  
\codes{\Csg:~\COri:~\CInc,~\CAlgor}
\itemspace

\author{Ernesto W.\ De Luca}						%DE LUCA
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Emanuele Delucchi}						%DELUCCHI
\authorspace
\bsgitem{2007a}{}  Nested set complexes of Dowling lattices and complexes of Dowling trees.  \emph{J.\ Algebraic Combin.}\ 26 (2007), no.\ 4, 477--494.
\MR{2341861} (2008i:05190).  \Zbl{1127.05107}.					
\annot{} Studies Dowling trees (\cf\ \xref{Hultman}{2007a}{Axel Hultman}).
	
\codes{\Cgg:~\CMatrd:~\CInvar}
\itemspace

\author{Emanuele Delucchi, Noriane Girard, \& Giovanni Paolini}
\authorspace
\bsgitem{2019a}{}  Shellability of posets of labeled partitions and arrangements defined by root systems.  \emph{Electronic J.\ Combin.}\ 26 (2019), no.\ 4, art.\ 4.14, 22 pp. 
\MR{4025418}.  \Zbl{1422.05108}.  \arXiv{1706.06360}.					
\annot{} \Cf\ \xref{Bibby and Gadish}{2018a}{Christin Bibby \& Nir Gadish}.
	
\codes{\Cgg:~\CMatrd(\CGen)}%
\itemspace

\author{Renata Raposo Del-Vecchio}					%DEL-VECCHIO
\authorx{Renata R.\ Del-Vecchio}
\authorspace
\authref  See \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de Freitas} and \hyperlink{Rafael Esteves Mansano}{R.E.~Mansano}.
\itemspace

\authorx{Erik D.\ Demaine}						%DEMAINE
\author{Erik D.\ Demaine, Dotan Emanuel, Amos Fiat, \& Nicole Immorlica}
\authorspace
\bsgitem{2006a}{}  Correlation clustering in general weighted graphs.  Approximation and Online Algorithms.  \emph{Theoretical Computer Sci.}\ 361 (2006), no.\ 2-3, 172--187.  
\MR{2252576} (2008e:68157).  \Zbl{1099.68074}.					
\annot{} Clustering in a weighted signed graph; \cf\ \xref{Bansal, Blum, and Chawla}{2002a}{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}, \xrefd{2004a}{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}.  
An $O(\log n)$-approximation algorithm based on linear-program\-ming rounding and region growing.  
An $O(r^3)$-approximation algorithm for graphs without a $K_{r, r}$-minor [e.g., planar, if $r=3$].  Equivalent to minimum multicut, hence hard to approximate better than $\Theta(\log n)$. 
	
\dated{13 Sept 2009} 
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}%
\itemspace

\author{Erik D.\ Demaine, MohammadTaghi Hajiaghayi, \& Ken-ichi Kawarabayashi}
\authorspace
\bsgitem{2010a}{}  Decomposition, approximation, and coloring of odd-minor-free graphs.  
In:  Moses Charikar, ed., \emph{Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '10, Austin, Tex., 2010), pp.\ 329--344.  Society for Industrial and Appl.\ Math.\, Philadelphia, 2010.
\MR{2809679} (2012i:05276).  \Zbl{1288.05053}.  
\annot{} Graphs excluding $-\Gamma$: structure and algorithms.  Dictionary: ``$\Gamma$ has odd minor $H$'' = $-\Gamma$ has minor $-H$.
\dated{4 Feb 2021} 
\par
\codes{\Csg:~\CPar:~\CStr,~\CAlgor}%
\itemspace

\author{Erik Demaine \& Nicole Immorlica}
\authorspace
\bsgitem{2003a}{}  Correlation clustering with partial information. 
In: \emph{Proceedings of the 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM-APPROX 2003)} (Princeton, N.J., 2003), pp.\ 1--13.  \LNCS 2764.  Springer, Berlin, 2003.
\MR{2080776} (2005c:68291).  \Zbl{1202.68479}.					
\annot{} Conference version of \xref{Demaine, Emanuel, Fiat, and Immorlica}{2006a}{Erik D.\ Demaine, Dotan Emanuel, Amos Fiat, \& Nicole Immorlica}.
\dated{13 Sept 2009} 
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}%
\itemspace

\author{Jacques Demongeot}						%DEMONGEOT
\authorspace
\authref  See also \hyperlink{Julio Aracena}{J.~Aracena}, \hyperlink{Olivier Cinquin}{O.~Cinquin}, \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, and \hyperlink{Loic Forest}{L.~Forest}.
\authrefspace

\author{Jacques Demongeot, Julio Aracena, Samia Ben Lamine, Sylvain Meignen, Arnaud Tonnelier, \& Rene Thomas}
\authorx{Jacques Demongeot, Julio Aracena, Samia Ben Lamine, Sylvain Meignen, Arnaud Tonnelier, \& Ren\'e Thomas}
\authorspace
\bsgitem{2001a}{}  Dynamical systems and biological regulations.  In:  Eric Goles and Servet Mart\'{\i}nez, eds., \emph{Complex Systems}, pp.\ 105--149.  Nonlinear Phenomena and Complex Systems, Vol.\ 6.  Kluwer, 2001.
\MR{1886355} (2003b:37040).  \Zbl{1333.37011}.  
\annot{} Part I:  Various definitions of an attractor tend not to be logically comparable.  
Part II, ``Biological regulations'':  \S7.1, ``Positive regulation circuits and memory''.  \S8, ``Interaction matrices and ubiquitory genes''.
\codes{\Csd:~\CAdj:~\CDyn:~\CExp}%
\itemspace

\author{Jacques Demongeot, Julio Aracena, Florence Thuderoz, Thierry-Pascal Baum, \& Olivier Cohen}
\authorspace
\bsgitem{2003a}{}  Genetic regulation networks: circuits, regulons and attractors.  R\'eseaux de r\'egulation g\'en\'etique : circuits, r\'egulons, attracteurs.  \emph{C.R.\ Biologies} 326 (2003), 171--188.
\codes{\CSD,~\CBiol:~\CDyn}%
\itemspace

\author{Jacques Demongeot, Adrien Elena, Mathilde Noual, Sylvain Sen\'e, \& Florence Thuderoz}
\authorx{Jacques Demongeot, Adrien Elena, Mathilde Noual, Sylvain Sene, \& Florence Thuderoz}
\authorspace
\bsgitem{2011a}{}  ``Immunetworks'', intersecting circuits and dynamics.  \emph{J.\ Theor.\ Biol.}\ 280 (2011), 19--33.
\MR{2975039} (no rev).  \Zbl{1397.92249}.  \HAL{hal-00666298}.  
\par
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Jacques Demongeot, Marcelle Kaufman, \& Ren\'e Thomas}
\authorx{Jacques Demongeot, Marcelle Kaufman, \& Rene Thomas}
\authorspace
\bsgitem{2000a}{}  Positive feedback circuits and memory.  \emph{C.R.\ Acad.\ Sci.\ Paris, Sci.\ vie/Life Sci.}\ 323 (2000), 69--79.
\codes{\CSD:~\CDyn}%
\itemspace

\author{Jacques Demongeot, Mathilde Noual, \& Sylvain Sen\'e}
\authorx{Jacques Demongeot, Mathilde Noual, \& Sylvain Sene}
\authorspace
\bsgitem{2012a}{}  Combinatorics of Boolean automata circuits dynamics.  \emph{Discrete Appl.\ Math.}\ 160 (2012), 398--415.
\MR{2876323}.  \Zbl{1238.37035}.  
\annot{} The effect of cycles and intersecting cycles (``circuits'') in signed digraphs representing the action of a Boolean automaton.  [``Automata'' should be ``automaton''.]  
	
Dictionary:  ``Boolean automata circuit'' = digraph that is a signed cycle.  ``Double Boolean automata circuit'' = digraph that is two signed cycles with one common vertex.
\dated{16 Jan 2015} 
\codes{\CSD:~\CDyn}%
\itemspace

\author{Jacques Demongeot \& Ren\'e Thomas}
\authorx{Jacques Demongeot \& Rene Thomas}
\authorspace
\bsgitem{1999a}{}  Positive regulation circuits and memory.  Circuits de r\'egulation positifs et m\'emoire.  In:  \emph{Neuronal Information Processing: From Biological Data to Modelling and Applications}, pp.\ 148--163.  1999.
\codes{\CSD:~\CDyn}%
\itemspace

\author{Jacques Demongeot, Ren\'e Thomas, \& Michel Thellier}
\authorx{Jacques Demongeot, Rene Thomas, \& Michel Thellier}
\authorspace
\bsgitem{2000a}{}  A mathematical model for storage and recall functions in plants.  \emph{C.R.\ Acad.\ Sci.\ Paris, Sci.\ vie/Life Sci.}\ 323 (2000), 93--97.
\annot{} \S2, ``The logical approach'':  Exposition of signed-digraph model of bioregulation.
\dated{23 Aug 2017} 
\codes{\CSD:~\CExp,~\CBiol}%
\itemspace

\authorx{Hanyuan Deng}						%DENG HY
\author{Hanyuan Deng \& He Huang}
\authorspace
\bsgitem{2013a}{}  On the main signless Laplacian eigenvalues of a graph.  \emph{Electronic J.\ Linear Algebra} 26 (2013), art.\ 25, 381--393.
\MR{3084649}.  \Zbl{1282.05109}.  \arXiv{1208.5835}.				
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Hongzhong Deng}						%DENG HZ
\author{Hongzhong Deng \& Peter Abell}
\authorspace
\bsgitem{2010a}{}  A study of local sign change adjustment in balancing structures.  \emph{J.\ Math.\ Sociology} 34 (2010), no.\ 4, 253--282.
\Zbl{1201.91166}.  
\annot{} A random signed graph has edge $vw$ with probability $d$, which is positive with probability $\alpha_0$.  Degree of balance is the proportion of triangles that are positive.  A triangle of type $T_i$ has $i$ positive edges.  They study the long-term proportions of triangle types in examples.
\S3, ``Balance adjustment under a local rule'':  A triangle $\triangle uvw$ and edge $uv$ are chosen at random; $uv$ changes sign iff $\triangle uvw$ is negative.  This ``myopic adjustment rule'' is iterated.  For $0<\alpha_0<1$, the proportions approach $37\
\S\S4--7:  The sign-change probability depends on the triangle type.  Probabilities are suggested by models of Harary--Cartwright, \xref{Davis}{1967a}{James A.\ Davis}, and others, in which different sets of triangle types are ``attractors''.  Analytical and example results are reported.
\annotctd{} Model based on \xref{Antal, Krapivsky, and Redner}{2006a}{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner}.  Successor to \xref{Abell and Ludwig}{2009a}{Peter Abell \& Mark Ludwig} and \xref{Kujawski, Abell, and Ludwig}{2010a}{Bernard Kujawski, Mark Ludwig, \& Peter Abell}.  
[See \xref{Barahona, Maynard, Rammal, and Uhry}{1982a}{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry} for modelling of planar grid graphs.]
\dated{6 Dec 2009} 
\codes{\CSG:~\CBal}%
\itemspace

\author{Hongzhong Deng, Peter Abell, Ji Li, \& Jun Wu}	
\authorspace
\bsgitem{2012a}{}  A study of sign adjustment in weighted signed networks. \emph{Social Networks} 34 (2012), no.\ 2, 253--263.
\codes{\CSG:~\CWG,~\CPsS}%
\itemspace

\author{Hongzhong Deng, Peter Abell, Jun Wu, \& Yuejin Tang}
\authorspace
\bsgitem{2016a}{}  The influence of structural balance and homophily/heterophobia on the adjustment of random complete signed networks.  \emph{Social Networks} 44 (2016), 190--201.
\codes{\CSG:~\CBal,~\CPsS:~\CKG}%
\itemspace

\author{Tristan Denley}						%DENLEY
\authorspace
\bsgitem{1997a}{}  The odd girth of the generalised Kneser graph.  \emph{European J.\ Combin.}\ 18 (1997), 607--611.
\MR{1468332} (98g:05080).  \Zbl{908.05059}.  
\annot{} ``Odd girth'' of $\Gamma$ = negative girth of $-\Gamma$.  [\Question.  Is there a generalization to (some, or many) signed Kneser and generalized Kenser graphs?  The high symmetry of the graph suggests this.]
\dated{19 Aug 2022} 
\codes{\Csg:~\Cpar:~\CCircles}%
\itemspace

\author{[Wouter de Nooy]}						%DE NOOY
\authorspace
\authref  See \hyperlink{Wouter de Nooy}{W.~de~Nooy} (under `N').
\itemspace

\author{[Arnout van de Rijt]}						%DE RIJT
\authorspace
\authref   See \hyperlink{Arnout van de Rijt}{A.~van~de~Rijt} (under `V').
\itemspace

\authorx{B.\ Derrida}							%DERRIDA
\author{B.\ Derrida, Y.\ Pomeau, G.\ Toulouse, \& J.\ Vannimenus}
\authorspace
\bsgitem{1979a}{}  Fully frustrated simple cubic lattices and the overblocking effect.  \emph{J.\ Physique} 40 (1979), 617--626.  
\annot{} Physics of the signed $d$-hypercube in which every plaquette is negative; specifically, [cleverly] construct $\Sigma_d = (Q_d,\sigma_d)$, $d>0$, as $\Sigma_{d-1} \times (+Q_1)$ with the second copy of $\Sigma_{d-1}$ negated.  Invariants of physical interest are computed and compared to the balanced case.	
	
Dictionary:  ``plaquette'' = square.
\codes{\CPhys:~\CSG}%
\itemspace

\bsgitem{1980a}{}  Fully frustrated simple cubic lattices and phase transitions.  \emph{J.\ Physique} 41 (1980), 213--221.  
\MR{0566063} (80m:82020).  
\codes{\CPhys:~\CSG}%
\itemspace

\authorx{Madhav Desai}						%DESAI M
\author{Madhav Desai \& Vasant Rao}
\authorspace
\bsgitem{1994a}{}  A characterization of the smallest eigenvalue of a graph.  \emph{J.\ Graph Theory} 18 (1994), no.\ 2, 181--194.
\MR{1258251} (95c:05084).  \Zbl{792.05096}.					
\annot{} $\psi(\Gamma) := \min_S (l(-\Gamma{:}S)+\#E(S,S^c)/\#S)$, over $\eset \subset S \subseteq V$, is a measure of nonbipartiteness of $\Gamma$.  $\mu_1 :=$ smallest eigenvalue of $\Lap(-\Gamma)$ satisfies $\psi(\Gamma)^2/4\Delta(\Gamma) \leq \mu_1 \leq 4\psi(\Gamma)$.  Their $e_{\min}(\Gamma) := l(-\Gamma)$.  
[See \xref{Fan and Fallat}{2012a}{Yi-Zheng Fan \& Shaun Fallat} for another eigenvalue connection with $l(-\Gamma)$.]
\dated{19 Sept 2010, 29 Dec 2012} 
\codes{\CPar:~\CEig,~\CFr}%
\itemspace

\author{Emilio De Santis}						%SANTIS	%DE SANTIS
\authorspace
\authref  See also \hyperlink{Federico Camia}{F.~Camia}.
\authrefspace

\bsgitem{2001a}{}  Strict inequality for phase transition between ferromagnetic and frustrated systems.  \emph{Electronic J.\ Probab.}\ 6 (2001), Paper no.\ 6, 1--27.
\MR{1825713} (2002c:82026).  \Zbl{1050.82020}.  
\codes{\CPhys,~\CSG:~\CRand}%
\itemspace

\author{E.\ De Santis \& A.\ Gandolfi}
\authorspace
\bsgitem{1999a}{}  Bond percolation in frustrated systems.  \emph{Ann.\ Probab.}\ 27 (1999), no.\ 4, 1781--1808.
\MR{1742888} (2000k:60199).  \Zbl{0968.60092}.  
\codes{\CSG:~\CPhys:~\CFr}%
\itemspace

\author{[L.\ de S\`eze]} 						%DE SEZE
\authorspace
\authref  See \hyperlink{L.\ de Seze}{L.~de~S\`eze} (under `S').   
\itemspace

\authorx{C.\ De Simone}						%DE SIMONE
\author{C.\ De Simone, M.\ Diehl, M.\ J\"unger, P.\ M\"utzel, G.\ Reinelt, \& G\ Rinaldi}
\authorx{C.\ De Simone, M.\ Diehl, M.\ Junger, P.\ Mutzel, G.\ Reinelt, \& G\ Rinaldi}
\authorspace
\bsgitem{1995a}{}  Exact ground states of Ising spin glasses:  New experimental results with a branch and cut algorithm.  \emph{J.\ Stat.\ Phys.}\ 80 (1995), 487--496.
\Zbl{1106.82323}.  
\annot{} Improves the algorithm of \xref{Barahona, Gr\"otschel, J\"unger,  and Reinelt}{1988a}{Francisco Barahona, Martin Grotschel, Michael Junger, \& Gerhard Reinelt} to find a switching with minimum $\#E^-$ ($= l(\Sigma)$) for signed toroidal square lattice graphs with an extra vertex (exterior magnetic field) and a fixed proportion of negative edges.  Applied to many signatures in order to find statistical properties.  Continued in \xrefd{1996a}{C.\ De Simone, M.\ Diehl, M.\ Junger, P.\ Mutzel, G.\ Reinelt, \& G\ Rinaldi}.
\dated{18 Aug 2012} 
\codes{\CPhys,~\CSG:~\CState(\Cfr):~\CAlgor}%
\itemspace

\bsgitem{1996a}{}  Exact ground states of two-dimensional $\pm J$ Ising spin glasses.  \emph{J.\ Stat.\ Phys.}\ 84 (1996), 1363--1371.
\annot{} Continuation of \xrefd{1995a}{C.\ De Simone, M.\ Diehl, M.\ Junger, P.\ Mutzel, G.\ Reinelt, \& G\ Rinaldi}.
\dated{18 Aug 2012} 
\par
\codes{\CPhys,~\CSG:~\CState(\Cfr):~\CAlgor}%
\itemspace

\authorx{A.H.\ Deutz}						%DEUTZ
\author{A.H.\ Deutz, A.\ Ehrenfeucht, \& G.\ Rozenberg}
\authorspace
\bsgitem{1994a}{}  Hyperedge channels are abelian.  \emph{Theor.\ Computer Sci.}\ 127 (1994), 367--393.
\MR{1275824} (96b:68023).  \Zbl{824.68011}.					
\codes{\CGH}
\itemspace

\author{Devaraj J}							%DEVARAJ	
\authorspace
\authref  See \hyperlink{Prem Delphy P}{P.~Delphy}.
\itemspace

\author{Lee DeVille}							%DEVILLE
\authorspace
\authref  See \hyperlink{Jared C.\ Bronski \& Lee DeVille}{J.C.~Bronski}.
\itemspace

\authorx{Vincent Devloo}						%DEVLOO
\author{Vincent Devloo, Pierre Hansen, \& Martine Labb\'e}
\authorspace
\bsgitem{2003a}{}  Identification of all steady states in large networks by logical analysis.  \emph{Bull.\ Math.\ Biol.}\ 65 (2003), 1025--1051.
\codes{\CSD:~\CDyn}%
\itemspace

\author{Matt DeVos}							%DEVOS
\authorspace
\authref  See also \hyperlink{Rong Chen}{R.~Chen}.
\authrefspace

\bsgitem{2000a}{}  \emph{Flows on Graphs}.  Doctoral thesis, Princeton Univ., 2000.
\annot{} See \xrefd{2004a}{Matt DeVos}, \xrefd{20xxa}{Matt DeVos}.
\dated{23 March 2010} 
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\bsgitem{2004a}{}  Flows on bidirected graphs.  Manuscript, 2004.
\annot{} See \xrefd{20xxa}{Matt DeVos}.  Proves, maybe, that a nowhere-zero 12-flow exists (if a nowhere-zero flow exists).  Corrected and extended in \xref{Raspaud and Zhu}{2011a}{Andre Raspaud \& Xuding Zhu} (\emph{q.v.}).
\dated{23 March 2010} 
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\bsgitem{20xxa}{}  Flows on bidirected graphs.  Submitted.  %
\arXiv{2013.8406}							
\annot{} A nowhere-zero 12-flow exists if any nowhere-zero flow exists.
\dated{18 Aug 2016} 
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\author{Matt DeVos \& Daryl Funk}
\authorspace
\bsgitem{2018a}{}  Almost balanced biased graph representations of frame matroids.  \emph{Adv.\ Appl.\ Math.}\ 96 (2018), 139--175.
\MR{3767506}.  \Zbl{1383.05218}.  \arXiv{1606.07370}.				
\par
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Matt DeVos, Daryl Funk, \& Irene Pivotto}
\authorspace
\bsgitem{2014a}{}  When does a biased graph come from a group labelling?  \emph{Adv.\ Appl.\ Math.}\ 61 (2014), 1--18.
\MR{3267062}.  \Zbl{1371.05113}.  \arXiv{1403.7667}.				
\codes{\CGG}%
\itemspace

\bsgitem{2017a}{}  On excluded minors of connectivity $2$ for the class of frame matroids.  \emph{European J.\ Combin.}\ 61 (2017), 167--196.
\MR{3588716}.  \Zbl{1352.05099}.  \arXiv{1502.06896}.				
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Matt DeVos, Jiaao Li, You Lu, Rong Luo, Cun-Quan Zhang, \& Zhang Zhang}
\authorspace
\bsgitem{2021a}{}  Flows on flow-admissible signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 149 (2021), 198--221.
\MR{4247027}.  \Zbl{1466.05085}.  \arXiv{1908.10853}.  
\annot{} All such graphs admit a nowhere-zero 11-flow.  \Cf\ \xref{Bouchet}{1983a}{Andre Bouchet}.
\dated{6 Jul 2022} 
\codes{\CSG:~\CFlows}%
\itemspace

\author{Matt DeVos, Edita Rollov\'a, \& Robert \v{S}\'amal}
\authorx{Matt DeVos, Edita Rollova, \& Robert Samal}
\authorspace
\bsgitem{2019a}{\dag}   A note on counting flows in signed graphs.  \emph{Electronic J.\ Combin.}\ 26 (2019), no.\ 2, art.\ P2.38, 7 pp.
\MR{3962234}.  \Zbl{1441.05096}.  \arXiv{1701.07369}.				
\annot{} For each $k\geq0$, the number of nowhere-zero flows in an abelian group of order $m$ that has $k$ factors $\bbZ_2$ is a polynomial function $f_k(m/2^k)$.  (Generalizes $k=0$ by \xref{Beck and Zaslavsky}{2006b}{Matthias Beck \& Thomas Zaslavsky}.)  [Further theory in \xref{Ren and Qian}{2019a}{Xiangyu Ren \& Jianguo Qian}.]
\dated{14 Aug 2017}
\codes{\CSG:~\CFlows:~\CInvar}%
\itemspace

\author{Sean Dewar}							%DEWAR
\authorspace
\bsgitem{2021a}{}  Flexible placements of periodic graphs in the plane.  \emph{Discrete Comput.\ Geom.}\ 66 (2021), 1286--1329.
\MR{4333293}.  \Zbl{1478.05032}.  \arXiv{1911.05634}.  
\codes{\CGG:~\CGeom,~\CCov,~\CSw}%
\itemspace

\authorx{Michel Marie Deza}						%DEZA*
\author{M.\ Deza, V.P.\ Grishukhin, \& M.\ Laurent}
\authorspace
\bsgitem{1991a}{}  The symmetries of the cut polytope and of some relatives.  
In:  Peter Gritzman and Bernd Sturmfels, eds., \emph{Applied Geometry and Discrete Mathematics:  The Victor Klee Festschrift}, pp.\ 205--220.  DIMACS Ser.\ Discrete Math.\ Theor.\ Comput.\ Sci., Vol.\ 4.  American Math.\ Soc., Providence, R.I., 1991.
\MR{1116350} (92e:52019).  \Zbl{748.05061}.  
\annot{} Switching (on coordinates) is an important symmetry of the cut polytope $P_n$ (of $K_n$); see p.\ 206.  [See \xref{Deza and Laurent}{1997a}{Michel Marie Deza \& Monique Laurent}.]  Thm.\ 2.6:  $\Aut P_n = \fD_n$, the Weyl group [$ = \SwAut (\pm K_n)$, the switching automorphism group].  
\Question\ (p.\ 207):  For the cut polytope $P_c(\Gamma)$, does $\Aut P_c(\Gamma) = \SwAut (\pm \Gamma)$?  [Edge signs and $\SwAut$ are not stated as such.]
\dated{12 Jun 2012} 
\codes{\Csg:~\Cpar:~\CKG:~\CGeom,~\Csw}%
\itemspace

\author{Michel Marie Deza \& Monique Laurent}
\authorspace
\bsgitem{1997a}{}  \emph{Geometry of Cuts and Metrics}.  Algorithms and Combin., Vol.\ 15.  Springer, Berlin, 1997.
\MR{1460488} (98g:52001).  \Zbl{885.52001}.					
\annot{} A main object of interest is the cut polytope, which is the bipartite subgraph polytope (see \xref{Barahona, Gr\"otschel, and Mahjoub}{1985a}{Francisco Barahona, Martin Grotschel, \& Ali Ridha Mahjoub}) of $K_n$, i.e., the balanced subgraph polytope (\xref{Poljak and Turz\'{\i}k}{1987a}{Svatopluk Poljak \& Daniel Turzik}) of $-K_n$.  \S4.5, ``An application to statistical physics'', briefly discusses the spin glass application.  
\S26.3, ``The switching operation'', discusses graph switching and its generalization to sets.  \S30.3, ``Circulant inequalities'', mentions \xref{Poljak and Turz\'{\i}k}{1987a}{Svatopluk Poljak \& Daniel Turzik}, \xrefd{1992a}{Svatopluk Poljak \& Daniel Turzik}.  No explicit mention of signed graphs.
\codes{\Csg:~\Cpar:~\CKG:~\Cfr,~\Csw~\CGeom:~\CExp}%
\itemspace

\author{Ayushi Dhama}						%DHAMA
\authorspace
\authref  See also \hyperlink{Deepa Sinha}{D.~Sinha}.
\authrefspace

\bsgitem{2013a}{}  \emph{Contributions to the Theory of Signed Graphs}.  Doctoral thesis, Banasthali University, 2013.
\codes{\CSG}%
\itemspace

\author{Dhananjayamurthy B.}						%DHANANJAYAMURTHY
\authorspace
\authref  See \hyperlink{Sophia Shalini G.B.}{S.~Shalini}.
\itemspace

\author{Inderjit S.\ Dhillon}						%DHILLON
\authorspace
\authref  See \hyperlink{Cho-Jui Hsieh}{C.-J.~Hsieh} and \hyperlink{Kai-Yang Chiang}{K.-Y.~Chiang}.
\itemspace

\author{Massimiliano Di Ventra}						%DI VENTRA  %VENTRA
\authorspace
\authref  See \hyperlink{Haik Manukian}{H.~Manukian} and \hyperlink{Yan Ru Pei}{Y.R.~Pei}.
\itemspace

\author{[F.\ di Liberto]}						%DI LIBERTO
\authorspace
\authref  See \hyperlink{F.\ di Liberto}{F.~di~Liberto} (under ``L'').
\itemspace

\author{Persi Diaconis}						%DIACONIS
\authorspace
\authref  See \hyperlink{Kenneth S.\ Brown}{K.S.~Brown}.
\itemspace

\authorx{Yuanan Diao}						%DIAO
\author{Yuanan Diao \& G\'abor Hetyei}
\authorx{Yuanan Diao \& Gabor Hetyei}
\authorspace
\bsgitem{2010a}{}  Relative Tutte polynomials for coloured graphs and virtual knot theory.  \emph{Combin.\ Probab.\ Computing} 19 (2010), no.\ 3, 343--369.
\MR{2607372} (2011f:05146).  \Zbl{1202.05064}.  
\codes{\CSGc:~\CInvar,~\CKnot}%
\itemspace

\author{Yuanan Diao, G\'abor Hetyei, \& Kenneth Hinson}
\authorx{Yuanan Diao, Gabor Hetyei, \& Kenneth Hinson}
\authorspace
\bsgitem{2009a}{}  Tutte polynomials of tensor products of signed graphs and their applications in knot theory.  \emph{J.\ Knot Theory Ramifications} 18 (2009), no.\ 5, 561--589.
\MR{2527677} (2010c:57010).  \Zbl{1185.05083}.  \arXiv{math/0702328}.			
\par
\codes{\CSGc:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{2011a}{}  A Tutte-style proof of Brylawski's tensor product formula.  \emph{European J.\ Combin.}\ 32 (2011), 775--781.
\MR{2821550}.  \Zbl{1229.05067}.  
\codes{\CSGc:~\CInvar}%
\itemspace

\authorx{Alicia Dickenstein}						%DICKENSTEIN
\author{Alicia Dickenstein \& Mercedes P\'erez Mill\'an}
\authorx{Alicia Dickenstein \& Mercedes Perez Millan}
\authorspace
\bsgitem{2011a}{\dag}  How far is complex balancing from detailed balancing?  \emph{Bull.\ Math.\ Biol.}\  73 (2011), 811--828.
\MR{2785146} (2012d:92019).  \Zbl{1214.92036}.  
\annot{} From a multiplicative gain digraph $\vec\Phi := (\vec\Gamma,\vec\phi,\fG)$ where $\vec\Gamma$ is a symmetric digraph, construct a gain graph $\Phi := (\Gamma,\phi,\fG)$ and $\fG=\bbR^\times_{>0}$, where $\Gamma$ has an edge $e_{ij}$ for each arc pair $(i,j),(j,i)$ and $\phi_{ij}(e_{ij}) := \vec\phi(i,j)/\vec\phi(j,i)$.  If $\Phi$ is balanced, $\vec\Phi$ is called ``formally balanced''.  This property and related ones are studied.  [The construction $\vec\Phi\mapsto\Phi$ is known in papers on chemical reaction graphs.  This paper is more gain-graphic than most though the gain graph $\Phi$ is not explicit. ]
	
\annotctd{} [A circle $C=e_{12}\cdots e_{l-1,l}e_{l1}$ is balanced iff $\vec\phi(\vec C)=\vec\phi(\vec C\inv)$, where $\vec C = \vec e_{12}\cdots \vec e_{l-1,l}\vec e_{l1}$.  \Question. What is the general theory of gain graphs derived from $\vec\Phi$ of the above type with a general abelian gain group?  In general define gains on a symmetric digraph: let $\vec\Gamma$ have arcs $\vec e_{ij}$, possibly with multiple arcs and loops, with a pairing ${}^*: \vec E \leftrightarrow \vec E$ such that $\vec e_{ij}{}^*$ is an $\vec e_{ji}$.  Example 1:  For a gain graph, let $\vec E := \bigcup \{\vec e_{ij},\ \vec e_{ji}=\vec e_{ij}^{\;-1} : e_{ij} \in E\}$ and $\vec\phi(\vec e_{ij}) := \phi_{ij}(e_{ij})$; then each (nonloop) pair is a balanced digon.  Example 2:  Only one of each pair has identity gain; this seems inequivalent to Example 1 (\Question:  Is it?), so arbitrary gains on symmetric digraphs seem more general than such gains from gain graphs and more structured than general gain digraphs.]
\dated{2 Apr 2016} 
\codes{\CGD,~\Cgg:~\CBal,~\CChem;~\CRef}%
\itemspace

\authorx{Gilles Didier}						%DIDIER
\author{Gilles Didier \& Elisabeth Remy}
\authorspace
\bsgitem{2012a}{}  Relations between gene regulatory networks and cell dynamics in Boolean models.  \emph{Discrete Appl.\ Math.}\ 160 (2012), no.\ 15, 2147--2157.
\MR{2954757}.  \Zbl{1291.92065}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Gilles Didier, Elisabeth Remy, \& Claudine Chaouiya}
\authorspace
\bsgitem{2011a}{}  Mapping multivalued onto Boolean dynamics.  \emph{J.\ Theor.\ Biol.}\ 270 (2011), 177--184.
\MR{2974862} (no rev).  \Zbl{1331.92051}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{M.\ Diehl}							%DIEHL
\authorspace
\authref  See \hyperlink{C.\ De Simone}{C.~De~Simone}.
\itemspace

\author{Hung T.\ Diep}						%DIEP
\authorspace
\authref  See also \hyperlink{Ojiro Nagai}{O.~Nagai}.
\authrefspace

\author{H.T.\ Diep, ed.}
\authorspace
\bsgitem{2004a}{}  \emph{Frustrated Spin Systems}.  World Scientific, Hackensack, N.J., 2004.
\Zbl{1104.82002}.  
\annot{} Expository.  Any chapter might inspire interesting mathematics of signed graphs, esp.\ 
Ch.\ 1:  \xref{Diep and Giacomini}{2004a}{H.T.\ Diep \& H.\ Giacomini};  
Ch. 2:  \xref{Nagai, Horiguchi, and Miyashita}{2004a}{Ojiro Nagai, Tsuyoshi Horiguchi, \& Seiji Miyashita};  
Ch. 5:  \xref{Misguich and Lhuillier}{2004a}{Gregoire Misguich and Claire Lhuillier}.  
\dated{13 Aug 2018} 
\codes{\CSG,~\CPhys:~\CExp,~\CRef}%
\itemspace

\author{H.T.\ Diep \& H.\ Giacomini}
\authorspace
\bsgitem{2004a}{}  Frustration - Exactly solved frustrated models.  In:  H.T.\ Diep, ed., \emph{Frustrated Spin Systems}, Ch.\ 1, pp.\ 1--58.  World Scientific, Hackensack, N.J., 2004.
\Zbl{1104.82002} (book).  \arXiv{1912.13042}.  
\annot{} Frustration on signed graph with Ising ($\pm1$) and vector spins (implicitly).  \S\S1.1--1.2 introduce frustration and physics concerns, e.g., ``degeneracy'' = multiple ground states.  Later, various periodic signed lattice graphs (\cf\ \xref{Liebmann}{1986a}{Rainer Liebmann}) are solved and diagrammed for Ising and XY ($S^1$) spins, illustrating spin frustration.
\dated{9 Aug 2018}
\par
\codes{\CSG,~\CPhys:~\CFr:~\CExp,~\CRef}%
\itemspace

\author{Hung T.\ Diep, P.\ Lallemand, \& O.\ Nagai}
\authorspace
\bsgitem{1985a}{}  Simple cubic fully frustrated Ising crystal by Monte Carlo simulations.  \emph{J.\ Appl.\ Phys.}\ 57 (1985), 3309--3311.
\annot{} Physics of fully frustrated $3$-dimensional cubic lattice (\cf\ \xref{Chui, Forgacs, and Hatch}{1982a}{S.T.\ Chui, G.\ Forgacs, \& D.M.\ Hatch}), but the negative edges are specifically chosen to form three orthogonal families of straight lines, alternating along each plane.  As signed lattice has a $2\times2$ fundamental domain, there are 8 translational symmetry types of vertex, each forming a double-sized sublattice.  The sublattices exhibit somewhat differentiated behavior.
\dated{18 Jun 2012} 
\codes{\CPhys,~\CSG,~\Csw}%
\itemspace

\bsgitem{1985b}{}  Critical properties of a simple cubic fully frustrated Ising lattice by Monte Carlo method.  \emph{J.\ Phys.\ C} 18 (1985), 1067--1078.
\annot{} Simulations on the the signed graph of \xrefd{1985a}{Hung T.\ Diep, P.\ Lallemand, \& O.\ Nagai}.  The 8 sublattices are equivalent in pairs.  
\dated{18 Jun 2012} 
\codes{\CPhys,~\CSG:~\CFr}%
\itemspace

\author{Jana Diesner}						%DIESNER
\authorspace
\authref  See \hyperlink{Ly Dinh}{L.~Dinh}.
\itemspace

\author{V.\ Di Giorgio} 						%DIGIORGIO
\authorspace
\bsgitem{1974a}{} 2-modules dans un graphe: equilibre et coequilibre d'un bigraphe---application taxonomique.  \emph{Bull.\ Math.\ Soc.\ Sci.\ Math.\ R.\ S.\ Roumanie (N.S.)} 18 (66) (1974), 81--102 (1975). 
\MR{0476564} (57 \#16124).  \Zbl{324.05127}.
\codes{\CSG:~\CBal}%
\itemspace

\author{Wil Dijkstra}							%DIJKSTRA
\authorspace
\bsgitem{1979a}{}  Response bias in the survey interview; an approach from balance theory.  \emph{Social Networks} 2 (1979--1980), no.\ 3, 285--304.
\annot{} Extends signed graphs to sign set $\{\pm1,0\}$ and extends the notion of (degree of) cycle balance.  A circle $C$ is ``balanced'' if its sign product $\sigma(C)=+1$.  Degree of balance = average sign product of all circles.  Degree of local balance at $X \subseteq V$ is the average sign of all circles that contain $X$.  Given a length weight function $1 \geq f(2) \geq f(3) \geq \cdots \geq 0$, the weighted degree of balance is the average value of $f(l(C))\sigma(C)$.  [\Cf\ kinds of cycle balance in \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}, \xref{Morrissette}{1958a}{Julian O.\ Morrissette}, \xref{Norman and Roberts}{1972a}{Robert Z.\ Norman \& Fred S.\ Roberts}, \xrefd{1972b}{Robert Z.\ Norman \& Fred S.\ Roberts}.]
\codes{\CSG:~\CBal,~\CFr}%
\annotctd{} It is assumed [!]\ that answers have probability dependent on weighted degree of local balance at $\{p,y\}$ where $p$ = respondent and $y$ = answer.  Speculation about choice of functions $f$ \etal\ \  One post-hoc application.
\par
\codes{\CSG:~\CPsS}%
\itemspace

\author{Genhong Ding}						%DING GH
\authorspace
\authref  See \hyperlink{Xiaobin Ma}{X.B.~Ma}.
\itemspace

\authorx{Ly Dinh}							%DINH
\author{Ly Dinh, Rezvaneh Rezapour, Lan Jiang, \& Jana Diesner}
\authorspace
\bsgitem{20xxa}{}  Structural balance in signed digraphs: considering transitivity to measure balance in graphs constructed by using different link signing methods.  Submitted.  \emph{}\ 
\arXiv{2006.02565}.  
\codes{\CSD:~\CBal:~\CAlgor}%
\itemspace

\authorx{Yvo M.I.\ Dirickx}		 				%DIRICKX
\author{Yvo M.I.\ Dirickx \& M.R.\ Rao}
\authorspace
\bsgitem{1974a}{}  Networks with gains in discrete dynamic programming.  \emph{Management Sci.}\ 20 (1974), No.\ 11 (July, 1974), 1428--1431.  
\MR{0359827} (50 \#12279).  \Zbl{303.90052}. 
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{Divya T}							%DIVYA
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\author{Ajit A.\ Diwan}						%DIWAN
\authorspace
\authref  See also \hyperlink{Manas Joglekar}{M.~Joglekar}.
\authrefspace

\author{Ajit A.\ Diwan, Josh B.\ Frye, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}
\authorspace
\bsgitem{2011a}{}  A sufficient condition for the existence of an anti-directed 2-factor in a directed graph.  \emph{Discrete Math.}\ 311 (2011), no.\ 21, 2556--2562.
\MR{2832152} (2012i:05118).  \Zbl{1238.05209}.  
\annot{} An antidirected circle is a balanced circle in the poise gains of a digraph.  [For early bidirected work see \xref{Andersen and Grant}{1981a}{Lars Dovling Andersen \& Douglas D.\ Grant}.  For connected 2-factor see \xref{Busch, Jacobson, Morris, Plantholt, and Tipnis}{2013a}{Arthur H.\ Busch, Michael S.\ Jacobson, Timothy Morris, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}.  For Hamilton paths in $K_n$ see \xref{El Sahili and Abi Aad}{2018a}{Amine El Sahili \& Maria Abi Aad}.]  [\Question.  How does this generalize to bidirected graphs?]
\dated{20, 30 May 2018} 
\codes{\Cgg:~\CStr}\codesctd{\Csg:~\Cpar:~\COri}%
\itemspace

\author{Daniel B.\ Dix}						%DIX
\authorspace
\bsgitem{2006a}{}  Polyspherical coordinate systems on orbit spaces with applications to biomolecular shape.  \emph{Acta Appl.\ Math.}\ 90 (2006), 247--306.
\MR{2248745} (2007k:92043).  \Zbl{1182.92028}.					
\codes{\CGG:~\CAppl}
\itemspace

\author{Vlastimil Dlab}						%DLAB
\authorspace
\bsgitem{1980a}{}  \emph{Representations of Valued Graphs}.  S\'em.\ Math.\ Sup\'erieures, 73.  D\'ep.\ Math.\ Stat., Universit\'e de Montr\'eal.  Les Presses de l'Universit\'e de Montr\'eal, Montr\'eal, 1980.
\MR{0586769} (82k:16037).  \Zbl{478.16026}.					
\annot{} A valued graph is a simple symmetric digraph with $\bbZ_{>0}$-gains, such that $\phi(e_{ij})/\phi(e_{ji}) = f(v_j)/f(v_i)$ for some $f: V \to \bbZ_{>0}$ [equivalently, cycle gains invert by reversing direction].  $\phi(e_{ij})$ represents the dimension of an $(F_i,F_j)$-bimodule corresponding to $e_{ij}$, where $F_i$ is an algebra associated to $v_i$.  [There is no gain-graph theory.]  [Based on 3 articles; see Zbl.]
\dated{26 Dec 2015} 
\codes{\Cgd:~\CAlgeb}%
\itemspace

\authorx{Duong D.\ Doan}						%DOAN
\author{Duong D.\ Doan \& Patricia A.\ Evans}
\authorspace
\bsgitem{2011a}{}  Haplotype inference in general pedigrees with two sites.  6th Int.\ Symp.\ Bioinformatics Res.\ Appl.\ (ISBRA'10, Storrs, Conn., 2010).  \emph{BMC Proc.}\ 5 (2011), Suppl.\ 2, 56, 10 pp.
\annot{} A pedigree is a kind of signed graph with $< n$ edges, with 3-colored vertices.  Frustration index (``line index'') $l$ = minimum number of necessary recombinations.  
Elementary relations among $l$, vertex cuts, and switching.  Reduction rules, including the negative-subdivision trick, to test $l \leq k$.  
[\Question.  Does sparseness reduce the hardness of testing $l \leq k$?]
\dated{29 Apr 2012} 
\codes{\CBiol:~\CSG:~\CFr,~\CAlgor,~\Csw}%
\itemspace

\bsgitem{2011b}{}  An FPT haplotyping algorithm on pedigrees with a small number of sites.  \emph{Algorithms Molecular Biol.}\ 6 (2011), no.\ 8, 8 pp.
\annot{} See \xrefd{2011a}{Duong D.\ Doan \& Patricia A.\ Evans}.  This problem adds parity constraints.
\dated{29 Apr 2012} 
\codes{\CBiol:~\CSG:~\CFr,~\CAlgor}%
\itemspace

\authorx{R.L.\ Dobrushin}						%DOBRUSHIN
\author{R.L.\ Dobrushin \& S.B.\ Shlosman}
\authorspace
\bsgitem{1985a}{}  The problem of translation invariance of Gibbs states at low temperatures.  Transl.\ Morton Hamermesh.  In:  S.P.\ Novikov, ed., \emph{Mathematical Physics Reviews}, Vol.\ 5 (1985), pp.\ 53--195.  Soviet Sci.\ Rev., Sect.\ C.  Harwood, Chur, Switz., 1985.
\MR{0852217 (87j:82013)}.  \Zbl{0613.76010}.  
\annot{} Partly expository.  The problems are existence of nonperiodic ground states and stability for nonferromagnetic interactions, e.g., signed graphs that are not all positive.  (``Ground state'' means a state at temperature 0; a ``state'' is a probabilistic mixture of what are usually called states, here called ``configurations''.) The graphs are infinite hypercubic lattice graphs $\bbZ^\nu$, $\nu\ge1$.  The set of ``configurations'' $\sigma:\bbZ^\nu \to S$, where $S$ is a fixed set (usually finite; $S=\{\pm1\}$ for Ising model, etc.), is $\Omega$.  ``State'': a probability measure on $\Omega$ w.r.t.\ $\cB$, the finitely cylindrical $\sigma$-algebra on $\bbZ^\nu$.  ``Interactions'' between vertices have finite range, not necessarily only adjacent.
\annotctd{} \S2, ``Ground states'':  \S2.8, ``The symmetric ferromagnetic Ising model'', describes ground configurations in terms of the hypercubic lattice facets dual to frustrated edges of the configuration.  3-Dimensional examples.
\S2.9, ``The antiferromagnetic Ising model''' with external field strength $h$.  There are 1 and 2 ground configurations for $|h|>2\nu$ and $<2\nu$.  Some nonperiodic ground states are described.
At $h=\pm2\nu$ ground states (not only configurations) exist, discussed in \S\S2.11--12.
\S2.11, ``Random ground states--the antiferromagnetic case''.  
\S2.12, ``Nonperiodic random ground states''.
App.\ I, ``Ground states of the model of a one-dimensional lattice antiferromagnet'', by S.E.\ Burkov and Ya.G.\ Sinai.  
App.\ II, ``On random ground states of one-dimensional antiferromagnetic model'', by A.A.\ Kerimov.  
Dictionary:  ``ferromagnetic'' = all-positive; ``antiferromagnetic'' = all-negative; ``periodic'' = toroidal; ``external field'' = extra vertex $v_0$, positive, with $N(v_0)=V$ and arbitrary interaction strength.
\dated{8 Mar, 23 May, 3 Jun 2015} 
\par
\codes{\CPhys,~\Csg:~\Cbal:~\CState(\Cfr)}%
\codesctd{\CPhys,~\Csg:~\Cpar:~\CState(\Cfr)}%
\itemspace

\author{Ebrahim Dodongeh}						%DODONGEH
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Benjamin Doerr}						%DOERR
\authorspace
\bsgitem{2000a}{}  Linear discrepancy of basic totally unimodular matrices.  \emph{Electronic J.\ Combin.}\ 7 (2000), Research Paper R48, 4 pp.
\MR{1785144} (2001e:15017).  \Zbl{996.15012}.
\annot{} The linear discrepancy of the transposed incidence matrix of a balanced signed graph.
\codes{\Csg:~\Cbal:~\CInc}%
\itemspace

\authorx{B.G.S.\ Doman}						%DOMAN
\author{B.G.S.\ Doman \& J.K.\ Williams}
\authorspace
\bsgitem{1982a}{}  Low-temperature properties of frustrated Ising chains.  \emph{J.\ Phys.\ C} 15 (1982), 1693--1706.
\annot{} \S2, ``The random bond model at low temperatures'':  A signed path with magnetic field $B$ [interpretable as an extra all-positive dominating vertex with edge weights $B$; \cf\ \xref{Barahona}{1982a}{Francisco Barahona}].
\S3, ``Frustrated periodic bond model'':  A path with edges signed ${+}{-}{-}{-}$ periodically.  Describes ground states (``allowed states''), which depend on $B$.
\dated{28 Aug 2012} 
\codes{\CPhys,~\CSG,~\CWG:~\CState(\Cfr)}%
\itemspace

\author{Janelle Domantay}						%DOMANTAY
\authorspace
\authref  See \hyperlink{Alvaro Carbonero}{A.~Carbonero}.
\itemspace

\author{Eytan Domany}						%DOMANY
\authorspace
\authref  See \hyperlink{Guy Hed}{G.~Hed} and \hyperlink{Daniel Kandel}{D.~Kandel}.
\itemspace

\authorx{Mirela Domijan}						%DOMIJAN
\author{Mirela Domijan \& Elisabeth P\'ecou}
\authorspace
\bsgitem{2012a}{}  The interaction graph structure of mass-action reaction networks.  \emph{J.\ Math.\ Biol.}\ 65 (2012), 375--402.
\MR{2944515}.  \Zbl{1303.92038}.  
\par
\codes{\CBiol,~\CChem:~\CSD:~\CDyn}%
\itemspace

\author{Bing-can Dong}						%DONG BC
\authorspace
\authref  See \hyperlink{Ruilin Li}{R.L.~Li}.
\itemspace

\author{Chun-Long Dong}						%DONG CL
\authorspace
\authref  See \hyperlink{Yi-Zheng Fan, Wen-Xue Du, \& Chun-Long Dong}{Y.-Z.~Fan}.
\itemspace

\author{Fengming Dong}						%DONG FM
\authorspace
\authref  See \hyperlink{Shengjie He}{S.J.~He}.
\itemspace

\authorx{Jiu-Gang Dong}						%DONG JG
\author{Jiu-Gang Dong \& Lin Lin}
\authorspace
\bsgitem{2016a}{}  Laplacian matrices of general complex weighted directed graphs.  \emph{Linear Algebra Appl.}\ 510 (2016), 1--9.
\MR{3551615}.  \Zbl{1352.05112}.  
\codes{\CGD:~\CLap}%
\itemspace

\author{Wenkuan Dong}						%DONG WK
\authorspace
\authref  See \hyperlink{Dijian Wang}{D.-J.~Wang}.
\itemspace

\author{Alfredo Donno}						%DONNO
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo} and \hyperlink{Matteo Cavaleri}{M.~Cavaleri}.
\itemspace

\author{Michael Doob} 						%DOOB
\authorspace
\authref  See also \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}.
\authrefspace

\bsgitem{1970a}{} A geometric interpretation of the least eigenvalue of a line graph.  
In: \emph{Proceedings of the Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications} (1970), pp.\ 126--135.
University of North Carolina at Chapel Hill, Chapel Hill, N.C., 1970.  
\MR{0268060} (42 \#2959).  \Zbl{209.55403} (209, p.\ 554c).
\annot{} A readable, tutorial introduction to \xrefd{1973a}{Michael Doob} (without matroids).
\par
\codes{\Cecyc:~\CLG,~\CInc,~\CEig(\CLG)}%
\itemspace

\bsgitem{1973a}{}  An interrelation between line graphs, eigenvalues, and matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 15 (1973), 40--50.  
\MR{0439687} (55 \#12573).  \Zbl{245.05125}, (\Zbl{257.05132}).			
\annot{} Along with \xref{Sim\~oes-Pereira}{1973a}{J.M.S.\ Simoes-Pereira}, introduces to the literature the even-cycle matroid $\bfF(-\Gamma)$ [previously invented by Tutte, unpublished].  The multiplicity of $-2$ as an eigenvalue (in characteristic $0$) equals the number of independent even circles $= n - \rk \bfF(-\Gamma)$.  
In characteristic $p$ there is a similar theorem, but the pertinent matroid is $\bfM(\Gamma)$ if $p = 2$ and, when $p|n$, the matroid has rank $1$ greater than otherwise [a fact that mystifies me]. 
\codes{\CECyc:~\CLG,~\CInc,~\CEig(\CLG)}%
\itemspace

\bsgitem{1974a}{} Generalizations of magic graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 17 (1974), 205--217.  
\MR{0364019} (51 \#274). \Zbl{271.05128}, (\Zbl{287.05124}).				
\annot{} Thm.\ 3.2 is the theorem of \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}, supplemented by the observation that it remains true in any characteristic except 2.
	
	
	
\par
\codes{\CECyc:~\CInc}%
\itemspace

\bsgitem{1974b}{} On the construction of magic graphs.  
In:  F.\ Hoffman \etal, eds., \emph{Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing} (Boca Raton, 1974), pp.\ 361--374.  Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1974.  
\MR{0409279} (53 \#13039).  \Zbl{325.05123}.
\codes{\Cecyc:~\CInc}%
\itemspace

\bsgitem{1978a}{} Characterizations of regular magic graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 25 (1978), 94--104.  
\MR{0505855} (58 \#21840).  \Zbl{384.05054}.
\codes{\Cecyc:~\CInc}%
\itemspace

\author{Michael Doob \& Drago\v{s} Cvetkovi\'c} 
\authorx{Michael Doob \& Dragos Cvetkovic} 
\authorspace
\bsgitem{1979a}{}  On spectral characterizations and embeddings of graphs.  \emph{Linear Algebra Appl.}\ 27 (1979), 17--26.  
\MR{0545719} (81d:05050).  \Zbl{417.05025}.					
\codes{\Csg:~\CLG,~\CEig(\CLG)}%
\itemspace

\author{Patrick Doreian} 						%DOREIAN
\authorspace
\authref  See also \hyperlink{Michael Brusco}{M.~Brusco}, \hyperlink{Norman P.\ Hummon}{N.P.~Hummon} and \hyperlink{Andrej Mrvar}{A.~Mrvar}.
\authrefspace

\bsgitem{1971a}{}  \emph{Mathematics and the Study of Social Relations}.  Schocken, New York, 1971.
\annot{} Ch.\ 5, ``Structural balance'':  Signed digraphs, sometimes ignoring direction.  The \xref{Cartwright--Harary}{1956a}{Dorwin Cartwright \& Frank Harary} model with signed digraphs.  The \xref{Abelson--Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} model with ambiguous arcs [repeating Abelson--Rosenberg's error about balance via $R(\Sigma)$].  
\S 5.1, ``Balance theory''.  \S 5.2, ``The structure theorems'' (of balance and clusterability).  \S 5.3, ``Measures of balance''.  \S 5.4, ``Construction of signed structures'': distinguishes between balance of a signed digraph (as in Cartwright--Harary) and balance as perceived by a vertex (a person) (as in \xref{Heider}{1946a}{Fritz Heider}).  \S 5.5, ``Structural balance as a deductive theory''.
\dated{29 Aug 2013} 
\codes{\CPsS,~\CSD,~\Csg:~\CBal,~\CFr,~\CClu,~\CAdj:~\CExp}%
\itemspace

\bsgitem{1985a}{}  Review of \emph{Structural Models in Anthropology} by Per Hage and Frank Harary.  \emph{J.\ Math.\ Sociology} 11 (1985), 283--285.
\annot{} Review of \xref{Hage and Harary}{1983a}{Per Hage \& Frank Harary}.
\codes{\CPsS:~\CSG:~\CBal}%
\itemspace

\bsgitem{2002a}{}  Event sequences as generators of social network evolution.  \emph{Social Networks} 24 (2002), 93--119.
\codes{\CSG:~\CBal,~\CPsS}%
\itemspace

\bsgitem{2004a}{}  Evolution of signed human networks.  \emph{Metodolo{\v y}ski Zvezki} 1 (2004), 277--293. 
\annot{} Reviews the development of balance and clustering theory for signed (di)graphs in social psychology, mainly \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}, \xref{Doreian and Krackhardt}{2001a}{Patrick Doreian \& David Krackhardt}, and especially \xref{Hummon and Doreian}{2003a}{Norman P.\ Hummon \& Patrick Doreian}.  The difference between \xref{Heider's}{1946a}{Fritz Heider} and \xref{Cartwright and Harary's}{1956a}{Dorwin Cartwright \& Frank Harary} models, and the need to combine them.
\dated{24 Apr 2009}
\par
\codes{\CPsS:~\CExp:~\CSD,~\CBal,~\CClu,~\CAlgor}%
\itemspace

\bsgitem{2008a}{}  A multiple indicator approach to blockmodeling signed networks.  \emph{Social Networks} 30 (2008), 247--258.
\annot{} Signed graphs $\Sigma_1,\ldots$ (``multiple indicators'') may be approximations of a hidden signed graph $\Sigma$.  Goals: detect whether $\Sigma$ exists, and find an optimal clustering of $\Sigma$.  Methods:    
(1)  Examine the $\Sigma_j$ for compatibility via statistical tests.  
(2)  Estimate $\Sigma$ by $\sum_j \sigma_j$.  
(3)  Applies the clusterability index and algorithm of \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}.  
((2) implies using weighted signed graphs.)  
This article treats examples, with analysis of the methods' success.  
\dated{27 Apr 2009} 
\codes{\CPsS,~\CSD:~\Csg:~\CClu}%
\itemspace

\bsgitem{2008b}{}  Clashing paradigms and mathematics in the social sciences.  \emph{Contemp.\ Sociology}\ 37 (2008), no.\ 6, 542--545.
\annot{} Two books on and the philosophy of mathematics and sociology.
\dated{27 Apr 2012} 
\codes{\CPsS:~\CSG,~\CSD}%
\itemspace

\author{Patrick Doreian, Vladimir Batagelj, \& Anu\v{s}ka Ferligoj}
\authorx{Patrick Doreian, Vladimir Batagelj, \& Anuska Ferligoj}
\authorspace
\bsgitem{2005a}{}  \emph{Generalized Blockmodeling}.  Structural Analysis in the Social Sciences, No.\ 25.  Cambridge Univ.\ Press, Cambridge, Eng., 2005.
\annot{} Ch.\ 10: ``Balance theory and blockmodeling signed networks''.  Thm.\ (pp.\ 305--306; proof by Martin Everett):  The sizes of the partitions of $V$ that minimize the clustering index (\xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}) are consecutive integers.
\codes{\CPsS,~\CSD:~\Csg:~\CClu,~\CBal}%
\itemspace

\author{Patrick Doreian, Roman Kapuscinski, David Krackhardt, \& Janusz Szczypula}
\authorspace
\bsgitem{1996a}{}  A brief history of balance through time.  \emph{J.\ Math.\ Sociology} 21 (1996), 113--131.  Repr.\ in Patrick Doreian and Frans N.\ Stokman, eds., \emph{Evolution of Social Networks}, pp.\ 129--147.  Gordon and Breach, Australia, Amsterdam, etc., 1997.
\Zbl{883.92034}.  
\annot{} \S2.3: ``A method for group balance''.  Describes the negation-minimal index of clusterability (generalized imbalance) from \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}.
\codes{\CSG:~\CBal,~\CClu:~\CFr(\CGen):~\CExp}%
\annotctd{} \S3.3: ``Results for group balance''.  Describes results from analysis of data on a small (social) group, in terms of frustration index $l$ and a clusterability index $\min_{k>2} 2 P_{k, .5}$ (slightly different from the index in \xref{Doreian and Mrvar}{1996a}{Patrick Doreian \& Andrej Mrvar}), finding both measures (but more so the latter) decreasing with time. 
\codes{\CPsS:~\CBal,~\CClu:~\CFr(\CGen)}%
\itemspace

\author{Patrick Doreian \& David Krackhardt}
\authorspace
\bsgitem{2001a}{}  Pre-transitive balance mechanisms for signed networks.  \emph{J.\ Math.\ Sociology} 25 (2001), no.\ 1, 43--67.
\Zbl{1017.91520}.  
\annot{} In a signed digraph from empirical social-group data, a tendency to transitivity of signs on directed edges $ij, ik, jk$ (i.e., $\sigma(ij)\sigma(jk)\sigma(ik)=+$) holds when $\sigma(ij)=+$ and fails when $\sigma(ij)=-$.  This suggests that balance is not a primary tendency and \xref{Harary's}{1953a}{Frank Harary} and \xref{Davis's}{1967a}{James A.\ Davis} theorems on balance and clusterability have limited relevance to social groups.  [Also, that undirected signed graphs have limited relevance.  Digraph sign transitivity properties are more relevant.]  [A thoughtful article.]  
\dated{13 Apr 2009}
	
\codes{\CPsS,~\Csd}%
\itemspace

\author{Patrick Doreian, Paulette Lloyd, \& Andrej Mrvar}
\authorspace
\bsgitem{2013a}{}  Partitioning large signed two-mode networks: Problems and prospects.  \emph{Social Networks} 35 (2013), 178--203.
\codes{\CSG:~\CBal,~\CFr,~\CPsS}%
\itemspace

\author{Patrick Doreian \& Andrej Mrvar} 
\authorspace
\bsgitem{1996a}{}  A partitioning approach to structural balance.  \emph{Social Networks} 18 (1996), 149--168.  
\annot{} They propose indices for clusterability (as in \xref{Davis}{1967a}{James A.\ Davis}) that generalize the frustration index of $\Sigma$.  Fix $k \geq 2$ and $\alpha \in [0, 1]$.  For a partition $\pi$ of $V$ into $k$ ``clusters'', they define $P(\pi) := \alpha n_- + (1-\alpha)n_+$, where $n_+ :=$ number of positive edges between clusters, $n_- :=$ number of negative edges within clusters, and $0 \leq \alpha \leq 1$ is fixed.  
The first proposed measure is $P_k := \min P(\pi)$, minimized over $k$-partitions.  
A second suggestion is the ``negation-minimal index of generalized imbalance'' [i.e., of clusterability], the smallest number of edges whose negation [equivalently, deletion] makes $\Sigma$ clusterable.  [Call it the `clusterability index' $Q(\Sigma)$.]  
[Note that $P(\pi)$ effectively generalizes the Potts Hamiltonian as given by \xref{Welsh}{1993a}{D.J.A.\ Welsh}.  \Question.  Does $P(\pi)$ fit into an interesting generalized Potts model?]  
[$P(\pi)$ also resembles the Potts Hamiltonian in \xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz} (\qv\ for a related research question).]
[The data in \xref{Doreian}{2008a}{Patrick Doreian}, and common sense, suggest that clusters should be allowed to overlap.  This is an open research direction.]  
\annotctd{} They employ a local optimization algorithm to evaluate $P_k(\alpha)$ and find an optimal partition: random descent from partition to neighboring partition, where $\pi$ and $\pi'$ are neighbors if they differ by transfer of one vertex or exchange of two vertices between two clusters.  
This was found to work well if repeated many times.  [A minimizing partition into at most $k$ clusters is equivalent to a ground state of the $k$-spin Potts model in the form given by \xref{Welsh}{1993a}{D.J.A.\ Welsh}, but not quite in that of \xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz}.]
\annotctd{}  Dictionary:  $P(\pi)$ is the ``criterion function''.  [More explicitly, call $Q(\Sigma,\pi;\alpha) := 2P(\pi)$ the `$\alpha$-weighted clusterability index of $\pi$', so the clusterability index is $Q(\Sigma) = \min_\pi Q(\Sigma,\pi;.5)$; and call $Q_k := 2P_k$ the `$k$-clusterability index' of $\Sigma$.].  Clusterability is ``$k$-balance'' or ``generalized balance''.  Clusters are ``plus-sets''.  Signed digraphs are employed in the notation but direction is ignored. 
\annotctd{}  [Further developments in \hyperlink{Patrick Doreian}{Doreian} \etal\ (various), \xref{Hummon and Doreian}{2003a}{Norman P.\ Hummon \& Patrick Doreian}, \xref{Bansal \etal\!\!}{2004a}{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}, \xref{Demaine \etal\!\!}{2006a}{Erik D.\ Demaine, Dotan Emanuel, Amos Fiat, \& Nicole Immorlica}, \xref{Mrvar and Doreian}{2009a}{Andrej Mrvar \& Patrick Doreian}.]  
\annotctd{}  [The data in \xref{Doreian}{2008a}{Patrick Doreian}, and common sense, suggest that clusters should be allowed to overlap.  This is an unplumbed direction.]  
\dated{rev.\ 22 Sept 2009}
\codes{\CSD:~\Csg:~\CBal,~\CClu:~\CFr(\CGen),~\CAlgor,~\CPsS}%
\itemspace

\bsgitem{1996b}{}  Structural balance and partitioning signed graphs.  In: A.\ Ferligoj and A.\ Kramberger, eds., \emph{Developments in Data Analysis}, pp.\ 195--208.  Metodolo\v{s}ki zvezki, Vol.\ 12.  FDV, Ljubljana, Slovenia, 1996.  
\annot{} Similar to \xrefd{1996a}{Patrick Doreian \& Andrej Mrvar}.  Some lesser theoretical detail; some new examples.  The $k$-clusterability index $P_k(\alpha)$ \xrefd{1996a}{Patrick Doreian \& Andrej Mrvar} is compared for different values of $k$, seeking the minimum.  [But for which value(s) of $\alpha$ is not stated.]  Interesting observation: optimal values of $k$ were small.  It is said that positive edges between parts are far more acceptable socially than negative edges within parts [thus, in the criterion function $\alpha$ should be rather near $1$].
\codes{\CSD:~\Csg:~\CBal,~\CClu:~\CFr(\CGen),~\CAlgor,~\CPsS}%
\itemspace

\bsgitem{2009a}{}  Partitioning signed social networks.  \emph{Social Networks} 31 (2009), no.\ 1, 1--11.
\annot{} Generalizes the ideas of \xrefd{1996a}{Patrick Doreian \& Andrej Mrvar} (\qv).  Given:  A signed digraph $(\vec\Gamma,\sigma)$; a ``criterion function'' $P(\pi,\rho) := \alpha n^+ + (1-\alpha)n^-$, where $\pi:=\{B_1,\ldots,B_k\}$ partitions $V$ into ``clusters'', $\rho: \pi \times \pi \to \{+,-\}$, $0 \leq \alpha \leq 1$ is fixed, and $n^\epsilon :=$ number of edges $\overarrow{B_iB_j}$ with sign $\epsilon$ for which $\rho(B_i,B_j) = -\epsilon$ (over all $i,j$).  Objective: $(\pi,\rho)$, or simply $k := \#\pi$, that minimizes $P(\pi,\rho)$.  What is new and most interesting is $\rho$; also new is using the edge directions.
\annotctd{} Call $(\vec\Gamma,\sigma)$ ``sign clusterable'' if $\exists\ (\pi,\rho)$ with $P(\pi,\rho) = 0$.  Clusterability is sign clusterability with $\rho = \rho_+$, where $\rho_+(B_i,B_j) := + \text{ if } i=j,\ - \text{ if } i\neq j$.  
Let $P(k) := \min\{ P(\pi,\rho) : \#\pi=k \}$.  
Thm.\ 4: $P(k)$ is monotonically decreasing.  [Thus, there is always an optimum $\pi$ with singleton clusters.  Why this does not vitiate the model is not addressed.]  
Thm.\ 5:  If $(\vec\Gamma,\sigma)$ is sign clusterable, then $(\vec\Gamma,-\sigma)$ also is.  If $(\vec\Gamma,\sigma)$ is clusterable, then $(\vec\Gamma,-\sigma)$ is not clusterable with the same $\pi$ [provided $E \neq \eset$].  If $(\vec\Gamma,\sigma)$ is sign clusterable with $\rho=-\rho_+$, then $(\vec\Gamma,-\sigma)$ is clusterable with the same $\pi$.  
``Relocation'': Shift one vertex, or exchange two vertices, between blocks so as to decrease $P$, as in \xrefd{1996a}{Patrick Doreian \& Andrej Mrvar}.  This is said (but not proved) to minimize $P$.  
\annotctd{} Refinements discussed: partially prespecified blocks; null blocks (without outgoing edges); criterion functions with special treatment of null blocks.
\annotctd{} Applications to standard test examples of social psychology.
\annotctd{} Dictionary: ``balanced'' = clusterable; ``relaxed balanced'' = sign clusterable; ``$k$-balanced'' = clusterable with $\#\pi=k$; ``relaxed structural balance blockmodel'' = this whole system.  
\dated{7 Feb 2009}
\par
\codes{\CSG:~\CBal,~\CClu,~\CPsS}%
\itemspace

\author{Florian D\"orfler}						%DORFLER
\authorx{Florian Dorfler}
\authorspace
\authref  See \hyperlink{Wenjun Mei}{W.-J.~Mei}.
\itemspace

\author{W.\ D\"orfler} 						%DORFLER
\authorx{W.\ Dorfler}
\authorspace
\bsgitem{1977a}{} Double covers of graphs and hypergraphs.  
In: \emph{Beitrage zur Graphentheorie und deren Anwendungen} (Proc.\ Int.\ Colloq., Oberhof, D.D.R., 1977), pp.\ 67--79.  Technische Hochschule, Ilmenau, 1977.  
\MR{0599766} (82c:05074).  \Zbl{405.05055}.
\codes{\CSG:~\CCov,~\CLG}\codesctd{\CSD,~\CSH:~\CCov}%
\itemspace

\bsgitem{1978a}{}  Double covers of hypergraphs and their properties.  \emph{Ars Combin.}\ 6 (1978), 293--313.  
\MR{0599766} (82d:05085).  \Zbl{423.050532}.
\codes{\CSH:~\CCov,~\CLG}%
\itemspace

\author{Tomislav Do\v{s}li\'c}						%DOSLIC
\authorx{Tomislav Doslic}						%DOSLIC
\authorspace
\authref  See also \hyperlink{Zahra Yarahmadi}{Z.~Yarahmadi}.
\authrefspace

\author{Tomislav Do\v{s}li\'c \& Damir Vuki\v{c}evic}
\authorx{Tomislav Doslic \& Damir Vukicevic}
\authorspace
\bsgitem{2007a}{}  Computing the bipartite edge frustration of fullerene graphs.  \emph{Discrete Appl.\ Math.}\ 155 (2007), 1294--1301.
\MR{2332321} (2008b:05086).  \Zbl{1118.05092}.  
\par
\codes{\CPar:~\CFr:~\CAlgor}%
\itemspace

\author{[Jos\'e Manuel dos Santos Sim\~oes-Pereira]}		    %DOS SANTOS SIMOES-PEREIRA
\authorx{Jose Manuel dos Santos Simoes-Pereira}
\authorspace
\authref  See \hyperlink{J.M.S.\ Simoes-Pereira}{J.M.S.\ Sim\~oes-Pereira}.
\itemspace

\author{Lynne L.\ Doty} 						%DOTY
\authorspace
\authref  See \hyperlink{Fred Buckley}{F.~Buckley}.
\itemspace

\author{Peter Doubilet} 						%DOUBILET
\authorspace
\bsgitem{1971a}{} Dowling lattices and their multiplicative functions.
In: \emph{M\"obius Algebras} (Proc.\ Conf., Waterloo, Ont., 1971), pp.\ 187--192.  University of Waterloo, Ont., 1971, reprinted 1975.  
\MR{0357137} (50 \#9605).  \Zbl{385.05008}.
\codes{\CGG:~\CMatrd}
\itemspace

\author{Peter Doubilet, Gian-Carlo Rota, \& Richard Stanley} 
\authorspace
\bsgitem{1972a}{}  On the foundations of combinatorial theory (VI):  The idea of generating function.  
In:  \emph{Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability} (Berkeley, Calif., 1970/71), Vol.\ II: \emph{Probability Theory}, pp.\ 267--318.  University of California Press, Berkeley, Calif., 1972.  
\MR{0403987} (53 \#7796).  \Zbl{267.05002}.  
Repr.\ in:  Gian-Carlo Rota, \emph{Finite Operator Calculus}, pp.\ 83--134.  Academic Press, New York, 1975.  
\MR{0379213} (52 \#119).  \Zbl{328.05007}.  
Repr.\ again in:  Joseph P.S.\ Kung, ed., \emph{Gian-Carlo Rota on Combinatorics:  Introductory Papers and Commentaries}, pp.\ 148--199.  Birkh\"auser, Boston, 1995.  
\MR{1392961} (99b:01027).  \Zbl{841.01031}.  
\annot{} \S5.3:  Brief treatment of Dowling lattices via symmetric gain digraphs (not quite the same as gain graphs).
\dated{rev 20 Jan 2021}
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Thomas A.\ Dowling}						%DOWLING
\authorspace
\bsgitem{1971a}{}  Codes, packings, and the critical problem.  
In: \emph{Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Perugia, 1970)}, pp.\ 209--224.  Ist.\ Mat., Univ.\ di Perugia, Perugia, Italy, 1971. 
\MR{0337669} (49 \#2438).  \Zbl{231.05029}.  					
\annot{} Pp.\ 221--223:  The first intimations of Dowling lattices/geometries/ma\-troids, as in \xrefd{1973a}{Thomas A.\ Dowling}, \xrefd{1973b}{Thomas A.\ Dowling}, and their higher-weight relatives (see \xref{Bonin}{1993a}{Joseph E.\ Bonin}).
\codes{\Cgg,~\CGen:~\CMatrd}%
\itemspace

\bsgitem{1973a}{}  A $q$-analog of the partition lattice.  
In: J.N.\ Srivastava \etal, eds., \emph{A Survey of Combinatorial Theory} (Proc.\ Int.\ Sympos., Ft.\ Collins, Colo., 1971), Ch.\ 11, pp.\ 101--115.  North-Holland, Amsterdam, 1973.  
\MR{0366707} (51 \#2954).  \Zbl{259.05023}.					
\annot{} Linear-algebraic progenitor of \xrefd{1973b}{Thomas A.\ Dowling}.  Treats the Dowling lattice of group $\GF(q)^{\times}$ as naturally embedded in $\PG^{n-1}(q)$.  Interesting is p.\ 105, Remark:  One might generalize some results to any ambient (simple) matroid.
\codes{\Cgg:~\CGeom,~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{1973b}{\dag\dag}  A class of geometric lattices based on finite groups.  \emph{J.\ Combin.\ Theory Ser.\ B} 14 (1973), 61--86.  
\MR{0307951} (46 \#7066).  \Zbl{247.05019}. 					
Erratum.  \emph{Ibid.}\ 15 (1973), 211.  
\MR{0319828} (47 \#8369).  \Zbl{264.05022}.					
\annot{$Q_n(\fG)$} Introduces the Dowling lattices $Q_n(\fG)$ of a group, treated as lattices of group-labelled partial partitions.  Equivalent to the frame matroid of complete $\fG$-gain graph $\fG K_n\full$.  [The gain-graphic approach was known to \xref{Dowling}{1973a}{Thomas A.\ Dowling}, p.\ 109, but first published in \xref{Doubilet, Rota, and Stanley}{1972a}{Peter Doubilet, Gian-Carlo Rota, \& Richard Stanley}.]  Isomorphism, vector representation, Whitney numbers and characteristic polynomial.  [The first and still fundamental paper.]
\annotctd{} [The Dowling Whitney numbers and polynomials have given rise to a little field of their own; \cf\ \xref{Mez\H{o}}{2010a}{Istvan Mezo} and references.]
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace
 
\author{Thomas Dowling \& Hongxun Qin}
\authorspace
\bsgitem{2005a}{}  Reconstructing ternary Dowling geometries.  \emph{Adv.\ Appl.\ Math.}\ 34 (2005), no.\ 2, 358--365.
\MR{2110557} (2005j:05017).  \Zbl{1068.52017}.						
\annot{} Thm.\ 1.5:  The Dowling geometry $Q_r(\bbZ_2)$ is the only matroid of rank $r\geq4$ such that every contraction by a point is $Q_{r-1}(\bbZ_2)$.
\codes{\Csg:~\CMatrd}%
\itemspace


\author{Felix Dr\"axler}						%DRAXLER
\authorx{Felix Draxler}
\authorspace
\authref  See \hyperlink{Igor Balla}{I.~Balla}.
\itemspace

\author{[Pauline van den Driessche]}					%DRIESSCHE
\authorspace
\authref  See \hyperlink{Pauline van den Driessche}{P.~van~den~Driessche} (under `V').
\itemspace

\authorx{Fran\c{c}ois Dross}						%DROSS
\authorx{Francois Dross}
\authorspace
\authref  See also \hyperlink{Julien Bensmail}{J.~Bensmail}.
\authrefspace

\author{Fran\c{c}ois Dross, Florent Foucaud, Valia Mitsou, Pascal Ochem, \& Th\'eo Pierron}
\authorx{Francois Dross, Florent Foucaud, Valia Mitsou, Pascal Ochem, \& Theo Pierron}
\authorspace
\bsgitem{2020a}{}  Complexity of planar signed graph homomorphisms to cycles.  \emph{Discrete Appl.\ Math.}\ 284 (2020), 166--178.
\MR{4115466}.  \Zbl{1443.05084}.  \arXiv{1907.03266}.  \HAL{hal-02990576}.		
\codes{\CSG:~\CHom:~\CAlgor}%
\itemspace

\author{J.M.\ Drouffe} 						%DROUFFE
\authorspace
\authref  See \hyperlink{R.\ Balian}{R.~Balian}.
\itemspace

\authorx{K.\ Dr\"uhl}							%DRUHL
\authorx{K.\ Druhl}
\author{K.\ Dr\"uhl \& H.\ Wagner}
\authorx{K.\ Druhl \& H.\ Wagner}
\authorspace
\bsgitem{1982a}{}  Algebraic formulation of duality transformations for Abelian lattice models.  \emph{Ann.\ Phys.}\ 141 (1982), 225--253.
\MR{0673981} (84h:82064).  
\par
\codes{\CSG,~\CGG:~\CGen:~\CDual,~\CFr,~\CPhys}%
\itemspace

\author{L\'ucia Drummond}						%DRUMMOND
\authorx{Lucia Drummond}
\authorspace
\authref  See \hyperlink{Mario Levorato}{M.~Levorato}.
\itemspace

\author{Natasha D'Souza}						%D'SOUZA N
\authorx{Natasha D'souza e Jaques}
\authorspace
\authref  See \hyperlink{Tarkeshwar Singh}{T.~Singh}.
\itemspace

\author{Sabitha D'Souza [S.\ D'Souza]}					%D'SOUZA S
\authorx{Sabitha D'Souza}
\authorspace
\authref  See \hyperlink{Pradeep G.\ Bhat}{P.G.~Bhat}.
\itemspace

\author{Donglei Du}							%DU DL
\authorspace
\authref  See \hyperlink{Sai Ji}{S.~Ji}.
\itemspace

\author{Haifeng Du}							%DU HF
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai}.
\itemspace

\authorx{Hong Shan Du}						%DU HS
\author{Hong Shan Du, Qing Jun Ren, Hou Chun Zhou, \& Qing Yu Zheng}
\authorspace
\bsgitem{1998a}{}  The quasi-Laplacian permanental polynomial of a graph.  (In Chinese.)  \emph{Qufu Shifan Daxue Xuebao Ziran Kexue Ban} [\emph{J.\ Qufu Normal Univ., Nat.\ Sci.}] 24 (1998), no.\ 2, 59--62.
\MR{1655784} (no rev).  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Mingjun Du}							%DU MJ
\author{Mingjun Du, Baoli Ma, \& Deyuan Meng}
\authorspace
\bsgitem{2019a}{}  Algebraic criteria for structure identification and behaviour analysis of signed networks.  \emph{Int.\ J.\ Systems Sci.}\ 50 (2019), no.\ 12, 2333--2347.
\MR{4013308}.  \Zbl{1483.93083}.  
\par
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2019b}{}  Edge convergence problems on signed networks.  \emph{IEEE Trans.\ Cybernetics} 49 (2019), no.\ 11, 4029--4041.
\codes{\CSD,~\Csg:~\CDyn:~\CInc,~\CLG,~\CLap:~\CAppl}%
\itemspace

\bsgitem{2021a}{}  Further results for edge convergence of directed signed networks.  \emph{IEEE Trans.\ Cybernetics} 51 (2021), no.\ 12, 5659--5670.
\codes{\CSD,~\Csg:~\CDyn:~\CInc,~\CLG,~\CLap:~\CAppl}%
\itemspace

\author{Wenxue Du}							%DU WX
\authorspace
\authref  See also \hyperlink{Yi-Zheng Fan, Wen-Xue Du, \& Chun-Long Dong}{Y.-Z.~Fan}.
\authrefspace

\author{Wenxue Du, Xueliang Li, \& Yiyang Li}
\authorspace
\bsgitem{2010a}{}  Various energies of random graphs.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 64 (2010), no.\ 1, 251--260.
\MR{2677586} (2011k:05133).  
\annot{} Including a ``tight bound'' on signless Laplacian energy, of $\Lap(-\Gamma)$, and ``exact estimate'' of incidence energy, of $\Eta(-\Gamma)$.
\dated{24 Jan 2012} 
\par
\codes{\CPar:~\CEig,~\CInc}%
\itemspace

\author{Hangen Duan}						%DUAN HG
\authorspace
\authref  See \hyperlink{Shi-Cai Gong}{S.C.~Gong}.
\itemspace

\authorx{Wenqiang Duan}						%DUAN WQ
\author{Wenqiang Duan, Qinma Kang, Yunfan Kang, Jianwen Chen, \& Qingfeng Qin}
\authorspace
\bsgitem{2022a}{}  A simple and effective iterated greedy algorithm for structural balance in signed networks.  \emph{Int.\ J.\ Modern Phys.\ B}\ 36 (2022), No. 21, art.\ 2250129.
\par
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{M.\ Dub}							%DUB
\authorspace
\authref  See \hyperlink{Michael Doob}{M.~Doob}.
\itemspace

\authorx{Christopher Duffy}						%DUFFY
\author{Christopher Duffy, Fabien Jacques, Mickael Montassier, \& Alexandre Pinlou}
\authorspace
\bsgitem{2009a}{}  The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree.  Manuscript, 2009.
\arXiv{2009.05439}.  
\annot{} \Cf\ \xref{Jacques and Pinlou}{2022a}{Fabien Jacques \& Alexandre Pinlou}.
Dictionary:  ``2-edge-colored'' = signed, ``signed graph'' = switching class of signed graphs.
\codes{\CSG:~\CCol,~\CHom,~\CSw}%
\itemspace

\author{P.\ Robert Duimering}						%DUIMERING
\authorspace
\authref  See \hyperlink{Gbemisola Adejumo}{G.~Adejumo}.
\itemspace

\authorx{Richard A.\ Duke}
\author{Richard A.\ Duke, Paul Erd\"os, \& Vojt\v{e}ch R\"odl}			%DUKE
\authorx{Richard A.\ Duke, Paul Erdos, \& Vojtech Rodl}			%DUKE
\authorspace
\bsgitem{1991a}{}  Extremal problems for cycle-connected graphs.  
Proc.\ Twenty-Second Southeastern Conf.\ Combinatorics, Graph Theory, and Computing (Baton Rouge, La., 1991).  
\emph{Congressus Numer.}\ 83 (1991), 147--151.
\MR{1152087} (93a:05073).  \Zbl{772.05052}.						
\annot{} Results of the type in \xrefd{1992a}{Richard A.\ Duke, Paul Erdos, \& Vojtech Rodl} for polygon lengths $\leq 8$.  Thm.~2:  Given $d > 0$, constant.  Every $-\Gamma$ with $\#V = n$ and $\#E = dn^2$ has a subgraph $\Sigma'$ with $\#E' \geq d^2n^2(1-o(1))$ in which every two edges belong to a balanced polygon of length at most $8$.
\codes{\Cpar:~\Cbal(\CCircles)}%
\itemspace

\bsgitem{1992a}{}  Cycle-connected graphs.  \emph{Discrete Math.}\ 108 (1992), 261--278.
\MR{1189849} (94a:05106).  \Zbl{776.05057}.						
\annot{} All graphs are simple.  This is one of four related papers (including \xrefd{1991a}{Richard A.\ Duke, Paul Erdos, \& Vojtech Rodl}) that prove extremal results concerning subgraphs of $-\Gamma$ within which every two edges belong to a balanced circle of length at most $2k$, for all or particular $k$.
Typical theorem:  Let $F_l(n,m) =$ the largest number $m' = m'(n,m)$ such that every $-\Gamma$ with $\#V = n$ and $\#E \geq m$ has a subgraph $\Sigma'$ with $\#E' = m'$ in which every two edges belong to a balanced circle of length at most $l$.  For $m = m(n) \geq n^{3/2}$, there is a constant $c_3 > 0$ such that $F_l(n, m) \leq c_3m^2n^{-2}$ for all $l$.  (\S2, (2).)
[\Problem.  Extend these extremal results in an interesting way to arbitrary signed simple graphs, or to simply signed graphs (no repeated edges with the same sign).  (Merely allowing positive edges in addition to negative ones just makes the problem easier.  Something more is required.)]
\par
\codes{\Cpar:~\Cbal(\CCircles):~\CXtreml}%
\itemspace

\authorx{David M.\ Duncan}						%DUNCAN DM
\author{David M.\ Duncan, Thomas R.\ Hoffman, \& James P.\ Solazzo}
\authorspace
\bsgitem{2010a}{}  Equiangular tight frames and fourth root seidel matrices.  \emph{Linear Algebra Appl.}\ 432 (2010), 2816--2823.
\MR{2639246} (2011c:42081).  \Zbl{1223.05172}.					
\annot{} Adjacency matrices of fourth-root-of-unity gain graphs on $K_n$.  
Dictionary: ``Seidel matrix'' = adjacency matrix of such a gain graph.
\dated{20 Jun 2011} 
\codes{\Cgg:~\CGeom,~\Cadj:~\Ckg}%
\itemspace

\bsgitem{2011a}{}  Numerical measures for two-graphs.  \emph{Rocky Mountain J.\ Math.}\ 41 (2011), no.\ 1, 133--154.
\MR{2845937}.  \Zbl{1213.05165}.  \arXiv{0810.3189}.				
\annot{} Some measures to distinguish nonisomorphic two-graphs, i.e., switching-nonisomorphic signatures of $K_n$.
\dated{25 Oct 2012} 
\par
\codes{\Csg:~\Ckg,~\CTG:~\CInvar}%
\itemspace

\authorx{Yen Duong}							%DUONG
\author{Yen Duong, Joel Foisy, Killian Meehan, Leanne Merrill, \& Lynea Snyder}	
\authorspace
\bsgitem{2012a}{\dag}  Intrinsically linked signed graphs in projective space.  \emph{Discrete Math.}\ 312 (2012), 2009--2022.
\MR{2920861}.  \Zbl{1243.05101}.						
\codes{\CSG:~\CTop}%
\itemspace

\author{Arne D\"ur}							%DUR
\authorx{Arne Dur}
\authorspace
\bsgitem{1986a}{}  \emph{M\"obius Functions, Incidence Algebras and Power Series Representations}.   
\LNM 1202.  Springer-Verlag, Berlin, 1986.
\MR{0857100} (88m:\-05005).  \Zbl{592.05006}.
\annot{} Dowling lattices are an example of a categorial approach to incidence-algebra techniques in Ch.\ IV, \S7.  Computed are the characteristic polynomial and second kind of Whitney numbers.  Binomial concavity, hence unimodality of the latter [\cf\ \xref{Stonesifer}{1975a}{J.\ Randolph Stonesifer}] is proved by showing that a suitable generating polynomial has only distinct, negative zeros [\cf\ \xref{Benoumhani}{1999a}{Moussa Benoumhani}].
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Amit Dutta}							%DUTTA
\authorspace
\authref  See \hyperlink{Bikas K.\ Chakrabarti}{B.K.~Chakrabarti}.
\itemspace

\authorx{Luke Duttweiler}						%DUTTWEILER
\author{Luke Duttweiler \& Nathan Reff}
\authorspace
\bsgitem{2019a}{}  Spectra of cycle and path families of oriented hypergraphs.  \emph{Linear Algebra Appl.}\ 578 (2019) 251--271.
\MR{3953365}.  \Zbl{1419.05122}.  
\codes{\CSH:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{P.M.\ Duxbury}						%DUXBURY
\authorspace
\authref  See \hyperlink{M.J.\ Alava}{M.J.~Alava}.
\itemspace

\authorx{Zdenek Dvorak}						%DVORAK
\author{Zden\v{e}k Dvo\v{r}\'ak \& Luke Postle}
\authorx{Zdenek Dvorak \& Luke Postle}
\authorspace
\bsgitem{2018a}{}  Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8.  \emph{J.\ Combin.\ Theory Ser.\ B} 129 (2018), 38--54.
\MR{3758240}.  \Zbl{1379.05034}.  \arXiv{1508.03437}.				
\annot{} A combinatorial variant of gains: partial injections between lists of a list coloring problem.  [Short proof by \xref{Zaj\c{a}c}{20xxa}{Mariusz Zajac}.]
\dated{29 Jul 2019} 
\codes{\Cgg(\CGen):~\CCol}%
\itemspace

\author{Janusz Dybizba\'nski}						%DYBIZBANSKI
\authorx{Janusz Dybizbanski}
\authorspace
\bsgitem{2020a}{}  2-Edge-colored chromatic number of grids is at most $9$.  \emph{Graphs Combin.}\ 36 (2020), no.\ 3, 831--837.
\MR{4090528}.  \Zbl{1454.05037}.  
\codes{\CSGc:~\CHom}%
\itemspace

\author{Janusz Dybizba\'nski, Anna Nenca, \& Andrzej Szepietowski}
\authorx{Janusz Dybizbanski, Anna Nenca, \& Andrzej Szepietowski}
\authorspace
\bsgitem{2020a}{}  Signed coloring of 2-dimensional grids.  \emph{Inform.\ Processing Lett.}\ 156 (2020), art.\ 105918, 6 pp.
\MR{4056038}.  \Zbl{1481.05044}.  
\annot{} Dictionary:  ``signed coloring'' = switching homomorphism.  Bounds on target order.
\dated{3 Sept 2020} 
\codes{\CSG:~\CSw,~\CHom}%
\itemspace

\authorx{I.E.\ Dzyaloshinskii}						%DZYALOSHINSKII
\author{I.E.\ Dzyaloshinskii \& G.E.\ Volovik}
\authorspace
\bsgitem{1978a}{}  On the concept of local invariance in the theory of spin glasses.  \emph{J.\ Physique} 39 (1978), no.\ 6, 693--700.
\annot{} [Early attempt to apply frustration in physics.]  
\S2: Heisenberg spins $V(\Sigma) \to$ unit sphere $S^2$, applying ``frustration lines'' according to \xref{Toulouse}{1977a}{Gerard Toulouse}.  ``Local discrete invariance'' = switching.
\S8: Briefly, partial antiferromagnet (= signed graph); phase diagram varies with the proportion $c$ of negative edges.
\dated{6 Aug 2018} 
\codes{\CPhys:~\CSG:~\CFr,~\Csw}%
\itemspace


\label{LETTER E} \hypertarget{LETTER E}{}	% LETTER  E


\authorx{David Easley}						%EASLEY
\author{David Easley \& Jon Kleinberg}
\authorspace
\bsgitem{2010a}{}  \emph{Networks, Crowds, and Markets: Reasoning About a Highly Connected World}.  Cambridge Univ.\ Press, Cambridge, Eng., 2010.
\MR{2677125} (2011i:91001).  \Zbl{1205.91007}.					
\annot{} Ch.\ 5, ``Positive and negative relationships''.  \S\S5.1, ``Structural balance'', 5.2, ``Balanced networks and the Cartwright-Harary Theorem'':  Balance by triangles in signed complete graphs.  Proof of \xref{Harary's}{1953a}{Frank Harary} bipartition theorem for complete graphs (\S5.2).  \S5.3, ``Applications of structural balance'':  Applications to history.  Complete and incomplete graphs.  Alternatives to structural balance.  
\S5.4, ``A weaker form of structural balance'':  Clusterability (``weak balance'').  Proof of \xref{Davis's}{1967a}{James A.\ Davis} clusterability theorem.  
\S5.5, ``Advanced material: Generalizing the definition of structural balance'': Two parts.  \S5.5A, ``Structural balance in arbitrary (non-complete) networks'':  Proof of Harary's bipartition theorem for general signed graphs by finding connected components of the positive subgraph, then applying breadth-first search to sign the components.  
\dated{22 March 2010} 
\codes{\CSG:~\CBal:~\CExp,~\CExr}%
\annotctd{} \S5.5B, ``Approximately balanced networks'':  Thm.:  If the proportion of unbalanced triangles in a signed $K_n$ is $\leq \epsilon < \frac18$, and if $\delta := \surd[3]{\epsilon}$, then there are $(1-\delta) \#V$ vertices in which at most a fraction $\delta$ of the edges are negative, or there is a bipartition $V = X \cupdot Y$ such that at most a fraction $\delta$ of the edges in $X$ and also in $Y$ are negative and at most that fraction of the $XY$ edges are positive.
\dated{22 March 2010} 
\codes{\CSG:~\CBal,~\Cfr}%
\itemspace

\author{M.\ Ebrahimi}						%EBRAHIMI
\authorspace
\authref  See \hyperlink{A.\ Kargaran}{A.~Kargaran}.
\itemspace

\authorx{Paul H.\ Edelman}		 				%EDELMAN
\author{Paul H.\ Edelman \& Victor Reiner}
\authorspace
\bsgitem{1994a}{}  Free hyperplane arrangements between $A_{n-1}$ and $B_n$.  \emph{Math.\ Z.}\ 215 (1994), 347--365.  
\MR{1262522} (95b:52021).  \Zbl{793.05122}.					
\annot{} Characterizes all $\Sigma \supseteq +K_n$ whose frame matroid $\bfF(\Sigma)$ is supersolvable, free, or inductively free.  Essentially, iff the negative links form a threshold graph.  [Continued in \xref{Bailey}{20xxa}{G.\ David Bailey}.  Generalized in part to arbitrary gain groups in \xref{Zaslavsky}{2001a}{Thomas Zaslavsky}.]
\codes{\Csg:~\CMatrd,~\CGeom,~\Ccol}%
\itemspace

\bsgitem{1996a}{}  Free arrangements and rhombic tilings.  
\emph{Discrete Comput.\ Geom.}\ 15 (1996), no.\ 3, 307--340.
\MR{1380397} (97f:52019).  \Zbl{853.52013}.					
Erratum.  \emph{Ibid.}\ 17 (1997), no.\ 3, 359.
\MR{1432070} (97k:52013).  \Zbl{853.52013}.					
\itemspace

\author{Paul H.\ Edelman \& Michael Saks} 
\authorspace
\bsgitem{1979a}{}   Group labelings of graphs.  \emph{J.\ Graph Theory} 3 (1979), 135--140.  
\MR{0530300} (80j:05071).  \Zbl{411.05059}.
\annot{} Given $\Gamma$ and abelian group $\fA$.  Vertex and edge labellings $\lambda: V \to \fA$ and $\eta: E \to \fA$ are ``compatible'' if $\lambda(v) = \sum_e \eta(e)$ for every vertex $v$, the sum taken over all edges incident with $v$.  $\lambda$ is ``admissible'' if it is compatible with some $\eta$.  Admissible vertex labellings are characterized (differently for bipartite and nonbipartite graphs) and the number of edge labelings compatible with a given vertex labelling is computed.  [Dual in a sense to \xref{Gimbel}{1988a}{John Gimbel}.]
\codes{\CWG,~\CVS:~\CBal(\CDual),~\CEnum}%
\itemspace

\authorx{Herbert Edelsbrunner}						%EDELSBRUNNER
\author{Herbert Edelsbrunner, G\"unter Rote, \& Emo Welzl}
\authorx{Herbert Edelsbrunner, Gunter Rote, \& Emo Welzl}
\authorspace
\bsgitem{1987a}{}  Testing the necklace condition for shortest tours and optimal factors in the plane.  
In: T.\ Ottmann, ed., \emph{Automata, Languages and Programming} (Proc., Karlsruhe, 1987), pp.\ 364--375.  \LNCS 267.  Springer, Berlin, 1987.
\MR{0912722} (88k:90065).  \Zbl{636.68042}.
\annot{} Summary of \xrefd{1989a}{Herbert Edelsbrunner, Gunter Rote, \& Emo Welzl}.
\codes{\Cpar:~\Cori,~\CGeom:~\CAlgor}%
\itemspace

\bsgitem{1989a}{}  Testing the necklace condition for shortest tours and optimal factors in the plane.  \emph{Theor.\ Computer Sci.}\ 66 (1989), 157--180.
\MR{1019083} (90i:90042).  \Zbl{686.68035}.
\annot{} \S5.1, ``Testing the feasibility of the linear program (2) or (1)'': The dual linear program (4) belongs to an oriented all-negative signed graph.  Treated by expanding it to the graphic LP belonging to the canonical covering graph.
\codes{\Cpar:~\Cori,~\CGeom:~\CAlgor}%
\itemspace

\author{Jack Edmonds} 						%EDMONDS
\authorspace
\authref  See also \hyperlink{Julian Araoz}{J.~Ar\'aoz} and \xref{E.L.~Lawler}{1976a}{Eugene L.\ Lawler}.
\authrefspace

\bsgitem{1965a}{}  Paths, trees, and flowers.  \emph{Canad.\ J.\ Math.}\ 17 (1965), 449--467.  
\MR{0177907} (31 \#2165).  \Zbl{132.20903} (132, p.\ 209c).
\annot{} Followed up by much work, e.g., \xref{Witzgall and Zahn}{1965a}{C.\ Witzgall \& C.T.\ Zahn, Jr.}; see \xref{Ahuja, Magnanti, and Orlin}{1993a}{Ravindra K.\ Ahuja, Thomas L.\ Magnanti, \& James B.\ Orlin} for some references.
\par
\codes{\Cpar:~\Cori:~\Cinc,~\CAlgor}%
\itemspace

\bsgitem{1965b}{}  Maximum matching and a polyhedron with $0,1$-vertices.  \emph{J.\ Res.\ Nat.\ Bur.\ Standards (U.S.A.) Sect.\ B} 69B (1965), 125--130.  
\MR{0183532} (32 \#1012).  \Zbl{141.21802} (141, p.\ 218b).				
\annot{} Alludes to the polyhedron of \xref{Edmonds and Johnson}{1970a}{Jack Edmonds \& Ellis L.\ Johnson}.
\par
\codes{\Cpar:~\Cori:~\CInc,~\CGeom}%
\itemspace

\author{Jack Edmonds \& Ellis L.\ Johnson} 
\authorspace
\bsgitem{1970a}{\dag\dag}  Matching: a well-solved class of integral linear programs.  In:  Richard Guy \emph{et al}., eds., \emph{Combinatorial Structures and Their Applications} (Proc.\ Calgary Int.\ Conf., Calgary, 1969), pp.\ 89--92.  Gordon and Breach, New York, 1970.  
\MR{0267898} (42 \#2799).  \Zbl{258.90032}.					
\annot{} Introduces ``bidirected graphs''.  A ``matching problem'' is an integer linear program with nonnegative and possibly bounded variables and otherwise only equality constraints, whose coefficient matrix is the incidence matrix of a bidirected graph.  No proofs.  [See \xref{Ar\'aoz, Cunningham, Edmonds, and Green-Kr\'otki}{1983a}{Julian Araoz, William H.\ Cunningham, Jack Edmonds, \& Jan Green-Krotki} for further work.]
\par
\codes{\Csg:~\COri:~\CInc,~\CAlgor,~\CGeom}%
\itemspace

\bsgitem{2003a}{}  Matching: a well-solved class of integral linear programs.  
In:  Michael J\"unger, Gerhard Reinelt, and Giovanni Rinaldi, eds., \emph{Combinatorial Optimization---Eureka, You Shrink!  Papers Dedicated to Jack Edmonds} (5th Int.\ Workshop, Aussois, France, 2001), pp.\ 27--30.  \LNCS 2570, Springer-Verlag, Berlin, 2003.
\MR{2163946}.  \Zbl{1024.90505}.						
\annot{} Readably typeset reprint of \xrefd{1970a}{Jack Edmonds \& Ellis L.\ Johnson}.
\codes{\Csg:~\COri:~\CInc,~\CAlgor,~\CGeom}%
\itemspace

\authorx{S.F.\ Edwards}						%EDWARDS
\author{S.F.\ Edwards \& P.W.\ Anderson}
\authorspace
\bsgitem{1975a}{}  Theory of spin glasses.  \emph{J.\ Phys.\ F} 5 (1975), 965--974.
Repr.\ in Marc M\'ezard, Giorgio Parisi, and Miguel Angel Virasoro,  \emph{Spin Glass Theory and Beyond}, pp.\ 89--98.  World Scientific Lecture Notes in Physics, Vol.\ 9.  World Scientific, Singapore, 1987.
\annot{} Spins $s_i$ are (unit) vectors; edge weights and signs are $J_{ij}\in\bbR$; the interaction between spins is $J_{ij} s_i \cdot s_j$ with the scalar product.  [This seminal article leads to the appearance of signed graphs in physics.  Later developments refine the definitions, e.g., to $J_{ij}=\pm1$, $J_{ij}$ Gaussian random variables, $s_i\in\signnos$, etc.  \Cf\ esp.\ \xref{Sherrington and Kirkpatrick}{1975a}{David Sherrington \& Scott Kirkpatrick}.]
\codes{\CPhys:~\Csg,~\Cwg}%
\itemspace

\author{Yoshimi Egawa}						%EGAWA
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{Richard Ehrenborg}						%EHRENBORG
\authorspace
\bsgitem{2001a}{}  Counting faces in the extended Shi arrangement $\hat{\cA}^r_n$.  \emph{Conference Proceedings of the 13th Int.\ Conference on Formal Power Series and Algebraic Combin.}\ (FPSAC, Tempe, Ariz., 2001), pp.\ 149--158.
\annot{} Preliminary version of \xrefd{2019a}{Richard Ehrenborg}.
\dated{11 Mar 2011} 
\par
\codes{\Cgg:~\Ccol,~\CInvar,~\Cmatrd,~\CGeom}%
\itemspace

\bsgitem{2019a}{}  Counting faces in the extended Shi arrangement.  \emph{Adv.\ Appl.\ Math.}\ 109 (2019), 55--64.
\MR{3954085}.  \Zbl{418.52010}.  
\annot{} Calculates the characteristic (Cor.\ 2.5) and, implicitly, Whitney-number polynomials of $[-r+1,r]\vec K_n$ in terms of its affinographic hyperplane representation, the extended Shi arrangement.  The object is to count faces of the latter by dimension and dimension of the infinite part.  [\xref{Zaslavsky}{2003a}{Thomas Zaslavsky} has a different generalization of Shi faces.]
\dated{11 Mar 2011, 13 Jun 2020} 
\codes{\Cgg:~\Ccol,~\CInvar,~\Cmatrd,~\CGeom}%
\itemspace

\author{Richard Ehrenborg \& Margaret A.\ Readdy}
\authorspace
\bsgitem{1998a}{}  On valuations, the characteristic polynomial, and complex subspace arrangements.  \emph{Adv.\ Math.}\ 134 (1998), 32--42.
\MR{1612379} (98m:52018).  \Zbl{906.52004}.						
\annot{} An abstract additive approach to the characteristic polynomial $p(\lambda)$, applied in particular (\S3: ``The divisor Dowling arrangement'') to ``divisor Dowling'' hyperplane arrangements'' $\cB(m)$ and certain interpolating arrangements.  
[Let $\Phi = {\fG_1}K_1 \cup \cdots \cup {\fG_n}K_n$, where $V(K_i) = \{v_1, \ldots, v_i\}$ and $\bbZ_m = \fG_1 \geq \cdots \geq \fG_n$.  $\cB(m)$ is the complex hyperplane representation of $\Phi\full$.  Thus, $p_{\cB(m)}(\lambda) = \chi_{\Phi\full}(\lambda)$, the chromatic polynomial.  This is computable via gain-graph coloring when $\fG_1$ is any finite group.  The same is true for the other arrangements treated herein.]
\dated{25 Apr 2009}
\codes{\Cgg:~\Cmatrd:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{1999a}{}  On flag vectors, the Dowling lattice, and braid arrangements.  \emph{Discrete Comput.\ Geom.}\ 21 (1999), 389--403.
\MR{1672984} (2000a:52037).  \Zbl{941.52021}.						
\annot{} Canonical complex hyperplane representation of the Dowling lattice of $\bbZ_k$.  
P.\ 395: an interesting $EL$-labelling of the Dowling lattice by a [disguised lexicographic] ordering of atoms.  
Thm.\ 4.9 is a recursive formula for its $\text{\bf ab}$-index.  
Thm.\ 5.2: the $\text{\bf c}$-$2\text{\bf d}$-index of the face lattices in case $k=1,2$, i.e., those of the real root system arrangements $A^*_n$ and $B^*_n$.  
\S6 presents a combinatorial description of the face lattice of $B^*_n$ [which it is interesting to compare with that in \xref{Zaslavsky}{1991b}{Thomas Zaslavsky}].  
Dictionary: very confusingly, ``region'' = face.
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2000a}{}  The Dowling transform of subspace arrangements.  \emph{J.\ Combin.\ Theory Ser.\ A} 91 (2000), 322--333.
\MR{1780026} (2001k:52038).  \Zbl{962.05005}.						
\annot{} The group expansion of an ordinary graph is generalized to expansion of an $\bbR^\times_{>0}$-gain graph by a finite cyclic subgroup of $\bbC^\times$, with correspondingly generalized formulas for the chromatic polynomial.  The computations are technically incorrect; they should be done by gain-graph coloring.  [Dictionary: ``directed cycle'' = circle (not directed).]  [Generalized in \xref{Koban}{2004b}{Lori Koban}.]
\codes{\CGG:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2009a}{}  Exponential Dowling structures.  \emph{European J.\ Combin.}\ 30 (2009), 311--326.
\MR{2460236} (2010a:06007).  \Zbl{1157.05002}.					
\annot{} A generalization of Stanley's exponential structures, based on the partition lattice, to Dowling lattices.  \S2 defines Dowling lattices via partial partitions (``zero block'' = set of non-partitioned elements).  \S3 defines Dowling exponential structures and gives compositional identities via generating functions.  \S4: generating-function identities for the M\"obius invariant; structures with restricted block sizes---especially, block sizes divisible by $r$ with $K$ non-partitioned elements where $K \geq k$ and $K \equiv k\ (\mod r)$.
\codes{\Cgg:~\Cmatrd:~\CInvar,~\CEnum,~\CExp}%
\itemspace

\author{Andrzej Ehrenfeucht}						%EHRENFEUCHT 
\authorspace 
\authref  See also \hyperlink{A.H.\ Deutz}{A.H.~Deutz}.
\authrefspace

\author{Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, \& Grzegorz Rozenberg}
\authorspace
\bsgitem{2000a}{}  Complexity issues in switching classes of graphs.
In: Hartmut Ehrig \etal, eds., \emph{Theory and Applications of Graph Transformations (TAGT'98)} (Proc.\ 6th Int.\ Workshop, Paderborn, 1998), pp.\ 59--70.  \LNCS 1764.  Springer-Verlag, Berlin, 2000.
\MR{1794790} (no rev).  \Zbl{958.68133}.						
\par
\codes{\CTG:~\CSw:~\CAlgor}%
\itemspace

\bsgitem{2000b}{}  Pancyclicity in switching classes.  \emph{Inform.\ Processing Lett.}\ 73 (2000), 153--156.
\MR{1755051} (2001c:05081).  
\annot{} Every switching class of graphs except that of the edgeless graph contains a pancyclic graph.  Thus Hamiltonicity is polynomial-time for graph switching classes.
\codes{\CTG:~\CSw,~\CAlgor}%
\itemspace

\bsgitem{2004a}{}  Embedding in switching classes with skew gains.  In:  Hartmut Ehrig \etal, eds., \emph{Graph Transformations} (Proc.\ 2nd Int.\ Conf.\ (ICGT 2004), Rome, 2004), pp.\ 257--270.  \LNCS 3256.  Springer, Berlin, 2004.
\MR{2198112} (no rev).  \Zbl{1116.68553}.  
\annot{} \Cf\ \xrefd{2006a}{Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, \& Grzegorz Rozenberg}.
\codes{\CGG:~\CSw}%
\itemspace

\bsgitem{2006a}{}  The embedding problem for switching classes of graphs.  Special issue on ICGT 2004.  \emph{Fund.\ Inform.}\ 74 (2006), no.\ 1, 115--134.
\MR{2282895} (2007h:68104).  \Zbl{1106.68053}.					
\codes{\CGG:~\CSw}%
\itemspace

\author{Andrzej Ehrenfeucht, Tero Harju, \& Grzegorz Rozenberg}
\authorspace
\bsgitem{1996a}{}  Group based graph transformations and hierarchical representations of graphs.  
In: J.\ Cuny, H.\ Ehrig, G.\ Engels and G.\ Rozenberg, eds., \emph{Graph Grammars and Their Application to Computer Science} (5th Int.\ Workshop, Williamsburg, Va., 1994), pp.\ 502--520.  \LNCS 1073.  Springer-Verlag, Berlin, 1996.
\MR{1422047} (97h:68097).  
\annot{} The ``heierarchical structure'' of a switching class of skew gain graphs based on $K_n$.
\codes{\Cgg:~\CKG:~\CSw}%
\itemspace

\bsgitem{1997a}{}  2-Structures---A framework for decomposition and transformation of graphs.  
In: Grzegorz Rozenberg, ed., \emph{Handbook of Graph Grammars and Computing by Graph Transformation.  Vol.\ 1: Foundations}, Ch.\ 6, pp.\ 401--478.  
World Scientific, Singapore, 1997.  
\MR{1480952} (99b:68006) (book).  \Zbl{908.68095} (book).				
\annot{} A tutorial (with some new proofs).  The relevant sections, based on papers of Ehrenfeucht and Rozenberg with and without Harju, are those about dynamic labeled 2-structures, i.e., complete graphs with twisted gains.  \S6.7: ``Dynamic labeled 2-structures''.  \S6.8: ``Dynamic $\ell$2-structures with variable domains''.  \S6.9: ``Quotients and plane trees''.  \S6.10: ``Invariants'', concerns certain switching invariants called ``free invariants'' when the gains are not twisted.  
\codes{\Cgg:~\CKG:~\Csw:~\CExp,~\CRef}
\itemspace

\bsgitem{1997b}{}  Invariants of inversive $2$-structures on groups of labels.  \emph{Math.\ Structures Computer Sci.}\ 7 (1997), 303--327.
\MR{1460397} (98g:20089).  \Zbl{882.05119}.						
\annot{} Given a gain graph $(K_n, \phi, \fG)$, a word $w$ in the oriented edges of $K_n$ has a gain $\phi(w)$; call this $\psi_w(\phi)$.  A ``free invariant'' is a $\psi_w$ that is an invariant of switching classes.
Thm.:  There is a number $d = d(K_n, \fG)$ such that the group of free invariants is generated by $\psi_w$ with $w = z_1^d \cdots z_k^d u_1 \cdots u_l$ where $w_i$ are triangular cycles (directed!) and $u_i$ are commutators.  
[The whole paper applies \emph{mutatis mutandis} to arbitrary graphs, the triangular cycles being replaced by any set of cycles containing a fundamental system.]
Dictionary:  ``Inversive $2$-structure'' = gain graph based on $K_n$.
\codes{\Cgg:~\CKG:~\CSw,~\CInvar}%
\itemspace

\bsgitem{1999a}{}  \emph{The Theory of 2-Structures:  A Framework for Decomposition and Transformation of Graphs}.  World Scientific, Singapore, 1999.
\MR{1712180} (2001i:05001).  \Zbl{981.05002}.					
\codes{\Cgg:~\CKG:~\Csw:~\CExp,~\CRef}
\itemspace

\bsgitem{2004a}{}  Transitivity of local complementation and switching on graphs.  \emph{Discrete Math.}\ 278 (2004), 45--60.
\MR{2035389} (2005d:05074).  \Zbl{1033.05052}.
\annot{} Let antilocal complementation at $v$ mean reversing the edges except within the neighborhood of $v$.  Let strictly antilocal complementation mean reversing the edges except within the closed neighborhood of $v$.  Every simple graph of order $n$ can be converted to every other one by antilocal complementations, and also by stricly antilocal complementations.
\codes{\CTG}%
\itemspace

\author{Andrzej Ehrenfeucht \& Grzegorz Rozenberg}
\authorspace
\bsgitem{1993a}{}  An introduction to dynamic labeled 2-structures.  
In:  Andrzej M.\ Borzyszkowski and Stefan Soko{\l}owski, eds., \emph{Mathematical Foundations of Computer Science 1993} (Proc., 18th Int.\ Sympos., MFCS '93, Gda\'nsk, 1993),  pp.\ 156--173.  \LNCS 711.  Springer-Verlag, Berlin, 1993.  
\MR{1265062} (95j:68126).  
\annot{} Extended summary of \xrefd{1994a}{Andrzej Ehrenfeucht \& Grzegorz Rozenberg}.  
\codes{\CGG(\CGen):~\CKG:~\CSw,~\CStr}%
\itemspace
 
\bsgitem{1994a}{}  Dynamic labeled 2-structures.  \emph{Math.\ Structures Comput.\ Sci.}\ 4 (1994), 433--455.  
\MR{1322183} (96j:68144).  \Zbl{829.68099}.					
\annot{} They prove that a complicated definition of ``reversible dynamic labeled 2-structure'' $G$ amounts to a complete graph with a set, closed under switching, of twisted gains in a gain group $\Delta$.  The twist is a gain-group automorphism $\alpha$ such that $\lambda(e;x,y) = [\alpha\lambda(e;y,x)]\inv$, $\lambda$ being the gain function.  
Dictionary:  their ``domain'' $D =$ vertex set, ``labeling function'' $\lambda$ (or equivalently, $g$) = gain function, ``alphabet'' = gain group, ``involution'' $\delta = \alpha \circ$ inversion, ``$\delta$-selector'' $\hat S =$ switching function, ``transformation induced by $\hat S$'' = switching by $\hat S$; a ``single axiom'' d.l.\ 2-structure consists of a single switching class.  
\annotctd{} Further, they investigate ``clans'' of $G$.  Given $g$ (i.e., $\lambda$), deleting identity-gain edges leaves isolated vertices (``horizons'') and forms connected components, any union of which is a ``clan'' of $g$.  A clan of $G$ is any clan of any $g \in G$.  
\codes{\CGG(\CGen):~\CKG:~\CSw,~\CStr}%
\itemspace

\bsgitem{1994b}{}  Dynamic labeled 2-structures with variable domains.  
In:  J.\ Karhum\"aki, H.\ Maurer, and G.\ Rozenberg, eds., \emph{Results and Trends in Theoretical Computer Science} (Proc.\ Colloq.\ in Honor of Arto Salomaa, Graz, 1994), pp.\ 97--123.  \LNCS 812.  Springer-Verlag, Berlin, 1994.  
\MR{1286959} (95m:68128).  
\annot{} Combinations and decompositions of complete graphs with twisted gains.
\codes{\CGG(\CGen):~\CKG:~\CStr,~\CSw}%
\itemspace

\authorx{George C.M.A.\ Ehrhardt}					%EHRHARDT
\author{George C.M.A.\ Ehrhardt, Matteo Marsili, \& Fernando Vega-Redondo}
\authorspace
\bsgitem{2005a}{}  On the rise and fall of networked societies.  
In: Joaqu{\i\i}n Marro, Pedro L.\ Garrido, and Miguel A.\ Mu\~noz, eds., \emph{Modeling Cooperative Behavior in the Social Sciences} (Proc.\ 8th Granada Lect., Granada, Spain, 2005), pp.\ 96--103.  AIP Conf.\ Proc., Vol.\ 779.  Amer.\ Inst.\ Physics, Melville, N.Y., 2005.
\arXiv{physics/0505019}.  
\annot{} \S~III, ``The effect of negative links'':  A random model where positive edges may appear, and may change to negative.  Negative edges disappear over time.
\dated{12 Aug 2012} 
\codes{\CSG:~\CPsS:~\CRand,~\CPhys}
\itemspace

\author{K.\ Ehsani}							%EHSANI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{M.\ Einollahzadeh}						%EINOLLAHZADEH
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Kurt Eisemann}						%EISEMANN
\authorspace
\bsgitem{1964a}{}  The generalized stepping stone method for the machine loading model.  \emph{Management Sci.}\ 11 (1964/65), No.\ 1 (Sept., 1964), 154--176.  
\Zbl{136.13901} (136, p.\ 139a).  
\codes{\CGN:~\CInc,~\CMatrd(\Cbases)}%
\itemspace

\authorx{Joyce Elam}			 			%ELAM
\author{Joyce Elam, Fred Glover, \& Darwin Klingman}
\authorspace
\bsgitem{1979a}{} A strongly convergent primal simplex algorithm for generalized networks.  \emph{Math.\ Operations Res.}\ 4 (1979), 39--59.  
\MR{0543608} (81g:90049).  \Zbl{422.90081}.
\par
\codes{\CGN:~\CMatrd(\Cbases),~\CInc}%
\itemspace

\author{Adrien Elena}						%ELENA
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{David P.\ Ellerman} 						%ELLERMAN
\authorspace
\bsgitem{1984a}{} Arbitrage theory:  A mathematical introduction.  \emph{SIAM Rev.}\ 26 (1984), 241--261. 
\MR{0738931} (85g:90024).  \Zbl{534.90014}.
\par
\codes{\CGG:~\CBal,~\CInc,~\CFlows:~\CAppl,~\CRef}%
\itemspace

\author{M.N.\ Ellingham} 						%ELLINGHAM
\authorspace
\bsgitem{1991a}{}  Vertex-switching, isomorphism, and pseudosimilarity.  \emph{J.\ Graph Theory} 15 (1991), 563--572.  
\MR{1133811} (92g:05136).  \Zbl{802.05057}.					
\annot{} Main theorem (\S2) characterizes, given two signings of $K_n$ (where $n$ may be infinite) and a vertex set $S$, when switching $S$ makes the signings isomorphic.  [\Problem\ 1.  Generalize to other underlying graphs.  \Problem\ 2.  Prove an analog for bidirected $K_n$'s.]  
\annotctd{} A corollary (\S3) characterizes when vertices $u, v$ of $\Sigma = (K_n, \sigma)$ satisfy $\Sigma^{\{u\}} \cong \Sigma^{\{v\}}$ and discusses when in addition no automorphism of $\Sigma$ moves $u$ to $v$.  All is done in terms of Seidel (graph) switching (here called ``vertex-switching'') of unsigned simple graphs.
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1996a}{}  Vertex-switching reconstruction and folded cubes.  \emph{J.\ Combin.\ Theory Ser.\ B} 66 (1966), 361--364.  
\MR{1376057} (96i:05120).  \Zbl{856.05071}.					
\annot{} Deepens the folded-cube theory of \xref{Ellingham and Royle}{1992a}{M.N.\ Ellingham \& Gordon F.\ Royle}.  Nicely generalizing \xref{Stanley}{1985a}{Richard P.\ Stanley}, the number of subgraphs of a signed $K_n$ that are isomorphic to a fixed signed $K_m$ is reconstructible from the $s$-vertex switching deck if the Krawtchouk polynomial $K_s^n(x)$ has no even zeros between $0$ and $m$.  (Closely related to \xref{Krasikov and Roditty}{1992a}{I.\ Krasikov \& Y.\ Roditty}, Theorems 5 and 6.)  Remark 4: balance equations (\xref{Krasikov and Roditty}{1987a}{I.\ Krasikov \& Y.\ Roditty}) and Krawtchouk polynomials both reflect properties of folded cubes.  All is done in terms of Seidel switching of unsigned simple graphs.  
[It seems clear that the folded cube appears because it corresponds to the effect of switchings on signatures of $K_n$ (or any connected graph), since switching by $X$ and $X^c$ have the same effect.  For the bidirected case (Problem 2 under \xref{Stanley}{1985a}{Richard P.\ Stanley}), the unfolded cube should play a similar role.  \Question.  When treating a general underlying graph $\Gamma$, will a polynomial influenced by $\Aut \Gamma$ replace the Krawtchouk polynomial?]
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\author{M.N.\ Ellingham \& Gordon F.\ Royle}
\authorspace
\bsgitem{1992a}{}  Vertex-switching reconstruction of subgraph numbers and triangle-free graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 54 (1992), 167--177.  
\MR{1152444} (93d:05112).  \Zbl{695.05053}, (\Zbl{748.05071}).			
\annot{} Reconstruction of induced subgraph numbers of a signed $K_n$ from the $s$-vertex switching deck, dependent on linear transformation and thence Krawtchouk polynomials as in \xref{Stanley}{1985a}{Richard P.\ Stanley}.  The role of those polynomials is further developed.  Done in terms of Seidel switching of unsigned simple graphs, with the advantage of reconstructing arbitrary subgraph numbers as well.  A gap is noted in \xref{Krasikov and Roditty}{1987a}{I.\ Krasikov \& Y.\ Roditty}, proof of Lemma 2.5.  
[Methods and results are closely related to \xref{Krasikov}{1988a}{I.\ Krasikov} and \xref{Krasikov and Roditty}{1987a}{I.\ Krasikov \& Y.\ Roditty}, \xrefd{1992a}{I.\ Krasikov \& Y.\ Roditty}.]
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\author{Joanna A.\ Ellis-Monaghan}					%ELLIS-MONAGHAN
\authorspace
\authref  See also \hyperlink{Lowell Abrams}{L.~Abrams}.
\authrefspace

\author{Joanna A.\ Ellis-Monaghan \& Iain Moffatt}
\authorspace
\bsgitem{2011a}{}  The Tutte--Potts connection in the presence of an external magnetic field.  \emph{Adv.\ Appl.\ Math.}\ 47 (2011), no.\ 4, 772--782. 
\MR{2832375}.  \Zbl{1232.05100}.  \arXiv{1005.5470} (extended version).			
\codes{\Csg:~\CTop,~\CDual}\codesctd{\CSGc:~\CGen:~\CInvar}
\itemspace

\bsgitem{2012a}{}  Twisted duality for embedded graphs.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 364 (2012), no.\ 3, 1529--1569.
\MR{2869185} (2012m:05102).  \Zbl{1238.05067}.  \arXiv{0906.5557}.			
\par
\codes{\Csg:~\Cori:~\CTop,~\CDual}%
\itemspace

\bsgitem{2013a}{}  A Penrose polynomial for embedded graphs.  \emph{European J.\ Combin.}\ 34 (2013), no.\ 2, 424--445.
\MR{2994409}.  \Zbl{1254.05080}.  \arXiv{1106.5279}.				
\par
\codes{\Csg:~\CTop,~\CDual,~\CInvar}%
\itemspace

\bsgitem{2013b}{}  \emph{Graphs on Surfaces: Dualities, Polynomials, and Knots}.  SpringerBriefs in Mathematics.  Springer, New York, 2013.
\MR{3086663}.  \Zbl{1283.57001}.						
\par
\codes{\CSG:~\CTop,~\CDual,~\CInvar}
\itemspace

\bsgitem{2015a}{}  Evaluations of topological Tutte polynomials.  \emph{Combin.\ Prob.\ Computing} 24 (2015), no.\ 3, 556--583. 
\MR{3326433}.  \Zbl{1371.05134}.  \arXiv{1108.3321}.				
\par
\codes{\CSG:~\CTop,~\CDual,~\CInvar}
\itemspace

\author{Joanna A.\ Ellis-Monaghan \& Irasema Sarmiento}
\authorspace
\bsgitem{2011a}{}  A recipe theorem for the topological Tutte polynomial of Bollob\'as and Riordan.  \emph{European J.\ Combin.}\ 32 (2011), no.\ 6, 782--794.
\MR{2821551} (2012j:05220).  \Zbl{1223.05039}.  \arXiv{0903.2643}.			
\codes{\Csg:~\CTop,~\CDual,~\CInvar}%
\itemspace

\author{Joanna Ellis-Monaghan \& Lorenzo Traldi}
\authorspace
\bsgitem{2006a}{}  Parametrized Tutte polynomials of graphs and matroids.  \emph{Combin.\ Probab.\ Comput.}, 15 (2006), no.\ 6, 835--854.
\MR{2271830} (2007j:05038).  \Zbl{1108.05024}.					
\annot{} A slight generalization of the multiplicative property of the strong parametrized Tutte functions in \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}, and a necessary and sufficient condition for their existence on an arbitrary minor-closed class of matroids inspired by \xref{Bollob\'as and Riordan}{1999a}{Bela Bollobas \& Oliver Riordan}.  Also, specialization to graphs.  Dictionary:  ``Tutte polynomial'' means a function, not a polynomial.
\dated{rev 26 Jun 2022}
\codes{\CSGc:~\CGen:~\CInvar}%
\itemspace


\authorx{Abdelhakim El Maftouhi}				%EL MAFTOUHI	%MAFTOUHI
\authorx{Hakim El Maftouhi}
\author{Abdelhakim El Maftouhi, Ararat Harutyunyan, and Yannis Manoussakis}
\authorspace
\bsgitem{2014a}{}  (as Hakim El Maftouhi, Ararat Harutyunyan, and Yannis Manoussakis)  Balance in random signed graphs.  In:  \emph{Bordeaux Graph Workshop 2014}, pp.\ 43--44.  LaBRI, Bordeaux, 2014.  
URL \url{http://bgw.labri.fr/2014/bgw2014-booklet.pdf}
\annot{} Extended abstract of \xref{El Maftouhi, Manoussakis, \& Megalakaki}{2012a}{A.\ El Maftouhi, Y.\ Manoussakis, \& O.\ Megalakaki} and \xrefd{2015a}{Abdelhakim El Maftouhi, Ararat Harutyunyan, and Yannis Manoussakis}.  ``Weak balance'' = clusterability.
\dated{19 Mar 2017} 
\codes{\CSG:~\CRand:~\CBal,~\CClu}%
\itemspace

\bsgitem{2015a}{}  Weak balance in random signed graphs.  \emph{Internet Math.}\  11 (2015), no.\ 2, 143--154.
\MR{3316860} (no rev).  
\codes{\CSG:~\CRand:~\CClu}%
\itemspace

\author{A.\ El Maftouhi, Y.\ Manoussakis, \& O.\ Megalakaki}
\authorspace
\bsgitem{2012a}{}  Balance in random signed graphs.  \emph{Internet Math.}\ 8 (21012), no.\ 4, 364--380.
\MR{3009999}.  \Zbl{1258.05110}.  
\codes{\CSG:~\CBal,~\CFr:~\CRand}%
\itemspace

\author{Amani Elrayes}						%ELRAYES
\authorspace
\authref  See \hyperlink{Shokry Nada}{S.~Nada}.
\itemspace

\author{Ashraf Elrokh}						%ELROKH
\authorspace
\authref  See \hyperlink{Shokry Nada}{S.~Nada}.
\itemspace

\authorx{Amine El Sahili}						%EL SAHILI	%SAHILI
\author{Amine El Sahili \& Maria Abi Aad}
\authorspace
\bsgitem{2018a}{}  Antidirected Hamiltonian paths and directed cycles in tournaments.  \emph{Discrete Math.}\ 341 (2018), 2018--2027.
\MR{3802155}.  \Zbl{1387.05136}.  
\annot{} Antidirected means coherent in the poise gains of a digraph.  [\Cf\ \xref{Diwan, Frye, Plantholt, and Tipnis}{2011a}{Ajit A.\ Diwan, Josh B.\ Frye, Michael J.\ Plantholt, \& Shailesh K.\ Tipnis}.]  [\Question.  How does this generalize to bidirected graphs?]
\dated{30 May 2018} 
\par
\codes{\Cgg:~\CKG:~\CStr}\codesctd{\Csg:~\CKG:~\Cpar:~\COri}%
\itemspace

\authorx{D.\ Emanuel}						%EMANUEL
\author{D.\ Emanuel \& A.\ Fiat}
\authorspace
\authref  See also \hyperlink{Erik D.\ Demaine}{E.~Demaine}.
\authrefspace

\bsgitem{2003a}{}  Correlation clustering---Minimizing disagreements on arbitrary weighted graphs.  
In: \emph{Algorithms---ESA 2003} (Budapest, 2003), pp.\ 208--220.  \LNCS 2832.  Springer, Berlin, 2003.
\MR{2085454}.  \Zbl{1266.68228}.						
\annot{} Conference version of \xref{Demaine, Emanuel, Fiat, and Immorlica}{2006a}{Erik D.\ Demaine, Dotan Emanuel, Amos Fiat, \& Nicole Immorlica}.
\dated{13 Sept 2009} 
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}%
\itemspace

\author{Frank Emmert-Streib}						%EMMERT-STREIB
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.H.~Yu}.
\itemspace

\author{German Andres Enciso}					%ENCISO
\authorspace
\authref  See also \hyperlink{Bhaskar DasGupta}{B.~DasGupta}.
\authrefspace

\author{German Enciso \& Eduardo D.\ Sontag}
\authorspace
\bsgitem{2005a}{}  Monotone systems under positive feedback: multistability and a reduction theorem.  \emph{Systems Control Lett.}\ 54 (2005), no.\ 2, 159--168.
\MR{2109582} (2005g:93055).  \Zbl{1129.93398}.  
\codes{\CDyn:~\CSD}%
\itemspace

\bsgitem{2006a}{}  Global attractivity, i/o monotone small-gain theorems, and biological delay systems.  \emph{Discrete Continuous Dynamical Systems} 14 (2006), no.\ 3, 549--578.
\MR{2171727} (2006g:93128).  \Zbl{1111.93071}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2008a}{}  Monotone bifurcation graphs.  \emph{J.\ Biol.\ Dynamics} 2 (2008), no.\ 2, 121--139.
\MR{2427522} (2009e:34092).  \Zbl{1141.92005}.					
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Mechthild Enderle}						%ENDERLE
\authorspace
\authref  See \hyperlink{J.D.\ Noh, H.\ Rieger, M.\ Enderle, \& K.\ Knorr}{J.D.~Noh}.
\itemspace

\author{Shin-ichi Endoh}						%ENDOH
\authorspace
\authref  See \hyperlink{Tota Nakamura}{T.~Nakamura}.
\itemspace

\authorx{Gernot M.\ Engel}		 				%ENGEL
\author{Gernot M.\ Engel \& Hans Schneider}
\authorspace
\bsgitem{1973a}{}  Cyclic and diagonal products on a matrix.  \emph{Linear Algebra Appl.}\ 7 (1973), 301--335.  
\MR{0323804} (48 \#2160).  \Zbl{289.15006}.
\codes{\Cgg:~\CSw}%
\itemspace

\bsgitem{1975a}{}  Diagonal similarity and equivalence for matrices over groups with 0.  \emph{Czechoslovak Math.\ J.}\ 25(100) (1975), 389--403.
\MR{0396615} (53 \#477).  \Zbl{329.15007}.
\codes{\Cgg:~\CSw}%
\itemspace

\bsgitem{1980a}{} Matrices diagonally similar to a symmetric matrix.  \emph{Linear Algebra Appl.}\ 29 (1980), 131--138.  
\MR{0562753} (81k:15017).  \Zbl{432.15014}.
\codes{\Cgg:~\CSw}%
\itemspace

\authorx{Michael Engquist}						%ENGQUIST
\author{Michael Engquist \& Michael D.\ Chang}
\authorspace
\bsgitem{1985a}{}  New labeling procedures for the basis graph in generalized networks.  \emph{Operations Res.\ Lett.}\ 4 (1985), no.\ 4, 151--155.
\MR{0821177} (87j:05137).  \Zbl{572.90095}.					
\annot{} Generalizing pure-network procedures to get fast computations.
\dated{4 Sept 2010} 
\codes{\CGN:~\CMatrd(\CBases)}%
\itemspace

\author{R.C.\ Entringer}						%ENTRINGER
\authorspace
\bsgitem{1985a}{}  A short proof of Rubin's block theorem.  
In:  B.R.\ Alspach and C.D.\ Godsil, eds., \emph{Cycles in Graphs}, pp.\ 367--368.  Ann.\ Discrete Math., Vol.\ 27.  North-Holland Math.\ Stud., Vol.\ 115.  
North-Holland, Amsterdam, 1985. 
\MR{0821538} (87f:05144).  \Zbl{576.05037}.
\annot{} See \xref{Erd\H{o}s, Rubin, \& Taylor}{1980a}{Paul Erdos, Arthur L.\ Rubin, \& Herbert Taylor}. 
\codes{\Cpar:~\Cbal}%
\itemspace

\author{H.\ Era} 							%ERA
\authorspace
\authref  See \hyperlink{Jin Akiyama}{J.~Akiyama}.
\itemspace

\author{Paul Erdos [Paul Erd\"os, Paul Erd\H{o}s, P\'al Erd\H{o}s]}
\authorx{Paul Erd\"os}
\authorx{Paul Erdos}
\authorspace
\authref  See also \hyperlink{Bela Bollobas}{B.~Bollob\'as} and \hyperlink{Richard A.\ Duke}{R.A.~Duke}.
\authrefspace

\bsgitem{1996a}{}  On some of my favourite theorems.  
In: D.\ Mikl\'os, V.T.\ S\'os and T.\ Sz\H{o}nyi, eds., \emph{Combinatorics, Paul Erd\H{o}s is Eighty} (Papers from the Int.\ Conf.\ on Combinatorics, Keszthely, 1993), Vol.\ 2, pp.\ 97--132.  Bolyai Soc.\ Math.\ Studies, 2.  J\'anos Bolyai Mathematical Society, Budapest, 1996.
\MR{1395856} (97g:00002).  \Zbl{0853.11001}.
\annot{} P.\ 119 mentions the theorem of \xref{Duke, Erd\H{o}s, \& R\"odl}{1991a}{Richard A.\ Duke, Paul Erdos, \& Vojtech Rodl} on even circles.
\annotctd{} Pp.\ 120--121 mention (amongst similar problems) a theorem of Erd\H{o}s and Hajnal (source not stated):  Every all-negative signed graph with chromatic number $\aleph_1$ contains every finite bipartite graph [i.e., every finite, balanced, all-negative signed graph].  [\Problem.  Find generalizations to signed graphs.  For instance:  \Conjecture.  Every signed graph with chromatic number $\aleph_1$, that does not become antibalanced upon deletion of any finite vertex set, contains every finite, balanced signed graph up to switching equivalence.]
[MR: ``this is one of the best collections of problems that Erdos has published.'']
\codes{\Cpar:~\Cbal:~\CExp,~\CRef}%
\itemspace

\author{P.\ Erd\"os, R.J.\ Faudree, A.\ Gy\'arf\'as, \& R.H.\ Schelp}
\authorx{P.\ Erdos, R.J.\ Faudree, A.\ Gyarfas, \& R.H.\ Schelp}
\authorspace
\bsgitem{1991a}{}  Odd cycles in graphs of given minimum degree.  
In:  Y.\ Alavi, G.\ Chartrand, O.R.\ Oellermann, and A.J.\ Schwenk, eds., \emph{Graph Theory, Combinatorics, and Applications} (Proc.\ Sixth Quadrennial Int.\ Conf.\ Theory Appl.\ Graphs, Kalamazoo, Mich., 1988), Vol.\ 1,  pp.\ 407--418.  Wiley, New York, 1991.  
\MR{1170794} (93d:05085).  \Zbl{840.05050}.
\annot{} A large, nonbipartite, $2$-connected graph with large minimum degree contains a circle of given odd length or is one of a single type of exceptional graph.  [\Question.  Can this be generalized to negative circles in unbalanced signed graphs?]
\codes{\Cpar,~\Csg:~\CCircles,~\CXtreml}%
\itemspace

\author{P.\ Erd\H{o}s, E.\ Gy\H{o}ri, \& M.\ Simonovits}
\authorx{P.\ Erdos, E.\ Gyori, \& M.\ Simonovits}
\authorspace
\bsgitem{1992a}{}  How many edges should be deleted to make a triangle-free graph bipartite?
In: G.\ Hal\'asz, L.\ Lov\'asz, D.\ Mikl\'os, and T.\ Sz\"onyi, eds., \emph{Sets, Graphs and Numbers} (Proc., Budapest, 1991), pp.\ 239--263.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, Vol.\ 60.  J\'anos Bolyai Math.\ Soc., Budapest, and North-Holland, Amsterdam, 1992.
\MR{1218193} (94b:05104).  \Zbl{785.05052}.
\annot{} Assume $|\Sigma|$ simple of order $n$ and $\nsupseteq$ a fixed graph $\Delta$.  Results on frustration index $l$ of antibalanced $\Sigma$ if $\Delta$ is $3$-chromatic, esp.\ $C_3$.  
Thm.: If $\#E > n^2/5 - o(n^2)$, then $l(\Sigma) < n^2/25 - o(n^2)$.  \Conjecture\ (Erd\H{o}s):  For $\Delta = C_3$ the hypothesis on $\#E$ is unnecessary.
[\Question\ 1(a).  Is the answer different when $\Sigma$ need not be antibalanced?
\Question\ 2(a).  Exclude a fixed signed graph whose signed chromatic number $= 1$.
\Question\ 3(a).  In particular, exclude $-K_3$.
\Question\ 4(a).  Exclude $-K_l$.
\Question\ 5(a).  Exclude an unbalanced $C_l$.
\Questions\ 1--5(b).  Even if $l(\Sigma)$ cannot be estimated, is there always an extremal graph that is antibalanced---as when no graph is excluded, by \xref{Petersdorf}{1966a}{M.\ Petersdorf}?]
\codes{\Cpar:~\CXtreml}%
\itemspace

\author{P.\ Erd\"os \& L.\ P\'osa}%	% VERIFY authors' names.
\authorx{P.\ Erdos \& L.\ Posa}%	% VERIFY authors' names.
\authorspace
\bsgitem{1965a}{}  On independent circuits contained in a graph.  \emph{Canad.\ J.\ Math.}\ 17 (1965) 347--352.
\MR{0175810} (31 \#86).  \Zbl{129.39904}.
\annot{} An upper bound on $l_0$, the vertex frustration number, in terms of vertex packing of unbalanced circles, in the contrabalanced case.  \Problem.  Find an analog for signed graphs and a generalization to biased graphs.
\par
\codes{\Cgg:~\Cfr,~\CCircles}%
\itemspace

\author{Paul Erd\"os, Arthur L.\ Rubin, \& Herbert Taylor}
\authorx{Paul Erdos, Arthur L.\ Rubin, \& Herbert Taylor}
\authorspace
\bsgitem{1980a}{}  Choosability in graphs.  
In:  \emph{Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing} (Arcata, Calif., 1979), pp.\ 125--157.  Congressus Numer., XXVI.  
Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1980.  
\MR{0593902} (82f:05038).  \Zbl{469.05032}.
\annot{} Rubin's block theorem (Thm.\ R, p.\ 136): a block graph, not complete or an odd circle, contains an induced even circle with at most one chord.  [\Cf\ \xref{Entringer}{1985a}{R.C.\ Entringer}.]  
[\Question.  Does this generalize to signed graphs, Rubin's block theorem being the antibalanced case?  Rubin's $2$-choosability theorem, p.\ 132, is also tantalizingly reminiscent of antibalanced graphs, but in reverse.]
\codes{\Cpar:~\CStr,~\Cbal}%
\itemspace

\author{Carolyn Eschenbach}						%ESCHENBACH
\authorspace
\authref  See also \hyperlink{Zhongshan Li}{Z.-S.~Li} and \hyperlink{Jeffrey Stuart}{J.~Stuart}.
\authrefspace

\bsgitem{1993a}{}  ldempotence for sign-pattern matrices.  \emph{Linear Algebra Appl.}\ 180 (1993), 153--165.
\MR{1206414} (94b:15010).  \Zbl{777.05032}.					
\annot{} Irreducible $A$ is sign-idempotent iff every entry is $+$.  Necessary and sufficient conditions for reducible $A$ to be sign-idempotent; in particular, it need not have nonnegative entries, but $V$ must partition into $V_i$ inducing no arcs or an all-positive complete symmetric digraph with loops.  
[Counterexamples in Rong Huang, Sign idempotent sign patterns similar to nonnegative sign patterns, \emph{Linear Algebra Appl.}\ 428 (2008), 2524--2535.  \MR{2416567} (2009c:15010).  \Zbl{1144.15014}.]
\dated{29 Sept 2012} 
\codes{\CQM:~\CSD:~\CAdj}%
\itemspace

\bsgitem{1993b}{}  Sign patterns that require exactly one real eigenvalue and patterns that require $n-1$ nonreal eigenvalues.  \emph{Linear Multilinear Algebra} 35 (1993), no.\ 3-4, 213--223.
\MR{1308691} (95k:15009).  \Zbl{787.15008}.  
\codes{\CQM:~\Csd}%
\itemspace

\author{Carolyn A.\ Eschenbach, Frank J.\ Hall \& Charles R.\ Johnson}
\authorspace
\bsgitem{1993a}{}  Self-inverse sign patterns.  In: Richard A.\ Brualdi, Shmuel Friedland, and Victor Klee, eds., \emph{Combinatorial and Graph-Theoretical Problems in Linear Algebra} (IMA Workshop, Minneapolis, 1991), pp.\ 245--256.
IMA Volumes in Mathematics and its Applications, 50.  Springer-Verlag, New York, 1993.
\MR{1240969} (94e:15004).  \Zbl{792.15008}.
\annot{} Sign matrices whose sign patterns are self-inverse are essentially adjacency matrices of signed graphs and are very few.
\dated{13 Apr 2009}
\codes{\Csg,~\CQM}%
\itemspace

\author{Carolyn A.\ Eschenbach, Frank J.\ Hall, Charles R.\ Johnson, \& Zhongshan Li}
\authorspace
\bsgitem{1997a}{}  The graphs of the unambiguous entries in the product of two $(+,-)$-sign pattern matrices.  \emph{Linear Algebra Appl.}\ 260 (1997), 95--118.
\MR{1448352} (98e:05075).  \Zbl{881.05089}.					
\annot{} $A,B$ are nowhere-zero sign-pattern matrices.  $(AB)_{ij}$ may be necessarily $+$, $-$, or ambiguous [\xref{Abelson and Rosenberg's}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} $p,n,a$].  Let $\cR =$ set of rows, $\cC =$ set of columns.  The graph $G(A,B) \subseteq K_{\cR(A),\cC(B)}$ has an edge $ij$ for each unambigous entry in $AB$.  The digraph $D(A^2)$ has an arc $(i,j)$ for each unambiguous entry in $A^2$.  Thm.\ 3.2:  $\Gamma$ is a $G(A,B)$ iff it is the disjoint union of bicliques and isolated vertices.  Characterizing $D(A^2)$ seems hard.  Results on special cases.  [$D$ and $G$ are signed by $p,n$.  Thm.\ 5.11:  $D(A^2)$, if a circle, is balanced iff the circle is positive.  \S6, ``Characterization of permutation graphs in $D_n$'', i.e., $D(A^2)$ that are permutation graphs.  
[\Problem.  Investigate $D(A^2)$ and $G(A,B)$ signed by $p,n$.  \Problem.  Generalize to allow 0 entries (thus working over Abelson--Rosenberg's algebra $\{p,n,a,o\}$.)] 
Dictionary:  ``signature similarity'' of matrices = switching of digraph, ``negative matching'' = entry $n$ in $AB$ = negative edge in $G(A,B)$.  
\dated{4 Nov 2011} 
\par
\codes{\CQM:~\Csd,~\Csw}%
\itemspace

\author{Carolyn A.\ Eschenbach, Frank J.\ Hall \& Zhongshan Li}
\authorspace
\bsgitem{1998a}{}  From real to complex sign pattern matrices.  \emph{Bull.\ Austral.\ Math.\ Soc.}\ 5 (1998), 159--172.
\MR{1623848} (99d:15003).  \Zbl{951.15021}.					
\annot{} $S := \{\alpha+i\beta : \alpha,\beta = 0,\pm\}$, the set of complex signs.  A  complex sign pattern matrix has complex signs as entries.  If it is square it has a digraph $D(A)$ with complex signs as gains.  
\S3, ``Cyclic nonnegativity'':  Cycles with gain $\pm$.  Switching (via matrices) by $\pm, \pm\mathrm{i}$.  Cor.\ 3.2:  $D(A)$ is balanced iff it switches to all $+$.  Thm.\ 3.3:  iff $D$ is balanced and $D(A+A^*)$ switches to all $+$.  
\S4, ``Stability'':  Lem.\ 4.1:  If $A$ is sign stable, every digon has real or purely imaginary gain.  Lem.\ 4.2:  If $A$ is sign stable it is sign nonsingular.  Thm.\ 4.4 (generalizing \xref{Quirk and Ruppert}{1965a}{James Quirk \& Richard Ruppert} and \xref{Maybee and Quirk}{1969a}{John Maybee \& James Quirk}):  Assume every vertex has a negative loop.  Then $A$ is sign stable iff all digons are negative and no longer cycles exist.  
Thm.\ 5.2:  Similar, for ray stability.  
Dictionary:  ``cyclically nonnegative'' = all cycle gains are $+$; ``cyclically positive'' = cyclically nonnegative and no zeros.
\dated{4 Nov 2011} 
\par
\codes{\CQM:~\CGen;~\Cgg,~\Csw;~\CSta}%
\itemspace

\author{Carolyn A.\ Eschenbach \& Charles R.\ Johnson}
\authorspace
\bsgitem{1990a}{\dag}  A combinatorial converse to the Perron-Frobenius theorem.  \emph{Linear Algebra Appl.}\ 136 (1990), 173--180.
\MR{1061544} (91d:15047).  \Zbl{739.15004}.					
\annot{} ``Perron property'':  spectral radius is an eigenvalue.  Thm.\ 1.1:  Sign pattern matrix $A$ requires Perron property iff it is cyclically nonnegative.  Some sign patterns that allow Perron property.  
Dictionary:  ``Cyclically nonnegative matrix'': its signed digraph is cycle balanced.  
\dated{20 Oct 2020} 
\codes{\CQM,~\CSD:~\CBal(\CCycles),~\Csw}%
\itemspace

\author{Carolyn A.\ Eschenbach \& Zhongshan Li}
\authorspace
\bsgitem{1999a}{}  Potentially nilpotent sign pattern matrices.  \emph{Linear Algebra Appl.}\ 299 (1999), 81--99.
\MR{1723710} (2000i:15043).  \Zbl{941.15012}.					
\annot{} Matrices with tree digraph.  Cycle sign = sign product, p.\ 82.  2-cycle signs in Thms.\ 5.3 (proof), 5.5, 5.7 (proof).
\dated{5 Nov 2011} 
\par
\codes{\CQM:~\Csd,~\Csw}%
\itemspace

\authorx{Pouya Esmailian}						%ESMAILIAN
\author{Pouya Esmailian, Seyed Ebrahim Abtahi, \& Mahdi Jalili}
\authorspace
\bsgitem{2014a}{}  Mesoscopic analysis of online social networks: The role of negative ties.  \emph{Phys.\ Rev.\ E} 90 (2014), art.\ 042817, 11 pp.
\arXiv{1411.6057}.  
\codes{\CSG:~\CClu}%
\itemspace

\author{Pouya Esmailian \& Mahdi Jalili}
\authorspace
\bsgitem{2015a}{}  Community detection in signed networks: the role of negative ties in different scales.  \emph{Sci.\ Reports} 5 (2015), art.\ 14339, 17 pp.
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Ernesto Estrada}						%ESTRADA
\authorspace
\bsgitem{2019a}{}  Rethinking structural balance in signed social networks..  \emph{Discrete Appl.\ Math.}\ 268 (2019), 70--90.
\MR{4007232}.  \Zbl{1425.91370}.  
\annot{} Axioms for measurement of imbalance.
\codes{\CSG:~\CBal,~\CFr:~\CPsS}
\itemspace

\author{Ernesto Estrada and Michele Benzi}
\authorspace
\bsgitem{2014a}{}  Walk-based measure of balance in signed networks: Detecting lack of balance in social networks.  \emph{Phys.\ Rev.\ E} 90 (2014), art.\ 042802, 10 pp.
\arXiv{1406.2132}.  
\annot{} For signed graph or digraph, $K(\Sigma) := \tr(\exp A(\Sigma)) / \tr(\exp A(\Sigma|))$, ``degree of balance'' based on walk signs.  Weighting a closed $l$-walk by $w(W)=1/l!$, degree of balance is $K = w(\Sigma)/w(|\Sigma|)$; then $0\leq K\leq1$; $K=1$ iff $\Sigma$ is cycle balanced.  Thm\ 1: For $\Sigma_n:=(K_n,\sigma)$ with $\Sigma_n^+=C_n$, $K\rightarrow0$, interpreted as $\Sigma_n$ being ``largely unbalanced''.  
Degree of balance of $v_i$ is $K_i := (\exp A(\Sigma))_{ii} / (\exp A(|\Sigma|))_{ii}$.  
\S III:  Replace $A$ by $\beta A$, $\beta>0$ (inverse ``temperature''), giving $K(\beta)$, interpreted as equilibrium constant of a graph with fluctuating signs.  
\S\S IV-VII:  Compares walk-based measure $K$ to circle-based measures in some real examples including social networks.  \S VII, ``Tuning balance in social networks'':  Varying $\beta$ has interesting effects on $K$.  
\annotctd{} Dictionary:  ``the balance'' = degree of balance;  ``balanced closed walk'' = positive closed walk.  
The history is not entirely reliable.
arXiv title: ``Are social networks really balanced?''
\dated{3 Feb 2018, rev 3 Dec 2020} 
\codes{\CSG,~\CSD:~\CFr,~\CAdj,~\CPsS,~\CPhys}%
\itemspace

\author{Ernesto Estrada \& Naomichi Hatano}
\authorspace
\bsgitem{2008a}{}  Communicability in complex networks.  \emph{Phys.\ Rev.\ E} (3) 77 (2008), art.\ 036111, 12 pp.
\MR{2495430} (2010i:91171).  
\codes{\CSG:~\CKG}%
\itemspace

\author{Ernesto Estrada, Desmond J.\ Higham, \& Naomichi Hatano}
\authorspace
\bsgitem{2008a}{}  Communicability and multipartite structures in complex networks at negative absolute temperatures.  \emph{Phys.\ Rev.\ E} 78 (2008), art.\ 026102, 7 pp.
\codes{\CSG:~\CKG:~\Cclu}%
\itemspace

\author{Ernesto Estrada \& Juan A.\ Rodr{\'\i}guez-Vel\'azquez}
\authorx{Ernesto Estrada \& Juan A.\ Rodriguez-Velazquez}
\authorspace
\bsgitem{2005a}{}  Spectral measures of bipartivity in complex networks.  \emph{Phys.\ Rev.\ E} (3) 72 (2005), no.\ 4, art.\ 046105, 6 pp.
\MR{2202758} (2006i:94124).  
\codes{\Cpar:~\CFr,~\CEig}%
\itemspace

\author{Khasheyar Etemadi}						%ETEMADI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Anthony B.\ Evans}						%EVANS
\authorspace
\authref  See \hyperlink{Yu Qing Chen}{Y.Q.~Chen}.
\itemspace

\author{Patricia A.\ Evans}						%EVANS
\authorspace
\authref  See \hyperlink{Duong D.\ Doan}{D.D.~Doan}.
\itemspace

\author{Cloyd L.\ Ezell} 						%EZELL
\authorspace
\bsgitem{1979a}{} Observations on the construction of covers using permutation voltage assignments.  \emph{Discrete Math.}\ 28 (1979), 7--20.
\MR{0542932} (81a:05040).  \Zbl{413.05005}.
\par
\codes{\CGG:~\CTop,~\CCov,~\Csw}%
\itemspace

\author{Peyman Ezzati}						%EZZATI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace


\label{LETTER F} \hypertarget{LETTER F}{}	% LETTER  F


\authorx{Giuseppe Facchetti}						%FACCHETTI
\author{Giuseppe Facchetti, Giovanni Iacono, \& Claudio Altafini}
\authorspace
\bsgitem{2011a}{}  Computing global structural balance in large-scale signed social networks.  \emph{Proc.\ Nat.\ Acad.\ Sci.}\ 108 (2011), no.\ 52, 20953--20958.
URL \url{http://www.pnas.org/cgi/doi/10.1073/pnas.1109521108}
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{Fran\c{c}ois Fages}						%FAGES
\authorx{Francois Fages}						%FAGES
\authorspace
\authref  See also \hyperlink{K.\ Sriram}{K.~Sriram}.
\authrefspace

\author{Fran\c{c}ois Fages \& Sylvain Soliman}
\authorx{Francois Fages \& Sylvain Soliman}
\authorspace
\bsgitem{2008a}{}  From reaction models to influence graphs and back: A theorem.  In: Jasmin Fisher, ed., \emph{Formal Methods in Systems Biology} (First Int.\ Workshop, FMSB 2008, Cambridge, Eng., 2008), pp.\ 90--102.  \LNCS 5054.  Springer, Berlin, 2008.
\MR{2497938} (2010e:92050).  \Zbl{1375.92021}.  
\codes{\Csd:~\CChem}%
\itemspace

\authorx{Ulrich Faigle}						%FAIGLE
\author{Ulrich Faigle \& Rainer Schrader}
\authorspace
\bsgitem{1990a}{}  Orders and graphs.  In:  G.\ Tinhofer, E.\ Mayr, H.\ Noltemeier and M.M.\ Sys{\l}o, eds., \emph{Computational Graph Theory}.  Computing Supplementum, 7.  Springer-Verlag, Vienna, 1990.
\MR{1059927} (91d:05085).  \Zbl{725.05045}.  					
\annot{} An example is threshold signed graphs (\cf\ \xref{Benzaken, Hammer, and de Werra}{1985a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}).
\dated{16 Jan 2012} 
\codes{\CSG}%
\itemspace

\authorx{M.\ Falcioni}						%FALCIONI
\author{M.\ Falcioni, E.\ Marinari, M.L.\ Paciello, G.\ Parisi, \& B.\ Taglienti}
\authorspace
\bsgitem{1981a}{}  Phase transition analysis in $Z_2$ and U(1) lattice gauge theories.  \emph{Phys.\ Lett.\ B} 105 (1981), no.\ 1, 51--54. 
\codes{\CSG:~\CPhys}
\itemspace

\author{Shaun Fallat}						%FALLAT
\authorspace
\authref  See also \hyperlink{Michael S.\ Cavers}{M.S.~Cavers} and \hyperlink{Yi-Zheng Fan}{Y.-Z.~Fan}.
\authrefspace

\author{Shaun Fallat \& Yi-Zheng Fan}
\authorspace
\bsgitem{2012a}{}  Bipartiteness and the least eigenvalue of signless Laplacian of graphs.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 9, 3254--3267.
\MR{2900713}.  \Zbl{1244.05142}.  
\annot{} ``Bipartiteness'' of $\Gamma$ [also known as biparticity] is $b(-\Gamma)$.  ``Algebraic bipartiteness'' is the smallest eigenvalue $\lambda_{\min}(\Lap(-\Gamma))$.  Rephrased in terms of antibalanced signed graphs:  
Thm.\ 2.1.  If $-\Gamma$ is unbalanced, $\lambda_{\min} \leq l_0(-\Gamma)$, the vertex frustration number.  
Thm.\ 2.4.  (1) $\Spec \Lap(\widetilde{-\Gamma}) = \Spec \Lap(\Gamma) \cup \Spec \Lap(-\Gamma).$  [A special case of \xref{Bilu and Linial}{2006a}{Yonatan Bilu \& Nathan Linial}, Lemma.]
(2--4) Elementary properties of $\widetilde{-\Gamma}$ [found in \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}].  
(4) If $-\Gamma$ is connected and unbalanced, $\lambda_2(\Lap(\widetilde{-\Gamma})) = \min\{\lambda_{\min}(\Gamma), \lambda_2(\Gamma)\} > 0$.  
\annotctd{} $\bar\psi(-\Gamma) := \min_S [2l(-\Gamma:S) + \#E(S,S^c)]/\#S$ ($S \neq \eset,V$) (\cf\ \xref{Desai and Rao}{1994a}{Madhav Desai \& Vasant Rao}.)  
Thm.\ 2.6.  If $\Gamma$ is connected, $\Delta :=$ max degree, $\lambda_{\min} \geq \Delta - \surd{\Delta^2-\bar\psi^2}.$  
Thm.\ 2.7.  $\lambda_{\min} \leq 2\bar\psi \leq 4l(-\Gamma)/n$.  (Strengthens \xref{Y.Y.\ Tan and Fan}{2008a}{Ying Ying Tan \& Yi Zheng Fan}.)
[\Conjecture.  The results must generalize to all $(\Gamma,\sigma)$.]
\dated{20 Jan 2012} 
\codes{\Csg:~\CPar:~\CEig,~\CCov}%
\itemspace

\author{Genghua Fan}						%FAN GH
\authorspace
\authref  See \hyperlink{Bo Bao}{B.~Bao} and \hyperlink{Jing Chen}{J.~Chen}.
\itemspace

\author{Yi-Zheng Fan}						%FAN YZ
\authorspace
\authref  See also \hyperlink{Lin Cui}{L.~Cui}, \hyperlink{Shaun Fallat}{S.~Fallat}, \hyperlink{Shi-Cai Gong}{S.C.~Gong}, \hyperlink{Bit-Shun Tam}{B.S.~Tam}, \hyperlink{Ying-Ying Tan}{Y.Y.~Tan}, \hyperlink{Yi Wang}{Y.~Wang}, \hyperlink{Miao-Lin Ye}{M.L.~Ye}, \hyperlink{Gui-Dong Yu}{G.-D.~Yu}, and \hyperlink{Jun Zhou}{J.~Zhou}.
\authrefspace

\bsgitem{2003a}{}  On spectral integral variations of mixed graphs.  \emph{Linear Algebra Appl.}\ 374 (2003), 307--316.
\MR{2008794} (2005h:05133).  \Zbl{1026.05076}.					
\annot{} The signed graphs (not necessarily simple) for which adding an edge changes only one eigenvalue of the Laplacian matrix $\Lap(\Sigma)$ and increases that by an integer.  
[Dictionary:  ``mixed graph'' = bidirected graph $\Beta$ where all negative edges are extraverted, in effect the signed graph $-\Sigma_\Beta$;  ``quasibipartite'' = balanced;  ``$e^c$'' = $e$ with reversed sign.  The article's sign $\sgn(e)$ equals $-\sigma_\Beta(e)$.  The entire article is really about signed graphs $\Sigma$ and $A(\Sigma)$ and $\Lap(\Sigma)$ and uses signed-graph matrices and methods.]
Thm.\ 1:  This eigenvalue property holds iff the column $x(e)$ of $e$ in $\Eta(\Sigma)$ is an eigenvector of $\Lap(\Sigma)$.  Corollaries give other criteria and identify the change in the one eigenvalue.  Lemma 5: $\Lap$ is singular iff $\Sigma$ is balanced [special case of \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Theorem 8A.4].
\dated{13 Apr 2009, 10 Feb 2012}
\par
\codes{\CSG:~\Cinc,~\CEig}%
\itemspace

\bsgitem{2004a}{}  On structure of eigenvectors of mixed graphs.  Sixth Int.\ Conf.\ Matrix Theory Appl.\ in China.  \emph{Heilongjiang Daxue Ziran Kexue Xuebao} (\emph{J.\ Nat.\ Sci.\ Heilongjiang Univ.}) 21 (2004), no.\ 4, 50--54.
\MR{2129072} (no rev).  \Zbl{1077.05061}.						
\annot{} Early version of \xrefd{2007a}{Yi-Zheng Fan}.
\dated{9 Jan 2013} 
\codes{\Csg:~\CEig}%
\itemspace

\bsgitem{2004b}{}  Largest eigenvalue of a unicyclic mixed graph.  \emph{Appl.\ Math.\ J.\ Chinese Univ.\ Ser.\ B} 19 (2004), no.\ 2, 140--148.
\MR{2063313} (no rev).  
\annot{} The ``mixed graphs'' are signed graphs with reversed signs; see \xrefd{2003a}{Yi-Zheng Fan}.  Graphs are simple.  The eigenvalues are those of the Laplacian $\Lap(\Sigma)$.  
Prop.\ 2.2:  Laplacian spectrum of negative circle.  [The first such proof.  Equivalent to the adjacency spectrum because $C_n$ is regular.]
Thm.\ 2.8:  The signed 1-trees with max and min $\lambda_{\min}(\Lap(\Sigma))$.  
Thm.\ 2.9:  Those with $\lambda_{\min}=n$.  Thm.\ 2.10:  Those with $\lambda_{\min} > n$.  (N.B.  Lem.\ 2.4:  $\lambda_{\min} \leq n+1$ from \xref{Hou, Li, and Pan}{2003a}{Yaoping Hou, Jiongsheng Li, \& Yongliang Pan}, Thm.\ 3.5(1), or \xref{X.D.~Zhang and Li}{2002a}{Xiao-Dong Zhang \& Jiong-Sheng Li}.)
\dated{10 Feb 2012}
\codes{\CSG:~\Cinc,~\CEig}%
\itemspace

\bsgitem{2005a}{}  On the least eigenvalue of a unicyclic mixed graph.  \emph{Linear Multilinear Algebra} 53 (2005), no.\ 2, 97--113.
\MR{2133313} (2005m:05145).  \Zbl{1062.05090}.					
\annot{} The ``mixed graphs'' are signed graphs with reversed signs; see \xref{Y.-Z.\ Fan}{2003a}{Yi-Zheng Fan}.  Graphs are simple.  The eigenvalue is that of $\Lap(\Sigma)$.  Eigenvector structure leads to results on minimum and maximum of the least eigenvalue, given order and girth.
\dated{9 Jan 2013} 
\codes{\Csg:~\CEig}%
\itemspace

\bsgitem{2007a}{}  On eigenvectors of mixed graphs with exactly one nonsingular cycle.  \emph{Czechoslovak Math.\ J.}\ 57 (2007), no.\ 4, 1215--1222.
\MR{2357587} (2008i:05117).  \Zbl{1174.05075}.					
\annot{} The ``mixed graphs'' are signed graphs with reversed signs; see \xref{Y.-Z.\ Fan}{2003a}{Yi-Zheng Fan}.  Graphs are simple.  The eigenquantities are those of the Laplacian $\Lap(\Sigma)$.    The eigenvector of the smallest eigenvalue is similar to that of the second smallest Laplacian eigenvalue of a graph.
\dated{9 Jan 2013} 
\codes{\Csg:~\CEig}%
\itemspace

\author{Yi-Zheng Fan, Wen-Xue Du, \& Chun-Long Dong}
\authorspace
\bsgitem{2014a}{}  The nullity of bicyclic signed graphs.  \emph{Linear Multilinear Algebra} 62 (2014), no.\ 2, 242--251.
\MR{3175412}.  \Zbl{1297.05144}.  \arXiv{1207.6765}.				
\annot{} 
See \xref{Y.-Z.~Fan, Y.~Wang, and Y.~Wang}{2013a}{Yi-Zheng Fan, Yue Wang, \& Yi Wang}.
\codes{\CSG:~\CEig}%
\itemspace

\author{Yi-Zheng Fan \& Shaun Fallat}
\authorspace
\bsgitem{2012a}{}  Edge bipartiteness and signless Laplacian spread of graphs.  \emph{Appl.\ Anal.\ Discrete Math.}\ 6 (2012), no.\ 1, 31--45.
\MR{2952601}.  \Zbl{1289.05281}.  
\annot{} \Cf\ \xref{M.H.~Liu and Liu}{2010a}{Muhuo Liu \& Bolian Liu}, \xref{Oliveira, de Lima, de Abreu, and Kirkland}{2010a}{Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, \& Steve Kirkland}.  Weak relations between $l(\Sigma)$ and spread of $\Lap(\Sigma)$.  Min spread of $\Lap(\Sigma)$ is $2+2\cos(\pi/n)$, attained only by a path and $-C_{\text{odd}}$.  Next smallest spread = 4, attained only by $-K_4$, $-\Gamma$ consisting of two triangles joined by an edge, $K_{1,3}$, $C_\text{even}$.  Proofs by cases: $l(\Sigma) \leq 1$ or $\geq 2$ (min spread is from $-K_4$).  
Dictionary:  ``edge bipartiteness'' = frustration index $l(-\Gamma)$; ``mixed graph'' = (oriented) signed graph with reversed signs (oriented edges are called negative); Laplacian matrix of mixed graph $G = D(|\Sigma|)-A(\Sigma)$.
[See \xref{Desai and Rao}{1994a}{Madhav Desai \& Vasant Rao} for another eigenvalue connection with $l(\Sigma)$.]
\dated{29 Dec 2012} 
\par
\codes{\CPar:~\CEig,~\CFr,~\CCov}%
\itemspace

\author{Yi-Zheng Fan, Shi-Cai Gong, Yi Wang, \& Yu-Bin Gao}
\authorspace
\bsgitem{2009a}{}  First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph.  \emph{Discrete Math.}\ 309 (2009), no.\ 8, 2479--2487.
\MR{2512565} (2010g:05212).  \Zbl{1182.05081}.					
\annot{} The ``mixed graphs'' are signed graphs with reversed signs; see \xref{Fan}{2003a}{Yi-Zheng Fan}.  Graphs are simple.  Eigenvalues are those of $\Lap(\Sigma)$.  
\codes{\Csg:~\CEig}%
\itemspace

\author{Yi-Zheng Fan, Shi-Cai Gong, Jun Zhou, Ying-Ying Tan, \& Yi Wang}
\authorspace
\bsgitem{2007a}{}  Nonsingular mixed graphs with few eigenvalues greater than two.  \emph{European J.\ Combin.}\ 28 (2007), no.\ 6, 1694--1702.
\MR{2339495} (2008f:05115).  \Zbl{1122.05058}.					
\annot{} The ``mixed graphs'' are signed graphs with reversed signs; see \xref{Fan}{2003a}{Yi-Zheng Fan}.  Assume $\Sigma$ is connected.  $m:=$ number of eigenvalues $> 2$.
Thm.\ 2.2:  $d:=$ longest path length, $\mu:=$ matching number.  (i) $m\geq \lfloor d/2 \rfloor$, (ii) $m\geq \mu$ if $n>2\mu$, (iii) $m\geq \mu-1$ if $n=2\mu$.  
Now assume $\Sigma$ is unbalanced.  
Thm.\ 3.4.  If $n\geq 7$, then $m=2$ iff $|\Sigma|$ is one of two general types and $\Sigma$ has a certain negative triangle.
Thm.\ 3.5.  If $n\geq 6$, then $m=1$ iff $\Sigma\sim-K_4$ or an unbalanced subgraph.
Dictionary:  See \xref{Bapat, Grossman, and Kulkarni}{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}.
\dated{13 Jan 2012} 
\codes{\Csg:~\CEig}%
\itemspace

\author{Yi-Zheng Fan, Hai-Yan Hong, Shi-Cai Gong, \& Yi Wang}
\authorspace
\bsgitem{2007a}{}  Order unicyclic mixed graphs by spectral radius.  \emph{Australas.\ J.\ Combin.}\ 37 (2007), 305--316.
\MR{2284395} (2007j:05137).  \Zbl{1122.05059}.					
\annot{} Finds the unicyclic signed graphs with first, second, and third largest spectral radii.  The ``mixed graphs'' are signed graphs with reversed signs; see \xref{Fan}{2003a}{Yi-Zheng Fan}.  Graphs are simple.  The eigenvalues are those of $\Lap(\Sigma)$.  
\dated{9 Jan 2013} 
\codes{\Csg:~\CEig}%
\itemspace

\author{Yi-Zheng Fan, Bit-Shun Tam, \& Jun Zhou}
\authorspace
\bsgitem{2008a}{}  Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order.  \emph{Linear Multilinear Algebra} 56 (2008), no.\ 4, 381--397.
\MR{2434109} (2009e:15070).  \Zbl{1146.05032}.					
\annot{} The maximal graphs are $K_4 \setminus e$ with $n-4$ pendant edges at one trivalent vertex.
\dated{9 Sept 2010}
\codes{\Cpar:~\CInc,~\CEig}%
\itemspace

\author{Yi-Zheng Fan, Yue Wang, \& Yi Wang}
\authorspace
\bsgitem{2013a}{}  A note on the nullity of unicyclic signed graphs.  \emph{Linear Algebra Appl.}\ 437 (2013), no.\ 3, 1193--1200. 
\MR{2997803}.  \Zbl{1257.05083}.  \arXiv{1107.0400}.					
\annot{} For $\rk A(\Sigma) \geq 2$, the cases where $\rk \leq 3$ are characterized.	
\Cf\ \xref{Y.-Z.~Fan, W.-X.~Du, and C.-L.~Dong}{2014a}{Yi-Zheng Fan, Wen-Xue Du, \& Chun-Long Dong}, \xref{X.-Z.~Tan and B.-L.~Liu}{2006a}{Xuezhong Tan \& Bolian Liu}.
\dated{17 Dec 2011} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{Yi-Zheng Fan \& Dan Yang}
\authorspace
\bsgitem{2009a}{}  The signless Laplacian spectral radius of graphs with given number of pendant vertices.  \emph{Graphs Combin.}\ 25 (2009), no.\ 3, 291--298. 
\MR{2534887} (2010j:05233).  \Zbl{1194.05085}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{E.\ Fanchon}							%FANCHON
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Mohammad Reza Farahani}					%FARAHANI
\authorspace
\authref  See \hyperlink{M.R.\ Rajesh Kanna}{M.R.~Rajesh Kanna}.
\itemspace

\author{Thomas J.\ Fararo}						%FARARO
\authorspace
\authref  See \hyperlink{Norman P.\ Hummon}{N.P.~Hummon}.
\itemspace


\authorx{Luerbio Faria}						%FARIA
\author{Luerbio Faria, Sulamita Klein, \& Mat\v{e}j Stehl\'ik}
\authorspace
\bsgitem{2012a}{}  Odd cycle transversals and independent sets in fullerene graphs.  \emph{SIAM J.\ Discrete Math.}\ 26 (2012), no.\ 3, 1458--1469.
\MR{3022147}.  \Zbl{1256.05116}.  
\annot{} If $\Gamma$ is a fullerene graph (cubic, plane, no isthmi, all faces are pentagons and hexagons), $l(-\Gamma) \leq \sqrt{12n/5}.$  Dictionary:  ``odd cycle transversal'' = balancing set of $-\Gamma$.
\dated{1 Oct 2012} 
\codes{\Csg:~\Cpar:~\CFr}%
\itemspace

\authorx{Arthur M.\ Farley}						%FARLEY
\author{Arthur M.\ Farley \& Andrzey Proskurowski}
\authorspace
\bsgitem{1981a}{}  Computing the line index of balance of signed outerplanar graphs.  
Proc.\ Twelfth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing (Baton Rouge, 1981), Vol.\ I.  \emph{Congressus Numer.}\ 32 (1981), 323--332.  
\MR{0681891} (83m:68119).  \Zbl{489.68065}.
\annot{} Calculating frustration index is NP-complete, since it is more general than max-cut.  However, for signed outerplanar graphs with bounded size of bounded faces, it is solvable in linear time.  [It is quickly solvable for signed planar graphs.  See \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai}, \xref{Barahona}{1981a}{Francisco Barahona}, \xrefd{1982a}{Francisco Barahona}, and more.]
\codes{\CSG:~\CFr}%
\itemspace

\author{Rashid Farooq}						%FAROOQ
\authorspace
\authref  See also \hyperlink{Sumaira Hafeez}{S.~Hafeez} and \hyperlink{Mehtab Khan}{M.~Khan}.
\authrefspace

\author{Rashid Farooq, Mehtab Khan, \& Sarah Chand}
\authorspace
\bsgitem{2019a}{}  On iota energy of signed digraphs.  \emph{Linear Multilinear Algebra} 67 (2019), no.\ 4, 705--724.
\MR{3914326}.  \Zbl{1411.05138}.  
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{Rashid Farooq, Sarah Chand, \& Mehtab Khan}
\authorspace
\bsgitem{2019a}{}  Iota energy of bicyclic signed digraphs.  \emph{Asian-European J.\ Math.}\ 12 (2019), no.\ 5, art.\ 1950078, 14 pp.
\MR{3999110}.  \Zbl{1419.05123}.  
\annot{} \Cf\ \xref{Hafeez and Khan}{2018a}{Sumaira Hafeez \& Mehtab Khan}.
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{M.\ Farzan} 							%FARZAN
\authorspace
\bsgitem{1978a}{} Automorphisms of double covers of a graph.  
In: \emph{Problemes Combinatoires et Theorie des Graphes} (Colloq.\ Int., Orsay, 1976), pp.\ 137--138.  Colloques Int.\ du CNRS, 260.  Editions du C.N.R.S., Paris, 1978.  
\MR{0539960} (81a:05063).  \Zbl{413.05064}.
\annot{} A ``double cover of a graph'' means the double cover of a signing of a simple graph.
\codes{\Csg:~\CCov,~\CAut}%
\itemspace

\author{G.H.\ Fath-Tabar}						%FATH-TABAR
\authorspace
\authref  See \hyperlink{E.\ Ghasemian}{E.~Ghasemian}.
\itemspace

\author{R.J.\ Faudree}						%FAUDREE
\authorspace
\authref  See also \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\authrefspace

\authorx{Ralph J.\ Faudree}						%FAUDREE
\author{Ralph J.\ Faudree, Evelyne Flandrin, Michael S.\ Jacobson, Jen\H{o} Lehel, \& Richard H.\ Schelp}
\authorx{Ralph J.\ Faudree, Evelyne Flandrin, Michael S.\ Jacobson, Jeno Lehel, \& Richard H.\ Schelp}
\authorspace
\bsgitem{2000a}{}  Even cycles in graphs with many odd cycles.  \emph{Graphs Combin.}\ 16 (2000), 399--410.
\MR{1804339} (2001k:05117).  \Zbl{984.05048}.  
\codes{\CPar:~\CCycles}%
\itemspace

\author{Katherine Faust}						%FAUST
\authorspace
\authref  See also \hyperlink{Stanley Wasserman}{S.~Wasserman}.
\authrefspace

\bsgitem{2007a}{}  Very local structure in social networks.  \emph{Sociological Methodology} 37 (2007), Ch.\ 7, pp.\ 209--256.
\annot{} A documented warning that properties of triads (vertex triples) in a graph or digraph tend to be heavily dependent on monad (single-vertex) or dyad (vertex pair) properties such as the density of edges or the degree distribution and therefore must be evaluated in comparison to expected triad properties given the distribution of dyad types.  The focus is on digraphs; pp.\ 9--10 mention structural balance, i.e., signed-graph models.  [\Problem\ 1.  Carry out a similar analysis for signed graphs, and in particular, signed complete graphs (equivalent to graphs).  \Problem\ 2.  The same, for switching classes of the preceding, in which the meaning of a dyad census is unclear.]
\dated{22 Aug 2014} 
\codes{\CPsS}%
\itemspace

\authorx{Siamak Fayyaz Shahandashti}					%FAYYAZ SHAHANDASHTI
\author{Siamak Fayyaz Shahandashti, Mahmoud Salmasizadeh, \& Javad Mohajeri}
\authorspace
\bsgitem{2005a}{}  A provably secure short transitive signature scheme from bilinear group pairs.  
In:  C.\ Blundo and S.\ Cimato, eds., \emph{Security in Communication Networks} (4th Int.\ Conf., SCN 2004, Amalfi), pp.\ 60--76.  \LNCS 3352.  Springer-Verlag, Berlin, 2005.  
\Zbl{1116.94320}.  
\annot{} Edges have ``signatures'' for encryption.  No edge signs!  [Irresistible.]
\dated{5 Mar 2011} 
\codes{\bf None}%
\itemspace

\author{N.T.\ Feather} 						%FEATHER
\authorspace
\bsgitem{1971a}{}  Organization and discrepancy in cognitive structures.  \emph{Psychological Rev.}\ 78 (1971), 355--379.  
\annot{} A suggestion for defining balance in weighted digraphs:  pp.\ 367--369.
\par
\codes{\CPsS:~\CBal:~\CExp}\codesctd{\CWD:~\CBal}%
\itemspace

\author{Martin Feinberg}						%FEINBERG
\authorspace
\authref  See \hyperlink{Gheorghe Craciun}{G.~Craciun} and \hyperlink{Guy Shinar}{G.~Shinar}.
\itemspace

\author{Anna Felikson}						%FELIKSON
\authorspace
\authref  See \hyperlink{Mathieu Dutour Sikiric}{M.D.~Sikiri\'c}.
\itemspace

\author{Mariusz Felisiak}						%FELISIAK
\authorspace
\authref  See also \hyperlink{Rafal Bocian}{R.~Bocian}.
\authrefspace

\bsgitem{2013a}{}  Computer algebra technique for Coxeter spectral study of edge-bipartite graphs and matrix morsifications of Dynkin type $\bbA_n$.  \emph{Fundamenta Inform.}\ 125 (2013), no.\ 1, 21--49.
\MR{3114057}.  \Zbl{1277.16040}.  
\codes{\CSG}%
\itemspace

\author{Mariusz Felisiak \& Daniel Simson}
\authorspace
\bsgitem{2015a}{}  Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs.  \emph{Discrete Appl.\ Math.}\ 192 (2015), 49--64.
\MR{3354818}.  \Zbl{1319.05142}.  
\codes{\CSG}%
\itemspace

\author{Michael R.\ Fellows}						%FELLOWS
\authorspace
\authref  See \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender}.
\itemspace

\authorx{Stefan Felsner}						%FELSNER
\author{Stefan Felsner \& Kolja Knauer}
\authorspace
\bsgitem{2011a}{}  Distributive lattices, polyhedra, and generalized flows.  \emph{European J.\ Combin.}\ 32 (2011), 45--59.
\MR{2727459} (2012a:52022).  \Zbl{1205.06007}.  
\annot{} ``Generalized flows'' are flows (conservative at each vertex, i.e., real 1-cycles) on a graph with positive real gains (``generalized network'').		
\dated{2 Apr 2013} 
\codes{\CGG:~\CGN:~\CInc}%
\itemspace

\authorx{Paul Fendley}						%FENDLEY
\author{Paul Fendley \& Vyacheslav Krushkal}
\authorspace
\bsgitem{2010a}{}  Link invariants, the chromatic polynomial and the Potts model.  \emph{Adv.\ Theor.\ Math.\ Phys.}\ 14 (2010), no.\ 2, 507--540.
\MR{2721654} (2011k:57015).  \Zbl{1207.82007}.  \arXiv{0806.3484}.			
\annot{} The Potts model treats a graph as all negative (``antiferromagnetic''; see the low-temperature expansion in \S3).
\dated{12 Jan 2012} 
\par
\codes{\Cpar:~\CInvar}%
\itemspace

\authorx{Derek Feng}						%FENG D
\author{Derek Feng, Randolf Altmeyer, Derek Stafford, Nicholas A.\ Christakis, and Harrison H.\ Zhou}
\authorspace
\bsgitem{2022a}{}  Testing for balance in social networks.  \emph{J.\ Amer.\ Statist.\ Assoc.}\ 117 (2022), no.\ 537, 156--174.
\MR{4399076}.  \arXiv{1808.05260}.  
\annot{} A new, statistical model for measuring imbalance using triangles.  [Hidden assumption:  triangles largely determine the graph.  Examples may not satisfy this.]
\dated{19 Dec 2020} 
\codes{\CSG:~\CKG:~\CPsS,~\CFr}%
\itemspace

\author{Gang Feng}							%FENG G
\authorspace
\authref  See \hyperlink{Yanzhi Wu}{Y.-Z.~Wu}.
\itemspace

\author{Lihua Feng}							%FENG LH
\authorspace
\authref  See also \hyperlink{Guihai Yu}{G.H.~Yu}.
\authrefspace

\bsgitem{2010a}{}  The signless Laplacian spectral radius for bicyclic graphs with $k$ pendant vertices.  \emph{Kyungpook Math.\ J.}\ 50 (2010), no.\ 1, 109--116.
\MR{2609079} (2011d:05221).  \Zbl{1205.05140}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Lihua Feng \& Guihai Yu}
\authorspace
\bsgitem{2009a}{}  On three conjectures involving the signless Laplacian spectral radius of graphs.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 85(99) (2009), 35--38. 
\MR{2536687} (2010i:\-05204).  \Zbl{1265.05365}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2009b}{}  The signless Laplacian spectral radius of unicyclic graphs with graph constraints.  \emph{Kyungpook Math.\ J.}\ 49 (2009), no.\ 1, 123--131.
\MR{2527378} (2011b:\-05148).  \Zbl{1201.05056}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  The signless Laplacian spectral radius of graphs with given diameter.  \emph{Utilitas Math.}\ 83 (2010), 265--276.
\MR{2742294} (2011i:05129).  \Zbl{1242.05162}.  
\annot{} The graphs with maximum spectral radius.
\dated{19 Nov 2011}
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Lihua Feng, Guihai Yu, \& Aleksandar Ili\'c}
\authorx{Lihua Feng, Guihai Yu, \& Aleksandar Ilic}
\authorspace
\bsgitem{2010a}{}  The Laplacian spectral radius for unicyclic graphs with given independence number.  \emph{Linear Algebra Appl.}\ 433 (2010), 934--944.
\MR{2658644} (2011f:05175).  \Zbl{1215.05102}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Lihua Feng, Guihai Yu, Aleksandar Ili\'c, \& Dragan Stevanovi\'c}
\authorx{Lihua Feng, Guihai Yu, Aleksandar Ilic, \& Dragan Stevanovic}
\authorspace
\bsgitem{2013a}{}  The signless Laplacian spectral radius of graphs on surfaces.  \emph{Linear Multilinear Algebra} 61 (2013), no.\ 5, 573--581.
\MR{3005638}.  \Zbl{1273.05134}.  
\codes{\Csg:~\Cpar:~\CLap:~\CEig,~\CTop}%
\itemspace

\authorx{Lin Feng}							%FENG L
\author{Lin Feng, Yan Hong Yao, Ji Ming Guo, \& Shang Wang Tan}
\authorspace
\bsgitem{2011a}{}  [On] The signless Laplacian spectral radius of unicyclic graphs with fixed girth.  (In Chinese.)  \emph{Appl.\ Math.\ J.\ Chinese Univ.\ Ser.\ A}  26 (2011), no.\ 1, 121--126.
\MR{2807616} (no rev).  \Zbl{1240.05190}.						
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Shasha Feng}						%FENG SS
\author{Shasha Feng, Li Wang, Yijia Li, Shiwen Sun, \& Chengyi Xia}
\authorspace
\bsgitem{2018a}{}  A nonlinear merging protocol for consensus in multi-agent systems on signed and weighted graphs.  \emph{Physica A} 490 (2018), 653--663.
\MR{3716395} (no rev).  
\codes{\CSG,~\CWG:~\CAlgor}%
\itemspace

\authorx{Zouhaier Ferchiou}						%FERCHIOU
\author{Zouhaier Ferchiou \& Bertrand Guenin}
\authorspace
\bsgitem{2020a}{}  A short proof of Shih's isomorphism theorem on graphic subspaces.  \emph{Combinatorica} 40 (2020), no.\ 6, 805--837.
\MR{4208098}.  \Zbl{1474.05169}.  
\annot{} \Cf\ \xref{C.-S. Shih}{1982a}{Ching-Hsien Shih}.
\codes{\Csg:~\Cmatrd}%
\itemspace

\author{Anu\v{s}ka Ferligoj}						%FERLIGOJ
\authorspace
\authref  See \hyperlink{Patrick Doreian}{P.~Doreian}.
\itemspace

\author{Lori Fern [Lori Koban]}						%FERN
\authorx{Lori Fern}
\authorspace
\authref  See also \hyperlink{Lori Koban}{L.~Koban}.
\authrefspace

\author{Lori Fern, Gary Gordon, Jason Leasure, \& Sharon Pronchik}
\authorspace
\bsgitem{2000a}{}  Matroid automorphisms and symmetry groups.  \emph{Combin.\ Probab.\ Comput.}\ 9 (2000), 105--123.
\MR{1762784} (2001g:05034).  \Zbl{960.05055}.						
\annot{} Consider a subgroup $W$ of the hyperoctahedral group $Oc_n$ that is generated by reflections.  Let $M(W)$ be the vector matroid of the vectors corresponding to reflections in $W$.  The possible direct factors of any automorphism group of $M(W)$ are $S_k$, $Oc_k$, and $Oc_k^+$.  The proof is stricly combinatorial, via signed graphs.
\codes{\CSG:~\CMatrd:~\CAut,~\CGeom}%
\itemspace

\author{Ros\'ario Fernandes}						%FERNANDES R
\authorx{Rosario Fernandes}
\authorspace
\bsgitem{2010a}{}  Location of the eigenvalues of weighted graphs with a cut edge.  \emph{Linear Multilinear Algebra} 58 (2010), no.\ 3, 305--322.
\MR{2663432} (2011d:15014).  \Zbl{1203.05092}.					
\annot{} The ``weights'' are skew gains [\cf\ \xref{J.~Hage}{1999a}{Jurriaan Hage} \etal] in $\bbC^\times$; the anti-involution is conjugation.  Identities satisfied by the eigenvalues.
\dated{11 Jan 2012} 
\codes{\CGG:~\CGen:~\CEig}%
\itemspace

\authorx{L.A.\ Fern\'andez}						%FERNANDEZ
\authorx{L.A.\ Fernandez}
\author{L.A.\ Fern\'andez, V.\ Martin-Mayor, G.\ Parisi, \& B.\ Seoane}
\authorx{L.A.\ Fernandez, V.\ Martin-Mayor, G.\ Parisi, \& B.\ Seoane}
\authorspace
\bsgitem{2010a}{}  Spin glasses on the hypercube.  \emph{Phys.\ Rev.\ B} 81 (2010), \#134403, 14 pp.
\arXiv{0911.4667}.  
\annot{} Average behavior of random signed subhypercubes $(\Gamma,\sigma)$, with spanning $\Gamma \subseteq Q_D$, with random spins $\zeta: V \to \signnos$.  Each $(\Gamma,\sigma,\zeta)$ is a ``sample''.  To avoid irregularities $\Gamma$ is $z$-regular (``connectivity $z$'') for a fixed $z$ (here, 6).
\dated{19 Jun 2012} 
\codes{\CPhys,~\CSG:~\CState}%
\itemspace

\author{Daniela Ferrero}						%FERRERO
\authorspace
\bsgitem{2008a}{}  Product line sigraphs.  
In:  \emph{The International Symposium on Parallel Architectures, Algorithms, and Networks} (i-span 2008), pp.\ 141--145.  IEEE Computer Soc., 2008.
\annot{} The product line graph [$= \Lambda_{\times}(\Sigma)$ in \xref{M.\ Acharya}{2009a}{Mukti Acharya}] is balanced.  [Immediate from \xref{Harary's}{1953a}{Frank Harary} balance theorem or \xref{Sampathkumar's}{1972a}{E.\ Sampathkumar}, \xrefd{1984a}{E.\ Sampathkumar} similar theorem.]
\dated{2008, 20 Dec 2010}
\par
\codes{\CSG:~\CLG,~\CBal}%
\itemspace

\author{A.\ Fiat}							%FIAT
\authorspace
\authref  See \hyperlink{Erik D.\ Demaine}{E.~Demaine} and \hyperlink{D.\ Emanuel}{D.~Emanuel}.
\itemspace

\author{Miroslav Fiedler} 						%FIEDLER
\authorspace
\bsgitem{1957a}{}  Uber qualitative Winkeleigenschaften der Simplexe.  \emph{Czechoslovak Math.\ J.}\ 7(82) (1957), 463--478. 
\MR{0094740} (20 \#1252).  \Zbl{093.33602} (93, p.\ 336b).				
\par
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1957b}{}  Einige Satze aus der metrischen Geometrie der Simplexe in euklidischen Raumen.  \emph{Schr.\ Forschungsinst.\ Math.}\ 1 (1957), 157. 
\MR{0087110} (19, 303d).  \Zbl{089.16706} (89, p.\ 167f).
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1961a}{}  Uber die qualitative Lage des Mittelpunktes der ungeschriebenen Hyperkugel im $n$-Simplex [On the qualitative location of the center of circumscribed hyperspheres in the $n$-simplex].  \emph{Comment.\ Math.\ Univ.\ Carolinae}\ 2 (1961), no.\ 1, 3--51.  
\Zbl{101.13205} (101, p.\ 132e).  
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1964a}{}  Some applications of the theory of graphs in matrix theory and geometry.  
In:  \emph{Theory of Graphs and Its Applications} (Proc.\ Sympos., Smolenice, 1963), pp.\ 37--41.  Publ.\ House Czechoslovak Acad.\ Sci., Prague, 1964.  
\MR{0175109} (30 \#5294).  \Zbl{163.45605} (163, 456e).
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1967a}{}  Graphs and linear algebra.  
In:  M.\ Fiedler, ed., \emph{Theory of Graphs: International Symposium} (Rome, 1966), pp.\ 131--134.  Gordon and Breach, New York; Dunod, Paris, 1967.  
\MR{0223265} (36 \#6313).  \Zbl{263.05124}.
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1969a}{}  Signed distance graphs.  \emph{J.\ Combin.\ Theory} 7 (1969), 136--149.  
\MR{0242705} (39 \#4034).  \Zbl{181.26001} (181, p.\ 260a).				
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1970a}{}  Pozn\'amka o distancnich grafech [A remark on distance graphs] (in Czech).
In:  \emph{Matematika (geometrie a teorie grafu)} [Mathematics (Geometry and Graph Theory)], pp.\ 85--88.  Univ.\ Karlova, Prague, 1970.  
\MR{0277410} (43 \#3143).  \Zbl{215.50203}.
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{1975a}{}  Eigenvectors of acyclic matrices.  \emph{Czechoslovak Math.\ J.}\ 25(100) (1975), 607--618.  
\MR{0387308} (52 \#8151).  \Zbl{325.15014}.					
\codes{\Csg:~{\bf Trees}:~\CEig}%
\itemspace

\bsgitem{1985a}{}  Signed bigraphs of monotone matrices.  
In:  Horst Sachs, ed., \emph{Graphs, Hypergraphs and Applications} (Proc.\ Int.\ Conf., Eyba, 1984), pp.\ 36--40.  
Teubner-Texte zur Math., B.\ 73.  B.G.\ Teubner, Leipzig, 1985.  
\MR{0869435} (87m:05121).  \Zbl{626.05023}.
\codes{\CSG:~\CEig:~\CExp}%
\itemspace

\bsgitem{1993a}{}  A geometric approach to the Laplacian matrix of a graph.  
In: Richard A.\ Brualdi, Shmuel Friedland, and Victor Klee, eds., \emph{Combinatorial and Graph-Theoretical Problems in Linear Algebra}, pp.\ 73--98.  
IMA Vols.\ Math.\ Appl., 50.  Springer-Verlag, New York, 1993. 
\MR{1240957} (94g:05055).  \Zbl{791.05073}.
\annot{} The signed bipartite graph of a normalized Gram matrix (pp.\ 85--86).  This is applied to study the types of angles in a geometric simplex.  
Dictionary:  $\Gamma(A) =$ the signed bipartite graph of a symmetric real matrix.
\par
\codes{\CSG}%
\itemspace

\bsgitem{1998a}{}  Additive compound graphs.  \emph{Discrete Math.}\ 187 (1998), 97--108.
\MR{1630684} (99c:05131).  \Zbl{958.05091}.						
\annot{} If $V(\Sigma)=[n]$, the $k$-th additive compound graph $\Sigma^{[k]}$ is defined for $k \in [n-1]$ via $A(\Sigma)$.  It respects connection, also edge-disjoint union and induced subgraphs.  $\Spec\Sigma^{[k]} = \{$sums of $k$ eigenvalues of $\Sigma$\}.  
Spectral radius: $\rho(\Sigma)\leq\rho(|\Sigma|)$ (Thm.\ 2.17).  For a path, $\rho(+P_n^{[k]}) = -1 + \sin\frac{2k+1}{n+1}\pi / \sin\frac{1}{n+1}\pi$.  
$\Sigma^{[n-k]}$ is $(-\Sigma^\zeta)^{[k]}$ where $\zeta$ switches odd-numbered vertices (Thm.\ 2.10, Rem.\ 2.11).  
$(+\Gamma)^{[k]}$ may be unbalanced, e.g.\ (Thm.\ 2.15, proof) $(+C_n)^{[2]}$ and $(+K_{1,3})^{[2]}$; however, $(\tilde C_n)^{[2]}$ is all-positive ($\tilde C_n$ = negative circle).  Thm.\ 2.15: For connected $\Gamma$, $(+\Gamma)^{[2]}$ is balanced iff $\Gamma=P_n$.
\dated{26 Jul 2013} 
\codes{\CSG:~\CLG(\CGen):~\CAdj,~\CSw}%
\itemspace

\author{Miroslav Fiedler \& Vlastimil Ptak} 
\authorspace
\bsgitem{1967a}{}  Diagonally dominant matrices.  \emph{Czechoslovak Math.\ J.}\ 17(92) (1967), 420--433.  
\MR{0215869} (35 \#6704).  \Zbl{178.03402} (178, p.\ 034b).				
\codes{\CGG:~\CSw,~\Cbal}%
\itemspace

\bsgitem{1969a}{}  Cyclic products and an inequality for determinants.  \emph{Czechoslovak Math.\ J.}\ 19(94) (1969), 428--451.  
\MR{0215869} (40 \#1409).  \Zbl{281.15014}.					
\codes{\Cgg:~\CSw}%
\itemspace

\authorx{B.\ Fierro}							%FIERRO
\author{B.\ Fierro, F.\ Bachmann, \& E.E.\ Vogel}
\authorspace
\bsgitem{2006a}{}  Phase transition in 2D and 3D Ising model by time-series analysis.  \emph{Physica B} 384 (2006), 215--217.
\annot{} Physical parameters calculated on a signed square lattice (``\xref{Edwards--Anderson}{1975a}{S.F.\ Edwards \& P.W.\ Anderson} model'') with 1/32 of edges negative (p.\  217).
\dated{10 Jan 2015} 
\codes{\CPhys:~\CSG}%
\itemspace

\authorx{Tara Fife}							%FIFE
\author{Tara Fife, Dillon Mayhew, James Oxley, \& Charles Semple}
\authorspace
\bsgitem{2020a}{}  The unbreakable frame matroids.  \emph{SIAM J.\ Discrete Math.}\ 34 (2020), no.\ 3, 1522--1537. 
\MR{4121881}.  \Zbl{1456.05028}.  
\codes{\CGG:~\CMatrd:~\CStr}%
\itemspace

\author{Rosa M.V.\ Figueiredo}						%FIGUEIREDO
\authorx{Rosa Figueiredo}
\authref  See also \hyperlink{Nejat Arinik}{N.~Ar{\i}n{\i}k}, \hyperlink{Mario Levorato}{M.~Levorato}, \hyperlink{Israel Mendonca}{I.~Mendon\c{c}a}, and \hyperlink{Eduardo Queiroga}{E.~Queiroga}.
\authrefspace

\author{Rosa Figueiredo \& Yuri Frota}
\authorspace
\bsgitem{2013a}{}  An improved Branch-and-cut code for the maximum balanced subgraph of a signed graph.  Manuscript, 2013.
\arXiv{1312.4345}.  \HAL{hal-02188356}.  
\annot{} Max order of balanced induced subgraph.
\dated{30 Mar 2021} 
\par
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\bsgitem{2014a}{}  The maximum balanced subgraph of a signed graph: Applications and solution approaches.  \emph{European J.\ Oper.\ Res.}\ 236 (2014), 473--487.
\MR{3179876}.  \Zbl{1317.90305}.  \HAL{hal-02181486}.  
\codes{\CVS:~\CFr:~\CAlgor}%
\itemspace

\author{Rosa Figueiredo, Yuri Frota, \& Martine Labb\'e}
\authorx{Rosa Figueiredo, Yuri Frota, \& Martine Labbe}
\authorspace
\bsgitem{2013a}{}  Solution of the maximum $k$-balanced subgraph problem.  In:  Giuseppe Nicosia and Panos Pardalos, eds., \emph{Learning and Intelligent Optimization} (Proc.\ 7th Int.\ Conf.\ Learning and Optimization, LION 7, Catania, Italy, 2013), pp.\ 266--271.  \LNCS 7997.  Springer, Berlin, 2013.
\annot{} Preliminary version of \xrefd{2019a}{Rosa Figueiredo, Yuri Frota, \& Martine Labbe}.
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\bsgitem{2019a}{}  A branch-and-cut algorithm for the maximum $k$-balanced subgraph of a signed graph.  \emph{Discrete Appl.\ Math.}\  261 (2019), 164--185. 
\MR{3958238}.  \Zbl{1410.05200}.  \HAL{hal-01937015}.  
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Rosa M.V.\ Figueiredo, Martine Labb\'e, \& Cid C.\ de Souza}
\authorx{Rosa M.V.\ Figueiredo, Martine Labbe, \& Cid C.\ de Souza}
\authorspace
\bsgitem{2011a}{}  An exact approach to the problem of extracting an embedded network matrix.  \emph{Computers Oper.\ Res.}\ 38 (2011), no.\ 11, 1483--1492.
\MR{2781542} (2012f:90223) (\qv).  \Zbl{1210.90038}.  
\codes{\CSG:~\CInc:~\CAlgor}%
\itemspace

\author{Rosa Figueiredo \& Gisele Moura}
\authorspace
\bsgitem{2013a}{}  Mixed integer programming formulations for clustering problems related to structural balance.  \emph{Social Networks} 35 (2013), no.\ 4, 639--651.
\HAL{hal-02171111}.  
\codes{\CSG:~\CClu:~\CAlgor,~\CAppl}%
\itemspace

\author{Joseph Fiksel} 						%FIKSEL
\authorspace
\bsgitem{1980a}{}  Dynamic evolution in societal networks.  \emph{J.\ Math.\ Sociology} 7 (1980), 27--46.  
\MR{0572489} (81g:92023) (\qv).  \Zbl{434.92022}.					
\codes{\CSG:~\CClu,~\CVS}%
\itemspace


\author{Miguel Angel Fiol}						%FIOL
\authorspace
\authref  See \hyperlink{C.\ Dalf'o}{C.~Dalf'o} and \hyperlink{Edwin R.\ van Dam}{E.R.~van~Dam}.
\itemspace

\author{Samuel Fiorini}						%FIORINI
\authorspace
\authref  See also \hyperlink{Nancy E.\ Clarke}{N.E.~Clarke} and \hyperlink{Michele Conforti}{M.~Conforti}.
\authrefspace

\author{Samuel Fiorini, Nadia Hardy, Bruce Reed, \& Adrian Vetta}
\authorspace
\bsgitem{2005a}{}  Approximate min-max relations for odd cycles in planar graphs.  
In:  M.\ J\"unger and V.\ Kaibel, eds., \emph{Integer Programming and Combinatorial Optimization} (11th Int.\ IPCO Conf., IPCO 2005, Berlin), pp.\ 35--50.  \LNCS 3509.  Springer, Berlin, 2005.
\MR{2210011} (2006j:90108).  \Zbl{1119.90360}.					
\annot{} See \xrefd{2007a}{Samuel Fiorini, Nadia Hardy, Bruce Reed, \& Adrian Vetta}.
\codes{\CSG:~\CFr,~\CCircles}%
\itemspace

\bsgitem{2007a}{}  Approximate min-max relations for odd cycles in planar graphs.  \emph{Math.\ Programming, Ser.\ B} 110 (2007), no.\ 1, 71--91.
\MR{2306131} (2008b:05087).  \Zbl{1113.05054}.					
\annot{} $\nu :=$ max number of vertex-disjoint negative circles; $\nu' :=$ edge analog.  $\rho :=$ min size of a transversal of negative face boundaries.  
Thm.\ 3 (\xref{Kr\'al and Voss}{2004a}{Daniel Kral' \& Heinz-Jurgen Voss}):  frustration index $l(\Sigma) \leq 2 \nu'$.  (Here, a shorter proof.)
Thm.\  4:  For an unbalanced signed plane graph, vertex frustration number $l_0(\Sigma) \leq 7\nu(\Sigma) + 3\rho(\Sigma) - 8$.  [Improved by \xref{Kr\'al', Sereni, and Stacho}{2012a}{Daniel Kral', Jean-Sebastien Sereni, \& Ladislav Stacho}.]
Cor.\ 2:  $l(\Sigma) \leq 10\nu(\Sigma)$.
Dictionary:  ``odd'' = negative, ``even'' = positive.
\dated{6 Feb 2011} 
\codes{\CSG:~\CFr,~\CCircles}%
\itemspace

\author{Samuel Fiorini \& Gwena\"el Joret}
\authorspace
\bsgitem{2009a}{}  On a theorem of Sewell and Trotter.  \emph{European J.\ Combin.}\ 30 (2009), 425--428.
\MR{2489274} (2010b:05131).  \Zbl{1229.05142}.  
\annot{} Short proof of \xref{Sewell and Trotter}{1993a}{E.C.\ Sewell \& L.E.\ Trotter, Jr.}.  
Dictionary:  ``totally odd $K_4$-subdivision'' = ``even subdivision of $K_4$'' (\xref{Sewell and Trotter}{1993a}{E.C.\ Sewell \& L.E.\ Trotter, Jr.}).
\dated{14 Feb 2013} 
\codes{\Csg:~\Cpar:~\CStr}%
\itemspace

\authorx{E.\ Fischer}							%FISCHER E
\author{E.\ Fischer, J.A.\ Makowsky, \& E.V.\ Ravve}
\authorspace
\bsgitem{2008a}{}  Counting truth assignments of formulas of bounded tree-width or clique-width.  \emph{Discrete Appl.\ Math.}\ 156 (2008), no.\ 4, 511--529.
\MR{2379082} (2009k:68090).  \Zbl{1131.68093}.					
\annot{} The incidence graph of clauses is a signed bipartite graph.  
\dated{16 Jan 2012}
\codes{\CSG}%
\itemspace

\authorx{Ilse Fischer}						%FISCHER I
\author{Ilse Fischer \& C.H.C.\ Little}
\authorspace
\bsgitem{2004a}{}  Even circuits of prescribed clockwise parity.  \emph{Electronic J.\ Combin.}\ 10 (2003), Research Paper 45, 20 pp.   
\MR{2014532} (2004h:05071).  \Zbl{1031.05073}.
\codes{\CSG}%
\itemspace

\authorx{K.H.\ Fischer}						%FISCHER KH
\author{K.H.\ Fischer \& J.A.\ Hertz}
\authorspace
\bsgitem{1991a}{}  \emph{Spin Glasses}.  Cambridge Studies in Magnetism, Vol.\  1.  Cambridge Univ.\ Press, Cambridge, Eng., 1991.
\MR{(93m:82019)} (93m:82019).  
\annot{} An excellent introduction to many aspects of physics (mainly theoretical) that often seem to be signed graph theory or to generalize it, e.g., by randomly weighting the edges.
\codes{\CPhys:~\Csg:~\Cfr:~\CExp,~\CRef}%
\annotctd{} \S2.5, ``Frustration'', discusses the spin glass Ising model (essentially, signed graphs) in square and cubical lattices, including the ``Mattis model'' (a switching of all positive signs), as well as a vector analog, the ``XY'' model (planar spins) and (p.\ 46) even a general gain-graph model with switching-invariant Hamiltonian. 
\codes{\CPhys:~\CSG:~\CFr,~\CSw:~\CExp,~\CRef}%
\annotctd{} Ch.~3 concerns the Ising and Potts models.  In \S3.7: ``The Potts glass'', the Hamiltonian (without edge weights) is $H = - \frac 12 \sum \sigma(e_{ij})(k \delta(s_i, s_j) - 1)$.  [It is not clear that the authors intend to permit negative edges.  If they are allowed, $H$ is rather like \xref{Doreian and Mrvar's}{1996a}{Patrick Doreian \& Andrej Mrvar} $P(\pi)$.  \Question.  Is there a worthwhile generalized signed and weighted Potts model with Hamiltonian that specializes both to this form of $H$ and to $P$?]  [Also \cf\ \xref{Welsh}{1993a}{D.J.A.\ Welsh} on the Ashkin--Teller--Potts model.]
\par
\codes{\CPhys:~\Csg,~\Cclu:~\CExp}%
\itemspace

\author{Steven D.\ Fischer}						%FISCHER SD
\authorspace
\bsgitem{1993a}{}  \emph{Signed Poset Homology and $q$-Analog M\"obius Functions.}  Doctoral thesis, University of Michigan, 1993.
\annot{} \S1.2: ``Signed posets''.  Definition of signed poset: a positively closed subset of the root system $B_n$ whose intersection with its negative is empty.  (Following Reiner (1990).)  Equivalent to a partial ordering of $\pm[n]$ in which negation is a self-duality and each dual pair of elements is comparable.  [This is really a special type of signed poset.  The latter restriction does not hold in general.]
\annotctd{} Relevant contents:  Ch.\ 2: ``Cohen-Macaulay signed posets'', \S2.2: ``EL-labelings of posets and signed posets'', and shellability.  Ch.\ 3: ``Euler characteristics'', and a fixed-point theorem.  \S5.1: ``The homology of the signed posets $S_\Pi$'' (a particular example).  App.\ A: ``Open problems'', several concerning signed posets.
\annotctd{} [Partially summarized by \xref{Hanlon}{1996a}{Phil Hanlon}.] 
\par
\codes{\CS:~\Csg,~\Cori,~\CGeom,~\CInvar}%
\itemspace

\authorx{M.\ Hamit Fi\c{s}ek}						%FISEK
\authorx{M.\ Hamit Fisek}
\author{M.\ Hamit Fi\c{s}ek, Robert Z.\ Norman, \& Max Nelson-Kilger}
\authorx{M.\ Hamit Fisek, Robert Z.\ Norman, \& Max Nelson-Kilger}
\authorspace
\bsgitem{1992a}{}  Status characteristics and expectation states theory: A priori model parameters and test.  \emph{J.\ Math.\ Sociology} 16 (1992), no.\ 4, 285--303.
\Zbl{741.92024}.  
\annot{} The ``strength'' of a path depends on the number of edges of each sign.
\dated{10 Nov 2012} 
\codes{\CPsS:~\CSG}%
\itemspace

\authorx{P.C.\ Fishburn}		 				%FISHBURN
\author{P.C.\ Fishburn \& N.J.A.\ Sloane}
\authorspace
\bsgitem{1989a}{} The solution to Berlekamp's switching game.  \emph{Discrete Math.}\ 74 (1989), 263--290.  
\MR{0992740} (90e:90151).  \Zbl{664.94024}.
\annot{} The maximum frustration index of a signed $K_{t,t}$, which equals the covering radius of the Gale--Berlekamp code, is evaluated for $t \leq 10$, thereby extending results of \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}.  See Table 1.  [Corrected and extended by \xref{Carlson and Stolarski}{2004a}{Jordan Carlson \& Daniel Stolarski}.]
\codes{\Csg:~\CFr}%
\itemspace

\author{Susanna Fishel}						%FISHEL
\authorspace
\bsgitem{2019a}{}  A survey of the Shi arrangement.  In:  H\'el\`ene Barcelo \etal, eds., \emph{Recent Trends in Algebraic Combinatorics}, pp.\ 75--113.  Assoc.\ Women Math.\ Ser., Vol.\ 16.  Springer, Cham, 2019. 
\MR{3969572}.  \Zbl{1417.52033}.  \arXiv{1909.01257}.  
\annot{} Much geometry, no gain graphs [though a gain graph is implicit; \cf\ e.g.\ \xref{Zaslavsky}{2003a}{Thomas Zaslavsky}].
\dated{22 Jan 2020} 
\codes{\CGeom:~\Cgg:~\CInvar}%
\itemspace

\authorx{Michael E.\ Fisher}						%FISHER ME
\author{Michael E.\ Fisher \& Rajiv R.P.\ Singh}
\authorspace
\bsgitem{1990a}{}  Critical points, large-dimensionality expansions, and the Ising spin glass.  In:  G.R.\ Grimmett and D.J.A.\ Welsh, eds., \emph{Disorder in Physical Systems:  A Volume in Honour of John M.\ Hammersley on the Occasion of His 70th Birthday}, pp.\ 87--111.  Clarendon Press, Oxford, 1990.
\MR{1064557} (91j:82021).  \Zbl{725.60111}.  
\annot{} Physics questions, e.g., phase transitions and high-temperature expansions, for signed lattice graphs ($\pm J$ spins) and with random weights (Gaussian edge weights).
\dated{24 Aug 2012} 
\codes{\Csg:~\CPhys:~\CFr:~\CExp}%
\itemspace

\author{Claude Flament} 						%FLAMENT
\authorspace
\bsgitem{1958a}{}  L'\'etude math\'ematique des structures psycho-sociales.  \emph{L'Ann\'ee Psychologique} 58 (1958), 119--131.  
\annot{} Signed graphs are treated on pp.\ 126--129. 
\codes{\CSG:~\CBal,~\CPsS:~\CExp}%
\itemspace

\bsgitem{1963a}{} \emph{Applications of Graph Theory to Group Structure}.  Prentice-Hall, Englewood Cliffs, N.J., 1963.  
\MR{0157785} (28 \#1014).  \Zbl{141.36301} (141, p.\ 363a).
\annot{} English edition of \xrefd{1965a}{Claude Flament}.  Ch.~3:  ``Balancing processes.'' 
\par
\codes{\CSG:~\CKG:~\CBal,~\CAlgor:~\CExp}%
\itemspace

\bsgitem{1965a}{} \emph{Th\'eorie des graphes et structures sociales}.  Math.\ et sci.\ de l'homme, Vol.\ 2.  Mouton and Gauthier-Villars, Paris, 1965.  
\MR{0221966} (36 \#5018).  \Zbl{169.26603} (169, p.\ 266c).   
\annot{} Ch.~III:  ``Processus d'\'equilibration.'' 
\codes{\CSG:~\CKG:~\CBal,~\CAlgor:~\CExp}%
\itemspace

\bsgitem{1970a}{}  \'Equilibre d'un graphe, quelques r\'esultats alg\'ebriques.  \emph{Math.\ Sci.\ Humaines}, No.\ 30 (1970), 5--10.  
\MR{0278978} (43 \#4704).  \Zbl{222.05124}.
\codes{\CBal}
\itemspace

\bsgitem{1979a}{} Independent generalizations of balance.  
In:  Paul W.\ Holland and Samuel Leinhardt, eds., \emph{Perspectives on Social Network Research} (Proc.\ Sympos., Dartmouth Coll.,  Hanover, N.H., 1975), Ch.\ 10, pp.\ 187--200.  Academic Press, New York, 1979.  
\codes{\CSG:~\CBal,~\CPsS:~\CExp}%
\itemspace

\author{Evelyne Flandrin}						%FLANDRIN
\authorspace
\authref  See \hyperlink{Ralph J.\ Faudree}{R.J.~Faudree}.
\itemspace

\author{Erica Flapan}						%FLAPAN
\authorspace
\bsgitem{1995a}{}  Intrinsic chirality.  \emph{J.\ Molecular Structure (Theochem)} 336 (1995), 157--164.
\annot{} Intrinsic chirality means the graph cannot be embedded in 3-space without a twist.  [\Question.  Can this be interpreted in terms of signed graphs?]
\Cf\ \xrefd{1998a}{Erica Flapan}, \xref{Flapan and Weaver}{1996a}{Erica Flapan \& Nikolai Weaver}, \xref{Hu and Qiu}{2009a}{Guang Hu \& Wen-Yuan Qiu}.
\dated{4 Nov 2010} 
\codes{\Csg:~\CTop:~\CChem}%
\itemspace

\bsgitem{1998a}{}  Knots and graphs in chemistry.  \emph{Chaos, Solitons \& Fractals} 9 (415) (1998), 547--560.
\MR{1628741} (99c:57017).  \Zbl{933.57002}.					
\annot{} A survey of chirality of 3-space embeddings.  See \xrefd{1995a}{Erica Flapan}.
\dated{4 Nov 2010} 
\codes{\Csg:~\CTop:~\CChem:~\CExp}%
\itemspace

\author{Erica Flapan \& Nikolai Weaver}
\authorspace
\bsgitem{1996a}{}  Intrinsic chirality of 3-connected graphs.  \emph{J.\ Combinatorial Theory Ser.\ B} 68 (1996), 223--232.
\MR{1417798} (97k:05058).  \Zbl{861.05023}.					
\annot{} See \xref{Flapan}{1995a}{Erica Flapan}.
\dated{4 Nov 2010} 
\codes{\Csg:~\CTop}%
\itemspace

\authorx{T.\ Fleiner}							%FLEINER
\author{T.\ Fleiner \& G.\ Wiener}
\authorspace
\bsgitem{2016a}{\dag}  Coloring signed graphs using DFS.  \emph{Optimization Lett.}\ 10 (2016), no.\ 4, 865--869.
\MR{3477382}.  \Zbl{1336.05055}.  \arXiv{1612.03280}.				
\annot{} An elegant short proof by depth-first search of the signed-graph Brooks' theorem of \xref{M\'a\v{c}ajov\'a, Raspaud, and \v{S}koviera}{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}.  [More general short proof by \xref{Zaj\c{a}c}{20xxa}{Mariusz Zajac}.]
\dated{17 Mar, 23 May 2017, rev 9 Feb 2020} 
\codes{\CSG:~\CCol:~\CAlgor}%
\itemspace

\author{Herbert Fleischner}						%FLEISCHNER
\authorspace
\bsgitem{1991a}{}  \emph{Eulerian Graphs and Related Topics}.  Part 1, Vol.\ 2.
Ann.\ Discrete Math., Vol.\ 50. North-Holland, Amsterdam, 1991.
\MR{1113484} (92f:05066).  \Zbl{792.05092}.					
\codes{\CSG:~\COri,~\CFlows}
\itemspace

\author{Laura Floresc}						%FLORESCU
\authorspace
\authref  See \hyperlink{Joel Spencer}{J.~Spencer}.
\itemspace

\author{Rigoberto Fl\'orez}						%FLOREZ
\authorx{Rigoberto Florez}
\authorspace
\authref  See also \hyperlink{Matthew Blair}{M.~Blair}.
\authrefspace

\bsgitem{2005a}{}  \emph{Four Studies in the Geometry of Biased Graphs}.  Doctoral dissertation, State University of New York at Binghamton, 2005.
\MR{2707450}.  
\annot{} Published as \xrefd{2006a}{Rigoberto Florez}, \xrefd{2009a}{Rigoberto Florez}, and \xref{Fl\'orez and Forge}{2007a}{Rigoberto Florez \& David Forge}, and in \xref{Fl\'orez and Zaslavsky}{2020a}{Rigoberto Florez \& Thomas Zaslavsky}.
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{2006a}{}  Lindstr\"om's conjecture on a class of algebraically non-representable matroids.  \emph{European J.\ Combin.}\ 27 (2006), no.\ 6, 896--905.
\MR{2226425} (2006m:05048).  \Zbl{1090.05010}.				
\annot{} Lindstr\"om conjectured that a certain matroid $M(n)$ is algebraically nonrepresentable if $n$ is nonprime.  Proved by showing that $M(n)$ extends by harmonic conjugation to $\bfL_0(\bbZ_n K_3)$, which in turn extends to a contradiction if $n$ is composite.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2009a}{}  Harmonic conjugation in harmonic matroids.  \emph{Discrete Math.}\ 309 (2009), no.\ 8, 2365--2372.
\MR{2510362} (2010f:05038).  \Zbl{1207.05027}.					
\annot{} In a harmonic matroid $H$, harmonic conjugates exist and are unique.  If $\bfL_0(\fG K_3) \subseteq H$ and $\fG = \bbZ$ or $\bbZ_p$, then the closure of $\bfL_0$ under harmonic conjugation is a projective plane over $\bbQ$ or $\GF(p)$, as appropriate.
\par
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Rigoberto Fl\'orez \& David Forge}
\authorx{Rigoberto Florez \& David Forge}
\authorspace
\bsgitem{2007a}{}  Minimal non-orientable matroids in a projective plane.  \emph{J.\ Combin.\ Theory Ser.\ A} 114 (2007), no.\ 1, 175--183.
\MR{2276967} (2007h:05031).  \Zbl{1120.52012}.					
\annot{} The minimal matroids are contained in lift matroids of $\bbZ_n K_3$.
\par
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2023a}{}  Activity from matroids to rooted trees and beyond.  \emph{J.\ Combin.\ Theory Ser.\ A}\ 198 (2023), art.\ 104755, 21 pp.
\arXiv{2209.03446}.  
\codes{\CMatrd:~\CInvar:~\CGG}%
\itemspace

\author{Rigoberto Fl\'orez \& Thomas Zaslavsky}
\authorx{Rigoberto Florez \& Thomas Zaslavsky}
\authorspace
\bsgitem{2019a}{}  Biased graphs.  VI.\  Synthetic geometry.  \emph{European J.\ Combin.}\ 81 (2019), 119--141.
\MR{3953388}.  \Zbl{1420.05030}.  \arXiv{1608.06021}.				
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{2020a}{}  Projective planarity of matroids of $3$-nets and biased graphs.  \emph{Australasian J.\ Combin.}\ 76 (2020), no.\ 2, 299--338.
\MR{4055697}.  \Zbl{1439.05046}.  \arXiv{1708.00095}.				
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\author{Joel Foisy}							%FOISY
\authorspace
\authref  See also \hyperlink{Yen Duong}{Y.~Duong}.
\authrefspace

\bsgitem{20xxa}{}  A spatial version of Tutte's conflict graph.  Submitted.  
\arXiv{2207.06244}.  
\annot{} Simplified description:  Embed $\Gamma$ in $\bbR^3$ with an $S^2$ that contains a maximal planar subgraph $M$.  The conflict graph is $\Sigma_M$ with $V(\Sigma_M) = E(\Gamma)\setm E(M)$.  $e,f \in V(\Sigma_M)$ may be obliged by certain criteria to be on the same side, or opposite sides, of $S^2$; then join them in $\Sigma_M$ by a positive, or negative, edge.  Thm.:  If every conflict graph is unbalanced, $\Gamma$ cannot be linklessly embedded in $S^3$.  There are complications.
\dated{19 Jul 2022} 
\codes{\CSG:~\CTop}%
\itemspace

\author{Joel Foisy \& Justin Raimondi}
\authorspace
\bsgitem{20xxa}{}  Conflict graphs of maximally planar subgraphs of Petersen family graphs.  Submitted.  
\arXiv{2207.06251}.  
\annot{} Technical propositions about maximal planar subgraphs and their signed conflict graphs that are used in \xref{Foisy}{20xxa}{Joel Foisy}.
\dated{19 Jul 2022} 
\codes{\CTop,~\CSG}%
\itemspace

\author{Wungkum Fong}						%FONG WK
\authorspace
\bsgitem{2000a}{}  \emph{Triangulations and Combinatorial Properties of Convex Polytopes}.  Doctoral dissertation, Massachusetts Inst.\ of Technology, 2000.
\annot{} A configuration consists of the vectors representing an acyclic orientation of a complete signed graph.  The volume of the pyramid over the configuration with apex at the origin.  
[\xref{Ohsugi and Hibi}{2003a}{Hidefumi Ohsugi \& Takayuki Hibi} treats a similar problem.  \Question.  Is there a connection with the chromatic polynomial?]
\dated{11 Apr 2011} 
\codes{\Csg:~\CGeom:~\CInvar}%
\itemspace

\author{Carlos M.\ da Fonseca}						%FONSECA
\authorspace
\authref  See \hyperlink{Abdullah Alazemi}{A.~Alazemi}, \hyperlink{Milica Andelic}{M.~An\'deli\'c}, and \hyperlink{Slobodan K.\ Simic}{S.K.~Simi\'c}.
\itemspace

\authorx{Angela Fontan}						%FONTAN
\author{Angela Fontan \& Claudio Altafini}
\authorspace
\bsgitem{2018a}{}  Achieving a decision in antagonistic multiagent networks: frustration determines commitment strength.  In:  \emph{2018 IEEE Conference on Decision and Control (CDC)} (Miami Beach, 2018), pp.\ 109--114.  IEEE, 2018.
\codes{\CSG:~\CDyn,~\CFr}%
\itemspace

\authorx{Lo{\"\i}c Forest}						%FOREST
\authorx{Loic Forest}
\author{Lo{\"\i}c Forest, Nicolas Glade, \& Jacques Demongeot}
\authorspace
\bsgitem{2007a}{}  Li\'enard systems and potential--Hamiltonian decomposition -- Applications in biology.  \emph{C.R.\ Biologies} 330 (2007), 97--106.
\annot{} P.\ 101 and Fig.\ 5 describe the ``regulon'', a signed digraph of order 2 with two stable states.
\dated{23 Aug 2017} 
\codes{\CBiol:~\CSD:~\CExp}%
\itemspace

\author{G.\ Forgacs}							%FORGACS
\authorspace
\authref  See also \hyperlink{S.T.\ Chui}{S.T.~Chui} and \hyperlink{B.W.\ Southern}{B.W.~Southern}.
\authrefspace

\bsgitem{1980a}{}  Ground-state correlations and universality in two-dimensional fully frustrated systems.  \emph{Phys.\ Rev.\ B}\ (3) 22 (1980), no.\ 9, 4473--4480.
\MR{0590596} (81i:82066).  
\annot{} 
Dictionary:  ``fully frustrated Ising model on a square lattice'' = signed grid (square lattice) graph in which every quadrilateral is negative; ``plaquette'' = ``square'' = region boundary = quadrilateral.
\codes{\CPhys:~\Csg}%
\itemspace

\author{G.\ Forgacs \& E.\ Fradkin}
\authorspace
\bsgitem{1981a}{}  Anisotropy and marginality in the two-dimensional fully frustrated Ising model.  \emph{Phys.\ Rev.\ B}\ 23 (3) (1981), no.\ 7, 3442--3447.
\MR{0607834} (82c:82094).  
\par
\codes{\CPhys:~\Csg}%
\itemspace

\author{David Forge}							%FORGE
\authorspace
\authref  See also \hyperlink{Pascal Berthome}{P.~Berthom\'e}, \hyperlink{Sylvie Corteel}{S.~Corteel}, and \hyperlink{Rigoberto Florez}{R.~Fl\'orez}.
\authrefspace

\bsgitem{2014a}{}  Linial arrangements and local binary search trees.  Manuscript, 2014.
\arXiv{1411.7834}.  
\codes{\CGG:~\CGeom}%
\itemspace

\author{David Forge \& Thomas Zaslavsky}
\authorspace
\bsgitem{2007a}{}  Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements.  \emph{J.\ Combin.\ Theory Ser.\ A} 114 (2007), no.\ 1, 97--109.
\MR{2275583} (2007i:52026).  \Zbl{1105.52014}.  \arXiv{math/0609051}.  
\annot{} The number of proper integral $m$-colorings of a rooted integral gain graph (root $v_0$ and a function $h: V \to \bbZ$ such that there are root edges $ge_{0i}$ for all $g\in(-\infty,h_i]$; the rest of the gain graph is finite).
\par
\codes{\CGG:~\CGeom,~\CInvar,~\CMatrd}%
\itemspace

\bsgitem{2016a}{}  Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 118 (2016), 186--227.
\MR{3471850}.  \Zbl{1317.05081}.  \arXiv{1306.6132}.  \HAL{hal-01724088}.		
\annot{} A weighted gain graph has lattice-ordered gain group and has vertex weights from an abelian semigroup acted upon by the gain group.  
The total dichromatic polynomial is a Tutte invariant (satisfying deletion-contraction and multiplicativity) with possibly uncountably many variables, but is not the universal one.  \Problem.  Find the universal Tutte invariant.  With integral gain group and integral weights, the integral chromatic function of \xrefd{2007a}{David Forge \& Thomas Zaslavsky} is an evaluation of the polynomial.  
Another special case is the polynomial of S.D.\ Noble and D.J.A.\ Welsh, A weighted graph polynomial from chromatic invariants of knots [Symposium \`a la M\'emoire de Fran\c{c}ois Jaeger (Grenoble, 1998).  \emph{Ann.\ Inst.\ Fourier (Grenoble)} 49 (1999), no.\ 3, 1057--1087].
\codes{\CGG:~\CInvar,~\CMatrd}%
\itemspace

\author{Robin Forman}						%FORMAN
\authorspace
\bsgitem{1993a}{}  Determinants of Laplacians on graphs.  \emph{Topology} 32 (1993), no.\ 1, 35--46.
\MR{1204404} (94g:58247).  \Zbl{780.05041}.  
\codes{\Cgg:~\CLap}%
\itemspace

\authorx{C.M.\ Fortuin}						%FORTUIN
\author{C.M.\ Fortuin \& P.W.\ Kasteleyn}
\authorspace
\bsgitem{1972a}{}  On the random cluster model.  I.  Introduction and relation to other models.  
\emph{Physica} 57 (1972), 536--564.
\MR{0359655} (50 \#12107).  
\annot{} Most of the paper recasts classical physical and other models (percolation, ferromagnetic Ising, Potts, graph coloring, linear resistance) in a common form that is generalized in \S7, ``Random cluster model''.  The ``cluster (generating) polynomial'' $Z(\Gamma; p, \kappa)$, where $p \in \bbR^E$ and $\kappa \in \bbR$, is a $1$-variable specialization of the general parametrized dichromatic polynomial.  In the notation of \xref{Zaslavsky}{1992b}{Thomas Zaslavsky} it equals $Q_\Gamma(q,p; \kappa,1)$, where $q_e = 1 - p_e$.  Thus it partially anticipates the general polynomials of \xref{Przytycka and Przytycki}{1988a}{Teresa M.\ Przytycka \& Jozef H.\ Przytycki}, \xref{Traldi}{1989a}{Lorenzo Traldi}, and \xref{Zaslavsky}{1992b}{Thomas Zaslavsky} that were based on \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} sign-colored Tutte polynomial.  A spanning-tree expansion is given only for the resistance model.  A feature [that seems not to have been taken up by subsequent workers] is the differentiation relation (7.7) connecting $\partial \ln Z/{\partial q_e}$ with [I think!] the expectation that the endpoints of $e$ are disconnected in a subgraph.  [\xref{Grimmett}{1994a}{G.\ Grimmett} summarizes subsequent work in the probabilistic direction.]
\codes{\Csgc:~\CGen:~\CInvar,~\CPhys}%
\itemspace

\author{Florent Foucaud}						%FOUCAUD
\authorspace
\authref  See also \hyperlink{Laurent Beaudou}{L.~Beaudou}, \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}, and \hyperlink{Francois Dross}{F.~Dross}.
\authrefspace

\author{Florent Foucaud, Herv\'e Hocquard, Dimitri Lajou, Valia Mitsou, \& Th\'eo Pierron}
\authorspace
\bsgitem{2019a}{}  Parameterized complexity of edge-coloured and signed graph homomorphism problems.  In:  \emph{14th International Symposium on Parameterized and Exact Computation}, art.\ 15, 16 pp.  LIPIcs.\ Leibniz Int.\ Proc.\ Informatics, Vol.\ 148.  Schloss Dagstuhl -- Leibniz-Zent.\ Inform., Wadern, 2019.
\MR{4042062}.  
\arXiv{1910.01099}.  
\codes{\CSG,~\CSGc:~\CHom}%
\itemspace

\bsgitem{2022a}{}  Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity.  \emph{Algorithmica} 84 (2022), no.\ 5, 1183--1212.
\MR{4414088}.  
\codes{\CSG,~\CSGc:~\CHom}%
\itemspace

\author{Florent Foucaud \& Reza Naserasr}
\authorspace
\bsgitem{2014a}{}  The complexity of homomorphisms of signed graphs and signed constraint satisfaction.  In:  Alberto Pardo and Alfredo Viola, eds., \emph{LATIN 2014: Theoretical Informatics} (Proc.\ 11th Latin American Symp., Montevideo, 2014), pp.\ 526--537.  \LNCS 8392.  Springer, Berlin, 2014.
\MR{3188136}.  \Zbl{06276036}.  
\codes{\CSG:~\CHom,~\CAlgor}%
\itemspace

\authorx{Louiza Fouli}						%FOULI
\author{Louiza Fouli \& Susan Morey}
\authorspace
\bsgitem{2012a}{}  Minimal reductions and cores of edge ideals.  \emph{J.\ Algebra} 364 (2012), 52--66.
\MR{2927047}.  \Zbl{1279.13012}.  \arXiv{1012.5746}.  
\codes{\CAlgeb:~\Csg:~\Cpar}%
\itemspace

\author{J.-L.\ Fouquet}						%FOUQUET
\authorspace
\authref  See \hyperlink{Claude Berge}{C.~Berge}.
\itemspace

\author{J.-C.\ Fournier}						%FOURNIER
\authorspace
\bsgitem{1979a}{}  \emph{Introduction \`a la notion de matro\"{\i}de (g\'eom\'etrie combinatoire)}.  Publ.\ Math.\ d'Orsay, [No.] 79-03.  D\'ep.\ Math., Universit\'e Paris-Sud, Orsay, 1979.  
\MR{0551494} (81a:05027).  \Zbl{424.05018}.
\annot{} \S 3.12: ``Matro\"{\i}des de Dowling'' (p.\ 52).  Definition by partial $\fG$-partitions and the linear representability theorem.
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\author{Patrick W.\ Fowler}						%FOWLER
\authorspace
\authref  See also \hyperlink{Nino Basic}{N.~Basic}.
\authrefspace

\bsgitem{2002a}{}  H\"uckel spectra of M\"obius $\pi$ systems.  \emph{Chem.\ Phys.\ Phys.\ Chem.}\ 4 (2002), no.\ 13, 2878--2883.
\annot{} \S3, ``Double covers and quotient surfaces'':  A remarkable theorem:  The spectrum of $A(\tilde\Sigma)$ is $\Spec A(\Sigma) \cup \Spec A(|\Sigma|)$.  Also, the eigenvectors of $A(\tilde\Sigma)$ are $(x,x)$ and $(x,-x)$ for $x$ an eigenvector of $A(|\Sigma|)$ and of $A(\Sigma)$, respectively.  
A nice topological proof using orientation embedding of $\Sigma$ in a surface $S$ and the lift to an embedding of $\tilde\Sigma$ in the orientable double cover of $S$.  [The graph theorem, not explicit, follows from the fact that every signed graph has an orientation embedding in some surface.]  
[Reproved independently, directly on the signed graph, by \xref{Bilu and Linial}{2006a}{Yonatan Bilu \& Nathan Linial}, \xref{Kalita and Pati}{2012a}{Debajit Kalita \& Sukanta Pati}, \xref{Gregory}{2012a}{David A.\ Gregory}.  Generalized to branched coverings in \xref{Butler}{2010a}{Steve Butler}.]  
\annotctd{} \S4, ``Chemical examples'':  (i), ``Monocyclic rings'':  The eigenvalues and eigenvectors of a positive or negative circle, derived from the Frost--Musulin circle.  [Reproved by \xref{Mathai and Zaslavsky}{2012a}{A.M.\ Mathai \& Thomas Zaslavsky} by a different method.  Also proved by others.]  An interesting discussion of the odd-order case.  (ii), ``In-plane $\pi$ systems'', (iii), ``Cyclic polyacenes'':  Chemical applications.
\dated{13 Jan 2015} 
\par
\codes{\Csg:~\CCov,~\CAdj:~\CEig,~\CChem}%
\itemspace

\bsgitem{2011a}{}  M\"obius systems and the Estrada index.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 66 (2011), no.\ 3, 751--764.
\MR{2884762} (2012m:05386).  \Zbl{1265.05366}.  
\par
\codes{\Csg:~\CEig,~\CCov,~\CChem}%
\itemspace

\author{Eduardo Fradkin}						%FRADKIN
\authorspace
\authref  See also \hyperlink{G.\ Forgacs}{G.~Forgacs}.
\authrefspace

\author{Eduardo Fradkin, B.A.\ Huberman, \& Stephen H.\ Shenker}
\authorspace
\bsgitem{1978a}{}  Gauge symmetries in random magnetic systems.  \emph{Phys.\ Rev.\ B} 18 (1978), no.\ 9, 4789--4814.
\annot{} Properties of physical interest of switching classes (``gauge symmetric'' properties) of signed graphs.
	
\dated{11 Jan 2015} 
\codes{\CPhys,~\CSG:~\CFr}%
\itemspace

\authorx{Aviezri S.\ Fraenkel}						%FRAENKEL
\author{Aviezri S.\ Fraenkel \& Peter L.\ Hammer}
\authorspace
\bsgitem{1984a}{}  Pseudo-Boolean functions and their graphs.  
In: \emph{Convexity and Graph Theory} (Jerusalem, 1981), pp.\ 137--146.  North-Holland Math.\ Stud., 87.  North-Holland, Amsterdam, 1984.  
\MR{0791023} (87b:90147).  \Zbl{557.94019}.
\codes{\Csh:~\Clg}%
\itemspace

\author{F.A.M.\ Fran\c{c}a}						%FRANCA
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Elisa Franco}							%FRANCO
\authorspace
\authref  See \hyperlink{Franco Blanchini}{F.~Blanchini}.
\itemspace

\author{Andr\'as Frank}						%FRANK A
\authorx{Andras Frank}
\authorspace
\bsgitem{1990a}{}  Packing paths, circuits, and cuts -- a survey.  
In:  B.~Korte, L.~Lov\'asz, H.J.~Pr\"o\-mel, and A.~Schrijver, eds., \emph{Paths, Flows, and VLSI-Layout}, pp.\ 47--100.  Algorithms and Combinatorics, Vol.\ 9.  Springer-Verlag, Berlin, 1990.
\MR{1083377} (91i:68116).  \Zbl{741.05042}.  
\annot{} Pp.\ 89--91:  Additively sign-weighted bipartite graphs.  Thms.\ 8.1$'$, 8.5$'$:  Criteria for negative circuit.  
[\Questions.  Is there a generalization to antibalanced signed graphs with additive sign-weights?  Does the existence of minors help?]  
Thms.\ 8.1$'$w, 8.1$''$:  Similar, for $\bbZ^+$-weighted or $\bbQ^+$-weighted graphs, not necessarily bipartite.  
Pp.\ 91--92 mention \xref{Gerards}{1990a}{A.M.H.\ Gerards} and graphs with a bipartizing vertex.
\dated{11 Jun 2012} 
\par
\codes{\CSGw,~\CGGw:~\COG}%
\itemspace

\bsgitem{1996a}{} A survey on $T$-joins, $T$-cuts, and conservative weightings.  
In:  D.\ Mikl\'os, V.T.\ S\'os, and T.\ Sz\H onyi, eds., \emph{Combinatorics, Paul Erd\H os is Eighty}, Vol.\ 2,  pp.\ 213--252.  Bolyai Soc.\ Math.\ Stud., 2.  J\'anos Bolyai Math.\ Soc., Budapest, 1996.  
\MR{1395861} (97c:05115).  \Zbl{846.05062}.					
\annot{} A ``conservative $\pm 1$-weighting'' of $G$ is an edge labelling by $+1$'s and $-1$'s so that in every circle the sum of edge weights is nonnegative.  It is a tool in several theorems.  [Related: \xref{Ageev, Kostochka, and Szigeti}{1995a}{A.A.\ Ageev, A.V.\ Kostochka, \& Z.\ Szigeti}, \xref{Seb\"o}{1990a}{Andras Sebo}.]
\codes{\CSGw:~\CStr,~\CAlgor:~\CExp,~\CRef}%
\itemspace

\authorx{Howard Frank}		 				%FRANK H
\author{Howard Frank \& Ivan T.\ Frisch}
\authorspace
\bsgitem{1971a}{}  \emph{Communication, Transmission, and Transportation Networks}.  Addison-Wesley, Reading, Mass., 1971.  
\MR{0347343} (49 \#12063).  \Zbl{281.94012}.  
\annot{} \S6.12:  ``Graphs with gains,'' pp.\ 277--288.
\codes{\CGN:~\CExp}%
\itemspace

\authorx{Ove Frank}		 					%FRANK O
\author{Ove Frank \& Frank Harary}
\authorspace
\bsgitem{1979a}{}  Balance in stochastic signed graphs.  \emph{Social Networks} 2 (1979/80), 155--163.  
\MR{0569277} (81e:05116).  
\annot{} An edge is present with probability $\alpha$ and positive with probability $p$.  They compute the expected values of two kinds of measures of imbalance: the number of balanced triangles (whose variance is also given), and the number of induced subgraphs of order $3$ having specified numbers of positive and negative edges.  [Related: \xref{\v{S}koviera}{1992a}{Martin Skoviera}, \xref{A.T.\ White}{1994a}{Arthur T.\ White}.]
\codes{\CSG:~\CRand,~\CFr}%
\itemspace

\author{Giancarlo Franzese}						%FRANZESE
\authorspace
\bsgitem{1996a}{}  Cluster analysis for percolation on a two-dimensional fully frustrated system.  \emph{J.\ Phys.\ A} 29 (1996), 7367--7375.
\Zbl{904.60081}.  
\annot{} The ``fully frustrated'' square lattice: alternate verticals are negative.  Extending \xref{Kandel, Ben-Av, and Domany}{1990a}{Daniel Kandel, Radel Ben-Av, \& Eytan Domany} by studying cluster properties in simulations, e.g., percolating clusters (that connect opposite sides of the lattice).  Illuminating diagrams.
\dated{18 Jun 2012} 
\par
\codes{\CPhys,~\CSG:~\CClu}%
\itemspace

\authorx{Maria Aguieiras A.\ de Freitas}					%FREITAS
\author{Maria Aguieiras A.\ de Freitas, Nair M.M.\ de Abreu, Renata R.\ Del-Vecchio, \& Samuel Jurkiewicz}
\authorspace
\bsgitem{2010a}{}  Infinite families of $Q$-integral graphs.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 9, 2352--2360.
\MR{2599865} (2011b:05150).  \Zbl{1219.05158}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Maria Freitas, Renata Del-Vecchio, \& Nair Abreu}
\authorspace
\bsgitem{2010a}{}  Spectral properties of $KK_n^j$ graphs.  \emph{Mat.\ Contemp.}\ 39 (2010), 129--134.
\MR{2962586}.  \Zbl{1251.05097}.  
\annot{} The graph is $K_n \cupdot K_n$ with $j$ additional edges.  Spectral properties of $\Lap(-\Gamma)$.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Maria Aguieiras A.\ de Freitas, Renata R.\ Del-Vecchio, Nair M.M.\ de Abreu, \& Steve Kirkland}
\authorspace
\bsgitem{2009a}{}  On $Q$-spectral integral variation.  LAGOS'09---V Latin-Amer.\ Algor.\ Graphs Optim.\ Sympos.  \emph{Electronic Notes Discrete Math.}\ 35 (2009), 203--208.
\MR{2579431}.  \Zbl{1268.05128}.						
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Maria Aguieiras A.\ de Freitas, Vladimir Nikiforov, \& Laura Patuzzi}
\authorspace
\bsgitem{2013a}{}  Maxima of the $Q$-index: forbidden 4-cycle and 5-cycle.  \emph{Electronic J.\ Linear Algebra}  26 (2013), 905--916.
\MR{3192408}.  \Zbl{1282.05166}.  
\annot{} Thm.\ 1.1.:  For $n\geq4$, $C_4 \not\subseteq \Gamma \implies \lambda_1(\Lap(-\Gamma)) \leq \lambda_1(F_n)$; $= \implies \Gamma=F_n$ ($F_n$ is the windmill of $(n-1)/2$ triangles for odd $n$ and is $F_{n-1}$ with a pendant edge on the center for even $n$).  Thm.\ 1.3:  For $n\geq6$, $C_5 \not\subseteq \Gamma \implies \lambda_1(\Lap(-\Gamma)) \leq \lambda_1(\Lap(-K_2 \vee \bar K_{n-2}))$; $= \implies \Gamma=K_2 \vee \bar K_{n-2}$.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Christian Fremuth-Paeger}					%FREMUTH-PAEGER
\author{Christian Fremuth-Paeger \& Dieter Jungnickel}
\authorspace
\bsgitem{1999a}{}  Balanced network flows.  I:  A unifying framework for design and analysis of matching algorithms.  \emph{Networks} 33 (1999), no.\ 1, 1--28.
\MR{1652254} (2000f:\-90005).  \Zbl{999.90005}.					
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{1999b}{}  Balanced network flows.  II:  Simple augmentation algorithms.  \emph{Networks} 33 (1999), no.\ 1, 29--41.
\MR{1652258} (2000g:90010).  \Zbl{999.90006}.					
\par
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{1999c}{}  Balanced network flows.  III:  Strongly polynomial augmentation algorithms.  \emph{Networks} 33 (1999), no.\ 1, 43--56.
\MR{1652262} (2000g:90011).  \Zbl{999.90007}.					
\par
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2001a}{}  Balanced network flows.  IV:  Duality and structure theory.  \emph{Networks} 37 (2001), no.\ 4, 194--201.
\MR{1837197} (2002k:90010).  \Zbl{1038.90007}.					
\par
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2001b}{}  Balanced network flows.  V:  Cycle-canceling algorithms.  \emph{Networks} 37 (2001), no.\ 4, 202--209.
\MR{1837198} (2002k:90011).  \Zbl{1038.90008}.					
\par
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2001c}{}  Balanced network flows.  VI:  Polyhedral descriptions.  \emph{Networks} 37 (2001), no.\ 4, 210--218.
\MR{1837199} (2002k:90012).  \Zbl{1040.90002}.					
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2002a}{}  Balanced network flows.  VII:  Primal-dual algorithms.  \emph{Networks} 37 (2002), no.\ 1, 35--42.
\MR{1871705} (2003d:90007).  \Zbl{1040.90003}.					
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2002b}{}   An introduction to balanced network flows.  In:  K.T.\ Arasu and \'A.\ Seress, eds., \emph{Codes and Designs} (Columbus, Ohio, 2000), pp.\ 125--144.  Ohio State Univ.\ Math.\ Res.\ Inst.\ Publ., 10.  Walter de Gruyter, Berlin, 2002.
\MR{1948139} (2004b:05160).  \Zbl{1009.05113}.					
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\bsgitem{2003a}{}  Balanced network flows.  VIII:  A revised theory of phase-ordered algorithms and the $O(\sqrt{n}m \log(n2/m)/ \log n)$ bound for the nonbipartite cardinality matching problem.  \emph{Networks} 37 (2003), no.\ 3, 137--142.
\MR{1970119} (2004f:90015).  \Zbl{1106.90013}.					
\codes{\Csg:~\Cpar:~\CFlows,~\Ccov}%
\itemspace

\author{Noah E.\ Friedkin}						%FRIEDKIN
\authorspace
\authref  See \hyperlink{Peng Jia}{P.~Jia}, \hyperlink{Wenjun Mei}{W.-J.~Mei}, and \hyperlink{Anton Proskurnikov}{A.~Proskurnikov}.
\itemspace

\author{Ivan T.\ Frisch} 						%FRISCH
\authorspace
\authref  See \hyperlink{Howard Frank}{H.~Frank}.
\itemspace

\author{Tobias Fritz}							%FRITZ
\authorspace
\bsgitem{2013a}{}  Velocity polytopes of periodic graphs and a no-go theorem for digital physics.  \emph{Discrete Math.}\ 313 (2013), 1289--1301.  
\MR{3061113}.  \Zbl{1279.05040}.  
Corrigendum.  \emph{Ibid.}\ 313 (2013), 2380.
\MR{3084285}.  \Zbl{1281.05081}.  
\annot{} A periodic graph is the (infinite) $\bbZ^d$-covering graph of a (finite) $\bbZ^d$-gain graph.
\codes{\CGG:~\CCov}%
\itemspace

\author{Yuri Frota}							%FROTA
\authorspace
\authref  See \hyperlink{Rosa M.V.\ Figueiredo}{R.M.V.~Figueiredo}, \hyperlink{Mario Levorato}{M.~Levorato}, and \hyperlink{Eduardo Queiroga}{E.~Queiroga}.
\itemspace

\author{Josh B.\ Frye}						%FRYE
\authorspace
\authref  See \hyperlink{Ajit A.\ Diwan}{A.A.~Diwan}.
\itemspace

\author{Toshio Fujisawa} 						%FUJISAWA
\authorspace
\bsgitem{1963a}{}  Maximal flow in a lossy network.  In:  J.B.\ Cruz, Jr., and John C.\ Hofer, eds., \emph{Proceedings, First Annual Allerton Conference on Circuit and System Theory}  (Monticello, Ill., 1963), pp.\ 385--393.  Dept.\ of Electrical Eng.\ and Coordinated Sci.\ Lab., University of Illinois, Urbana, Ill., [1963].  
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{Satoru Fujishige}						%FUJISHIGE
\authorspace
\authref  See \hyperlink{Kazutoshi Ando}{K.~Ando}.
\itemspace

\authorx{Shinya Fujita}						%FUJITA S
\authorspace
\authref  See also \hyperlink{Shuya Chiba}{Chiba}.
\authrefspace

\author{Shinya Fujita \& Ken-Ichi Kawarabayashi}
\authorspace
\bsgitem{2010a}{}  Non-separating even cycles in highly connected graphs.  \emph{Combinatorica} 30 (2010), no.\ 5, 565--580.
\MR{2776720} (2012b:05159).  \Zbl{1231.05146}.  
\annot{} If $\Gamma$ is $k$-connected ($k\geq5$), $-\Gamma$ has a positive circle $C$ such that $\Gamma \setminus V(C)$ is $(k-4)$-connected.  If $\Gamma$ has no triangles, we can say $(k-3)$-connected.  (\xref{Thomassen}{2001a}{Carsten Thomassen} conjectured the odd-circle analog.)  
[\Problem.  Generalize to signed graphs such that $|\Sigma|$ is $k$-connected.]
\dated{26 Dec 2012} 
\codes{\Csg:~\CPar:~\CCircles}%
\itemspace

\author{Shinya Fujita \& Colton Magnant}
\authorspace
\bsgitem{2011a}{}  Note on highly connected monochromatic subgraphs in 2-colored complete graphs.  \emph{Electronic J.\ Combin.}\ 18 (2011), art.\ P15, 5 pp..
\MR{2770120} (2012f:\-05097).  \Zbl{1205.05090}.  
\annot{} See \xref{{\L}uczak}{2016a}{Tomasz Luczak}.
\dated{24 Jan 2016} 
\codes{\Csg:~\CStr}%
\itemspace

\authorx{D.R.\ Fulkerson}						%FULKERSON
\author{D.R.\ Fulkerson, A.J.\ Hoffman, \& M.H.\ McAndrew}
\authorspace
\bsgitem{1965a}{}  Some properties of graphs with multiple edges.  \emph{Canad.\ J.\ Math.}\ 17 (1965), 166--177.
\MR{0177908} (31 \#2166).  \Zbl{132.21002}.						
\annot{} The ``odd-cycle condition'' is that any two odd circles without a common vertex are joined by an edge.  Assuming it, certain conditions are necessary and sufficient for a degree sequence to be realized by a submultigraph of $K_n$ with prescribed multiplicities.  The incidence matrix of $-K_n$ is employed in the geometrical proof.  
[\Problem.  Generalize to signed graphs.]
\dated{30 May 2011} 
\codes{\Csg:~\CPar:~\Cincid}%
\itemspace

\author{Edgar Fuller}							%FULLER
\authorspace
\authref  See \hyperlink{Xingqin Qi}{X.Q.~Qi}.
\itemspace

\authorx{Atsushi Funato}						%FUNATO
\author{Atsushi Funato, Nan Li, \& Akihiro Shikama}
\authorspace
\bsgitem{2014a}{}  Decomposable edge polytopes of finite graphs.  Manuscript, 2014.  
\arXiv{1406.1942}.  
\annot{} 
[This is the antibalanced case.  \Problem.  Generalize to signed graphs, including balanced graphs.]
\codes{\Csg:~\CPar:~\CGeom}%
\itemspace

\author{Daryl Funk}							%FUNK
\authorspace
\authref  See also \hyperlink{Nathan Bowler}{N.~Bowler}, \hyperlink{Rong Chen}{R.~Chen}, and \hyperlink{Matt DeVos}{M.~DeVos}.
\authrefspace

\bsgitem{2015a}{}  \emph{On Excluded Minors and Biased Graph Representations of Frame Matroids}.  Doctoral dissertation, Simon Fraser Univ., 2015. 
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Daryl Funk \& Dillon Mayhew}
\authorspace
\bsgitem{2018a}{}  On excluded minors for classes of graphical matroids.  \emph{Discrete Math.}\ 341 (2018), no.\ 6, 1509--1522.
\MR{3784773}.  \Zbl{1384.05143}.  \arXiv{1706.06265}.				
\par
\codes{\CGG:~\CMatrd:~\CStr}%
\itemspace

\author{Daryl Funk, Dillon Mayhew, \& Mike Newman}
\authorspace
\bsgitem{2022a}{}  Defining bicircular matroids in monadic logic.  \emph{Quart.\ J.\ Math.}\ 73 (2022), no.\ 1, 65--92.
\MR{4395073}.  
\arXiv{2005.04526}.  
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Daryl Funk, Irene Pivotto, \& Daniel Slilaty}
\authorspace
\bsgitem{2022a}{\dag}  Matrix representations of matroids of frame and lifted-graphic matroids correspond to gain functions.  \emph{J.\ Combin.\ Theory Ser.\ B} 155 (2022), 202--255.
\MR{4392273}.  \Zbl{1487.05045}.  \arXiv{1609.05574}.				
\par
\codes{\CGG:~\CMatrd:~\CGeom}%
\itemspace

\author{Martin J.\ Funk}						%FUNK
\authorspace
\authref  See \hyperlink{Marien Abreu}{M.~Abreu}.
\itemspace


\label{LETTER G} \hypertarget{LETTER G}{}	% LETTER  G


\author{H.N.\ Gabow}						%GABOW
\authorspace
\bsgitem{1983a}{}  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems.  In: \emph{Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing} (Boston, 1983), pp.\ 448--456.  Assoc.\ for Computing Machinery, New York, 1983.
\MR{0842673} (87g:68004) (book).  
\annot{} $O(m^{3/2})$ algorithm for max integral flow.  [See \xref{Babenko}{2006b}{Maxim A.\ Babenko} for improved time.]
\dated{9 Sept 2010} 
\codes{\Csg:~\COri:~\CAlgor}%
\itemspace

\author{Nir Gadish}							%GADISH
\authorspace
\authref  See \hyperlink{Christin Bibby}{C.~Bibby}.
\itemspace

\author{Stephen M.\ Gagola}                            	       			  %GAGOLA
\authorspace
\bsgitem{1999a}{}  Solution to Problem 10606.  \emph{Amer.\ Math.\ Monthly} 106 (June--July, 1999), no.\ 6, 590--591.
\annot{} Proposed by \xref{Zaslavsky}{1997c}{Thomas Zaslavsky}, \qv\ for statement of the problem and significance.
\codes{\Cgg}%
\itemspace

\author{Ling Gai}							%GAI L
\authorspace
\authref  See \hyperlink{Sai Ji}{Sai Ji}.
\itemspace

\author{Giovanni Gaiffi}						%GAIFFI
\authorspace
\bsgitem{2016a}{}  Exponential formulas for models of complex reflection groups.  \emph{European J.\ Combin.}\ 55 (2016), 149--168.
\MR{3474798}.  \Zbl{1333.05319}.  \arXiv{1507.02090}.				
\codes{\CAlgeb:~\Cgg:~\CMatrd}%
\itemspace

\author{Anah{\'\i} Gajardo}						%GAJARDO
\authorspace
\authref  See \hyperlink{Marco Montalva}{M.~Montalva}.
\itemspace

\author{David Gale} 							%GALE
\authorspace
\authref  See also \hyperlink{Alan J.\ Hoffman}{A.J.~Hoffman}.
\authrefspace

\author{David Gale \& A.J.\ Hoffman} 
\authorspace
\bsgitem{1982a}{}  Two remarks on the Mendelsohn--Dulmage theorem.  In:  Eric Mendelsohn, ed., \emph{Algebraic and Geometric Combinatorics}, pp.\ 171--177.  
North-Holland Math.\ Stud., 65.  Ann.\ Discrete Math., 15.  North-Holland, Amsterdam, 1982.  
\MR{0772593} (85m:05054).  \Zbl{501.05049}.
\codes{\Csg:~\CInc,~\CBal}%
\itemspace

\author{Joseph A.\ Gallian}						%GALLIAN
\authorspace
\bsgitem{2009a}{}  A dynamic survey of graph labeling.  \emph{Electronic J.\ Combin.}\ Dynamic Surveys in Combinatorics, \# DS6.  URL \url{http://www.combinatorics.org/issue/view/Surveys/}
\MR{1668059} (99m:05141).  \Zbl{953.05067}.					
\annot{} \S3.7, ``Cordial labelings'';  \S3.8, ``The friendly index--balance index''.  From $f: V \to \bbZ_2$ obtain balanced edge gains $f^*(uv) = f(u)+f(v)$.  $f$ is ``friendly'' if it has essentially equal numbers of each label, i.e., equal or differing by 1.  $f$ is ``cordial'' if $f$ and $f^*$ have essentially equal numbers of each label.  [\Cf\ \xref{Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.]  A great many references.  
[$(\Gamma,f)$ is like a balanced multiply signed graph but the questions are not gain-graphic.]
\dated{9 Oct 2010} 
\codes{\Cvs:~\CExp,~\CRef}%
\annotctd{} Vertex switching (switching one vertex) of some standard graph gives examples of many kinds of labellings.
\dated{2 Jan 2015}
\codes{\Ctg}
\itemspace

\author{Jean Gallier}							%GALLIER
\authorspace
\bsgitem{2016a}{}  Spectral theory of unsigned and signed graphs.  Applications to graph clustering: a survey.  Manuscript, 2016, 122 pp.
\arXiv{1601.04692}.  
\annot{} \S5, ``Signed graphs'', uses Laplacian $\Lap(\Sigma,w) := \diag(d) - A(\Sigma,w)$, $w$ = positive weight function, obtained from incidence matrix based on $\sqrt{w_{ij}}$ [= standard incidence matrix if $|w_{ij}|=1$].  
Dictionary:  ``signed degree'' := unsigned degree $d_i := \sum_j w_{ij}$.
\Cf\ alternatives in \xref{Kunegis, Schmidt, \etal\!}{2010a}{Jerome Kunegis, Stephan Schmidt, Andreas Lommatzsch, Jurgen Lerner, Ernesto W.\ De Luca, \& Sahin Albayrak}, \xref{Knyazev}{2018a}{Andrew Knyazev}.
\dated{8 Feb 2021} 
\par
\codes{\CSG,~\CWG:~\CClu:~\CLap(\CGen),~\CInc}\codesctd{\CExp}%
\itemspace

\authorx{Anna Galluccio}						%GALLUCCIO
\author{Anna Galluccio, Martin Loebl, \& Jan Vondr\'ak}
\authorx{Anna Galluccio, Martin Loebl, \& Jan Vondrak}
\authorspace
\authref  See also \hyperlink{J.\ Lukic}{J.~Lukic}.
\authrefspace

\bsgitem{2000a}{}  New algorithm for the Ising problem: Partition function for finite lattice graphs.  \emph{Phys.\ Rev.\ Lett.}\ 84 (2000), no.\ 26, 5924--5927.
\annot{} Describes \xrefd{2001a}{Anna Galluccio, Martin Loebl, \& Jan Vondrak}, emphasizing signed toroidal lattice graphs, i.e., toroidal lattice Ising models.
\dated{18 Aug 2012} 
\par
\codes{\CSG,~\CPhys:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2001a}{}  Optimization via enumeration: a new algorithm for the Max Cut Problem.  \emph{Math.\ Programming Ser.\ A} 90 (2001), 273--290.
\MR{1824075} (2002b:90057).  \Zbl{989.90127}.  
\annot{} An algorithm for the generating function of weighted cuts (= partition function of Ising model), hence for $\sum_\zeta x^{|E^-(\Sigma^\zeta)|}$ and frustration index $l(\Sigma)$, in polynomial time for graphs of bounded genus.
\dated{18 Aug 2012} 
\codes{\CSG:~\CFr:~\CAlgor,~\CPhys}%
\itemspace

\authorx{Huiling Gan}						%GAN HL
\author{Huiling Gan and Ellis L.\ Johnson}
\authorspace
\bsgitem{1989a}{}  Four problems on graphs with excluded minors.  \emph{Math.\ Programming}\ 45 (1989), 311--330.
\MR{1022812} (91a:90067).  \Zbl{681.90079}.					
\codes{\Csg:~\CStr,~\CAlgor}%
\itemspace

\author{Alberto Gandolfi}						%GANDOLFI
\authorspace
\authref  See also \hyperlink{E.\ De Santis \& A.\ Gandolfi}{E.~De~Santis}.
\authrefspace

\author{A.\ Gandolfi, C.M.\ Newman, \& D.L.\ Stein}
\authorspace
\bsgitem{2000a}{}  Zero-temperature dynamics of $\pm J$ spin glasses and related models.  \emph{Commun.\ Math.\ Phys.}\ 214 (2000), no.\ 2, 373--387.
\MR{1796026} (2001k:82072).  \Zbl{978.82098}.  
\codes{\CSG,~\CPhys:~\CFr}%
\itemspace

\authorx{Robert Ganian}						%GANIAN
\author{Robert Ganian, Petr Hlin\v{e}n\'y, \& Jan Obdr\v{z}\'alek}
\authorx{Robert Ganian, Petr Hlineny, \& Jan Obdrzalek}
\authorspace
\bsgitem{2010a}{}  Better algorithms for satisfiability problems for formulas of bounded rank-width.  Kamal Lodaya and Meena Mahajan, eds., \emph{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science} (FSTTCS 2010, 30th Int.\ Conf., Chennai, 2010), pp.\ 73--83.  
{LIPICS: Leibniz Int.\ Proc.\ Informatics, Vol.\ 8.}  Schloss Dagstuhl and Leibniz-Zent.\ Inform., Wadern, 2010.
\MR{2577943} (no rev).  \Zbl{1245.68108}.  \arXiv{1006.5621}.				
\par
\codes{\CSG:~\CAppl:~\CAlgor}%
\itemspace

\authorx{Hilal A.\ Ganie}						%GANIE
\author{Hilal A.\ Ganie \& Bilal A.\ Chat}
\authorspace
\bsgitem{2019a}{}  Bounds for the energy of weighted graphs.  \emph{Discrete Math.}\ 268 (2019), 91--101.
\MR{4007233}.  \Zbl{1419.05098}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Gao Hongzhu}						%GAO HZ
\authorspace
\authref  See \hyperlink{Zhiyun Cheng}{Cheng~Z.Y.}
\itemspace

\authorx{Hua Gao}							%GAO H
\author{Hua Gao, Zhijian Ji, \& Ting Hou}
\authorspace
\bsgitem{2018a}{}  Equitable partitions in the controllability of undirected signed graphs.  In:  \emph{2018 IEEE 14th International Conference on Control and Automation} (ICCA, Anchorage, Alaska, 2018), pp.\ 532--537.  IEEE, 2018.
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Jie Gao}							%GAO JE
\authorspace
\authref  See \hyperlink{Haotian Wang}{H.-T.~Wang}.
\itemspace

\author{Jiyang Gao}							%GAO JY
\authorspace
\bsgitem{20xxa}{}  Acyclic orientations and the chromatic polynomial of signed graphs.  Submitted.  
\codes{\CSG:~\CCol,~\CCov,~\COri}%
\itemspace

\authorx{Ruimei Gao}						%GAO RM
\author{Ruimei Gao \& Ying Chu}
\authorspace
\bsgitem{2018a}{}  Freeness of arrangements between the Weyl arrangements of types $A_{n?1}$
and $B_n$.  (In Chinese.  English summary.)  \emph{J.\ Shandong Univ., Nat.\ Sci.}\ 53 (2018), no.\ 6, 70--75.
\Zbl{1424.51007}.  
\annot{} Seems to be largely exposition of \xref{Edelman--Reiner}{1994a}{Paul H.\ Edelman \& Victor Reiner}.
\dated{10 Jun 2020} 
\codes{\CSG:~\CGeom:~\CExp}%
\itemspace

\author{Yu-Bin Gao}							%GAO YB
\authorspace
\authref  See also \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}, \hyperlink{Li Fang Huo}{L.F.\ Huo}, and \hyperlink{Yanling Shao}{Y.L.~Shao}.
\authrefspace

\author{Yubin Gao, Yihua Huang \& Yanling Shao}
\authorspace
\bsgitem{2009a}{}  Bases of primitive non-powerful signed symmetric digraphs with loops.  \emph{Ars Combin.}\ 90 (2009), 383--388.
\MR{2489540} (2010c:05049).  \Zbl{1224.05208}.					
\par
\codes{\CSD,~\Cqm}%
\itemspace

\author{Yubin Gao, Yanling Shao, \& Jian Shen}
\authorspace
\bsgitem{2009a}{}  Bounds on the local bases of primitive nonpowerful nearly reducible sign patterns.  \emph{Linear Multilinear Algebra} 57 (2009), no.\ 2, 205--215.
\MR{2492103} (2010b:05103).  \Zbl{1166.15008}.					
\codes{\CSD,~\CQM}%
\itemspace


\authorx{Marianne L.\ Gardner} 					%GARDNER
\author{Marianne L.\ Gardner [Marianne Lepp]}
\authorspace
\authref  See \hyperlink{Randy Shull}{R.~Shull}.
\itemspace

\authorx{T.\ Garel}							%GAREL
\author{T.\ Garel \& J.M.\ Maillard}
\authorspace
\bsgitem{1983a}{}  Study of a two-dimensional fully frustrated model.  \emph{J.\ Phys.\ A} 16 (1983), 2257--2265.
\MR{0713188} (85b:82069).  
\annot{} Physics approach.  Generalizes \xref{Southern, Chui, and Forgacs}{1980a}{B.W.\ Southern, S.T.\ Chui, \& G.\ Forgacs}'s square-lattice Ising model to four edge weights, symmetrically located, and reduces it to an all-positive graph with two weights.  \S3, ``Application to the Villain model'':  All weights equal [hence a signed graph]; further results on \xref{Villain}{1977a}{Jacques Villain}.
\dated{16 Jun 2012} 
\codes{\CPhys:~\Csg:~\Cwg}%
\itemspace

\author{Pravin Garg}							%GARG P
\authorspace
\authref  See also \hyperlink{Deepa Sinha}{D.~Sinha}.
\authrefspace

\bsgitem{2012a}{}  \emph{An Excursion to Some Emerging Frontiers of the Theory of Signed Graphs.}  Doctoral thesis, Banasthali University, 2012.
\codes{\CSG}%
\itemspace

\author{Vikas K.\ Garg}						%GARG VK
\authorspace
\authref  See \hyperlink{Priyanka Agrawal}{P.~Agrawal}.
\itemspace

\authorx{Michael L.\ Gargano} 						%GARGANO
\author{Michael Gargano \& Louis V.\ Quintas}
\authorspace
\bsgitem{1985a}{}  A digraph generalization of balanced signed graphs.  \emph{Congressus Numerantium} 48 (1985), 133--143.  
\MR{0830706} (87m:05095).  \Zbl{622.05027}.					
\annot{} Characterizes balance in abelian gain graphs.  [See \xref{Harary, Lindstr\"om, and Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom}.]  Very simple results on existence, for a given graph, of balanced nowhere-zero gains from a given abelian group.  [Elementary, if one notes that such gains exist iff the graph is $\#G$-colorable, $G$ being the gain group].  Comparison with the approach of \xref{Sampathkumar and Bhave}{1973a}{E.\ Sampathkumar \& V.N.\ Bhave}.  
Dictionary:  ``Symmetric $G$-weighted digraph'' = gain graph with gains in the (abelian) group $G$.  ``Weight'' = gain.  ``Non-trivial'' (of the gain function) = nowhere zero.
\codes{\CGG:~\CBal}%
\itemspace

\author{Michael L.\ Gargano, John W.\ Kennedy, \& Louis V.\ Quintas}
\authorspace
\bsgitem{1998a}{}  Group weighted balanced digraphs and their duals.  
Proc.\ Twenty-ninth Southeastern Int.\ Conf.\ on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1998).  \emph{Congressus Numer.}\ 131 (1998), 161--167.
\MR{1676483} (99j:05080).  \Zbl{951.05045}.					
\annot{} An abelian gain graph $\Phi$ is cobalanced (here called ``cut-balanced'') if the sum of gains on the edges of each coherently oriented cutset is $0$.  [This generalizes \xref{Kabell}{1985a}{Jerald A.\ Kabell}.]  Given $\Phi$ with $\|\Phi\|$ embedded in a surface, the surface dual graph is given gains by a right-rotation rule, thus forming a surface dual $\Phi^*$ of $\Phi$.  [This appears to require that the surface be orientable.  Note that cobalance generalizes to nonabelian gains on orientably embedded graphs, since the order of multiplication for the gain product on a cutset is given by the embedding.]  Thm.\ 3.2:  For a plane embedding of $\Phi$, $\Phi$ is cobalanced iff $\Phi^*$ is balanced.  Thm.\ 3.4 restates as criteria for cobalance of $\Phi$ the standard criteria for balance of $\Phi^*$, as in \xref{Gargano and Quintas}{1985a}{Michael Gargano \& Louis V.\ Quintas}.  More interesting are ``well-balanced'' graphs, which are both balanced and cobalanced.  \Problem.  Characterize them.  
Dictionary (also see \xref{Gargano and Quintas}{1985a}{Michael Gargano \& Louis V.\ Quintas}):  Balance is called ``cycle balance''.
\codes{\CGG:~\CBal(\CDual)}%
\itemspace

\author{Marcin G\c{a}siorek}						%GASIOREK
\authorx{Marcin Gasiorek}
\authorspace
\bsgitem{2013a}{}  Efficient computation of the isotropy group of a finite graph: a combinatorial approach.  
In:  Nikolaj Bj\"orner \etal, eds., \emph{15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2013, Timisoara, Romania, 2013), pp.\ 104--111.  IEEE, 2013.
\codes{\CSG}%
\itemspace

\author{Marcin G\c{a}siorek, Daniel Simson \& Katarzyna Zaj\c{a}c}
\authorx{Marcin Gasiorek, Daniel Simson \& Katarzyna Zajac}
\authorspace
\bsgitem{2012a}{}  On Coxeter spectral study of posets and a digraph isomorphism problem.  In:  \emph{14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2012, Timisoara, Romania), pp.\ 369--375.  IEEE, 2012.
\codes{\CSG:~\CAdj}%
\itemspace

\bsgitem{2014a}{}  On corank two edge-bipartite graphs and simply extended Euclidean diagrams.  
In:  Franz Winkler \etal, eds., \emph{The 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2014, Timisoara, Romania), pp.\ 66--73.  IEEE, 2014.
\codes{\CSG}%
\itemspace

\bsgitem{2015a}{}  On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams.  \emph{European J.\ Combin.}\ 48 (2015), 127--142.
\MR{3339018}.  \Zbl{1318.06004}.  
\codes{\CSG}%
\itemspace

\bsgitem{2015b}{}  Structure and a Coxeter--Dynkin type classification of corank two non-negative posets.  \emph{Linear Algebra Appl.}\ 469 (2015), 76--113.
\MR{3299057}.  \Zbl{1305.06002}.  
\codes{\CSG}%
\itemspace

\authorx{Gilles Gastou}						%GASTOU
\author{Gilles Gastou \& Ellis L.\ Johnson}
\authorspace
\bsgitem{1986a}{} Binary group and Chinese postman polyhedra.  \emph{Math.\ Programming} 34 (1986), 1--33.  
\MR{0819872} (88e:90060).  \Zbl{589.52004}.  
\annot{} \S10 introduces the co-postman and ``odd circuit'' problems, treated more thoroughly in \xref{Johnson and Mosterts}{1987a}{Ellis L.\ Johnson \& Sebastiano Mosterts} (\qv).  ``Odd'' edges and circuits are precisely negative edges and circles in an edge signing.  The ``odd circuit matrix'' represents $\bfL(\Sigma)$ (p.\ 30).  
The ``odd circuit problem'' is to find a shortest negative circle; a simple algorithm uses the signed covering graph (pp.\ 30--31).  The ``Fulkerson property'' may be related to planarity and $K_5$ minors [which suggests comparison with \xref{Barahona}{1990a}{Francisco Barahona}, \S5].
\codes{\CSG:~\CFr(\CGen),~\CCircles,~\CInc,~\CMatrd(\CBases),~\Ccov,~\CAlgor}%
\itemspace

\author{Atul Gaur}							%GAUR
\authorspace
\authref  See \xref{Acharya, Pranjali, Gaur, and Kumar}{2017a}{Mukti Acharya, Pranjali, Atul Gaur, \& Amit Kumar} and \hyperlink{Pranjali}{Pranjali}.
\itemspace

\authorx{Heather Gavlas}						%GAVLAS
\author{Heather Gavlas [Heather Jordon]}
\authorspace
\authref  See \hyperlink{Gary Chartrand}{G.~Chartrand}, \hyperlink{Dean Hoffman}{D.~Hoffman}, and \hyperlink{Heather Jordon}{H.~Jordon}. 
\itemspace

\authorx{Alexander L.\ Gavrilyuk}					%GAVRILYUK
\author{Alexander L.\ Gavrilyuk, Akihiro Munemasa, Yoshio Sano, \& Tetsuji Taniguchi}
\authorspace
\bsgitem{2021a}{}  Signed analogue of line graphs and their smallest eigenvalues.  \emph{J.\ Graph Theory}\ 98 (2021), no.\ 2, 309--325. 
\MR{4371442}.  
\arXiv{2003.05578}.  
\par
\codes{\CSG:~\CLG:~\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Premiys{\l}aw Gawro\'nski}					%GAWRONSKI
\authorx{Premiyslaw Gawronski}
\authorspace
\authref  See also \hyperlink{Forough Hassanibesheli}{F.~Hassanibesheli} and \hyperlink{Krzysztof Kulakowski}{K.~Ku{\l}akowski}.
\authrefspace

\author{P.\ Gawro\'nski, P.\ Gronek, \& K.\ Ku{\l}akowski}
\authorx{P.\ Gawronski, P.\ Gronek, \& K.\ Kulakowski}
\authorspace
\bsgitem{2005a}{}  The Heider balance and social distance.  \emph{Acta Phys.\ Polonica B} 36 (2005), no.\ 8, 2549--2558.
\arXiv{physics/0501160}.  
\annot{} Exposition of \xref{Ku{\l}akowski--Gawro\'nski--Gronek}{2005a}{Krzysztof Kulakowski, Premiyslaw Gawronski, \& Piotr Gronek}, and some numerical results.
\dated{24 Aug 2018} 
\codes{\CSG,~\CWG:~\CBal,~\CDyn,~\CExp}%
\itemspace

\author{Przemys{\l}aw Gawro\'nski, Ma{\l}gorzata J.\ Krawczyk, \& Krzysztof Ku{\l}akowski}
\authorx{Przemyslaw Gawronski, Malgorzata J.\ Krawczyk, \& Krzysztof Kulakowski}
\authorspace
\bsgitem{2015a}{}  Emerging communities in networks --- a flow of ties.  \emph{Acta Phys.\ Polonica B} 46 (2015), no.\ 5, 911--921.
\arXiv{1505.06295}.  
\annot{} \S2, ``Cognitive dissonance'':  A system of differential equations is used to find a switching with fewest negative edges.  Repeats \xref{Gawro\'nski and Ku{\l}akowski}{2005a}{P.\ Gawronski \& K.\ Kulakowski} with much more detail. 
\dated{19 Jan 2016} 
\codes{\CSG:~\Cfr:~\CAlgor}%
\itemspace

\author{P.\ Gawro\'nski \& K.\ Ku{\l}akowski}
\authorx{P.\ Gawronski \& K.\ Kulakowski}
\authorspace
\bsgitem{2005a}{}  Heider balance in human networks.  
In: Joaqu{\i\i}n Marro, Pedro L.\ Garrido, and Miguel A.\ Mu\~noz, eds., \emph{Modeling Cooperative Behavior in the Social Sciences} (Proc.\ 8th Granada Lect., Granada, Spain, 2005), pp.\ 93--95.  AIP Conf.\ Proc., Vol.\ 779.  Amer.\ Inst.\ Physics, Melville, N.Y., 2005.
\arXiv{physics/0503085}.  
\annot{} See \xref{Gawro\'nski, Krawczyk, and Ku{\l}akowski}{2015a}{Przemyslaw Gawronski, Malgorzata J.\ Krawczyk, \& Krzysztof Kulakowski}
\dated{21 Jan 2016} 
\codes{\CSG:~\Cfr:~\CAlgor}%
\itemspace

\bsgitem{2007a}{}  A numerical trip to social psychology: long-living states of cognitive dissonance.  
In:  Y.\ Shi \etal, eds., \emph{Computational Science -- ICCS 2007} (7th Int.\ Conf., Beijing, 2007), Part IV, pp.\ 43--50.  \LNCS 4490.  Springer, Berlin, 2007. 
\arXiv{physics/0611276}.  
\codes{\CSG:~\Cfr:~\CAlgor}
\itemspace

\author{Gayathri H}							%GAYATHRI
\authorspace
\authref  See \hyperlink{Reshma R}{Reshma R}.
\itemspace

\authorx{Chuanyuan Ge}						%GE CY
\author{Chuanyuan Ge \& Shiping Liu}
\authorspace
\bsgitem{20xxa}{}  Symmetric matrices, signed graphs, and nodal domain theorems.  Submitted.  \emph{}\ 
\arXiv{2201.00198}.  
\codes{\CSG:~\CLap,~\CBal,~\CSw}%
\itemspace

\authorx{Jim Geelen}
\author{Jim [James F.] Geelen}						%GEELEN
\authorx{James F.\ Geelen}
\authorspace
\authref  See also \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}.
\authrefspace

\bsgitem{2008a}{}  Some open problems on excluding a uniform matroid.  \emph{Adv.\ Appl.\ Math.}\ 41 (2008), 628--637.
\MR{2459453} (2009k:05050).  \Zbl{1172.05013}.  
\annot{} \S6, ``The basic classes'':  Frame matroids $\bfF(\Omega\full)$ of full biased graphs are called ``framed Dowling matroids'' and the submatroids $\bfF(\Omega)$ without the extra unbalanced edges are called ``Dowling matroids'' [despite these generalizations of Dowling geometries $Q_n(\fG)$ having been introduced in \xref{Zaslavsky}{1977a}{Thomas Zaslavsky}, \xrefd{1987a}{Thomas Zaslavsky}, \xrefd{1991a}{Thomas Zaslavsky}].  Bicircular matroids $\bfF(\Gamma,\eset)$ are mentioned as examples and are important in Conj.\ 6.1 due to Johnson, Robertson, and Seymour.  Thm.\ 6.2 has as one important class $\bfF(K_k,\eset)$.  There follows a nice list of four basic questions about frame matroids:  Find the excluded minors.  Can they be recognized in polynomial time?  Find the excluded minors for the frame matroids that have canonical representations (in the sense of \xref{Zaslavsky}{2003b}{Thomas Zaslavsky}) or frame representations (in the sense of papers of \hyperlink{Jim Geelen, Bert Gerards, \& Geoff Whittle}{Geelen, Gerards, \& Whittle}) over $\bbR$ [the projective representation Thm.\ 7.1 of \xref{Zaslavsky}{2003b}{Thomas Zaslavsky} might be relevant?].  Can those matroids be recognized in polynomial time?
\dated{26 Jan 2015}
\codes{\Cgg:~\CMatrd:~\CInc,~\CGeom}%
\annotctd{} In Conj.\ 8.2 about the fewest flats that cover a matroid, one special class is the bicircular matroids. 
\dated{26 Jan 2015}
\codes{\CBic}
\annotctd{} Spikes $\bfF(2C_n,\cB)$ are important (p.\ 630).
\dated{29 Apr 2012} 
\par
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2013a}{}  Graphical representations of matroids.  \emph{The Matroid Union} (2013), \url{http://matroidunion.org/?p=266}
\annot{} \Cf\ \xref{Geelen--Gerards--Whittle}{2018a}{Jim Geelen, Bert Gerards, \& Geoff Whittle}.
\dated{2 Jul 2019} 
\par
\codes{\CGG:~\CMatrd(\CGen):~\CExp}%
\itemspace

\author{James F.\ [Jim] Geelen \& A.M.H.\ [Bert] Gerards}
\authorspace
\bsgitem{2005a}{}  Regular matroid decomposition via signed graphs.  \emph{J.\ Graph Theory} 48 (2005), no.\ 1, 74--84.
\MR{2104582} (2005h:05037).  \Zbl{1055.05024}.  
\annot{} The lift matroid.	
\codes{\CSG:~\CMatrd:~\CStr}
\itemspace

\bsgitem{2009a}{}  Excluding a group-labelled graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), 247--253.
\MR{2467829} (2009k:05169).  \Zbl{1226.05213}.					
\annot{} Given finite, abelian $\fG$ and $\fG' \leq \fG$, and a $\fG$-gain graph $\Phi$ with a minor $\Psi \cong \fG'K_{4t}$ where $t = 8n\#\fG^2$.  Thm.\ 1.3:  Either $\exists\ X \subseteq V$ with $\#X<t$ such that in $\Phi \setminus X$ the block containing most of $\Psi$ is $\fG'$-balanced, or $\Psi$ has a minor $\cong \fG''K_n$ where $\fG' < \fG'' \leq \fG$.  $t$ may not be best possible.  Thm.\ 1.4:  $\forall\ n$, $\exists\ l(n)$ such that $\|\Phi\|$ has a $K_{l(n)}$ minor$\implies$$\Phi$ has a $0K_n$ minor.  
Dictionary:  ``Group-labelled graph'' = gain graph; $\Gamma$ means $\fG$; $G$ means $\Phi$; $\tilde G$ means $\|\Phi\|$; ``shifting'' means ``switching''; $\fG'$-balanced means switchable so all gains are in $\fG'$.
\codes{\CGG:~\CStr}%
\itemspace

\author{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}
\authorspace
\bsgitem{2009a}{}  On the odd-minor variant of Hadwiger's conjecture.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), no.\ 1, 20--29.
\MR{2467815} (2010f:05149).  \Zbl{1213.05079}.					
\annot{} An ``odd $K_m$-expansion'' in $\Gamma$ is a homeomorph of $-K_m$ in $-\Gamma$.
[\Problem.  Generalize to signed graphs.]
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\author{Jim Geelen, Bert Gerards, \& Geoff Whittle}
\authorspace

\bsgitem{2002a}{}  (as James F.\ Geelen, A.M.H.\ Gerards, \& Geoff Whittle)  Branch-width and well-quasi-ordering in matroids and graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 84 (2002), no.\ 2, 270--290.
\MR{1889259} (2003f:05027).  \Zbl{1037.05013}.  
\annot{} \MORE{Sect. no.}Vector representation of spikes $\bfL(2C_n,\cB)$ and tipped spikes $\bfL_0(2C_n,\cB)$.  Thm.\MORE{no.}:  Any representation of $\bfL(2C_n,\cB)$ extends to one of $\bfL_0(2C_n,\cB)$, if $n\geq4$.\MORE{Oxley 6.5.13.  Is it in this paper?}  [Special case of \xref{Zaslavsky}{2003b}{Thomas Zaslavsky}, Thm.\ 7.1.]  
Thm.\MORE{no.}:  All representations of $\bfL_0(2C_n,\cB)$ are of a specific form, up to projective transformations.  
\dated{18 Apr 2013} 
\codes{\Cgg:~\CMatrd,~\CInc}%
\itemspace

\bsgitem{2006a}{}  Matroid $T$-connectivity.  \emph{SIAM J.\ Discrete Math.}\ 20 (2006), no.\ 3, 588--596.
\MR{2272215} (2007j:05040).  \Zbl{1122.05023}.					
\annot{} The full bicircular matroid $\bfF(\Gamma\full,\eset)$ appears on p.\ 589.
\codes{\Cgg:~\Cbic}%
\itemspace

\bsgitem{2006b}{}  Towards a structure theory for matrices and matroids.  
In: Marta Sanz-Sol\'e \etal, eds., \emph{Proceedings of the International Congress of Mathematicians} (ICM, Madrid, 2006), Vol.\ III: \emph{Invited Lectures}, pp.\ 827--842.  European Mathematical Society, Z\"urich, 2006.
\MR{2275708} (2008a:05045).  \Zbl{1100.05016}.					
\annot{} See \xrefd{2007a}{Jim Geelen, Bert Gerards, \& Geoff Whittle}.
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{2007a}{}  Towards a matroid-minor structure theory.  
In:  Geoffrey Grimmett \etal, eds., \emph{Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh}, pp.\ 72--82.  Oxford Lect.\ Ser.\ Math.\ Appl., Vol.\ 34.  Oxford Univ.\ Press, Oxford, 2007.  
\MR{2314562} (2008d:05037) (\emph{q.v.}).  \Zbl{1130.05015}.				
\annot{} \Conjecture.  A minor-closed proper subclass of all $\GF{q}$-representable matroids is essentially constructible from frame matroids and their duals.  
Dictionary:  ``Dowling matroid'' = simple frame matroid, i.e., submatroid of \xref{Dowling's}{1973a}{Thomas A.\ Dowling}, \xrefd{1973b}{Thomas A.\ Dowling} matroids $\bfF(\fG K_n\full)$, for $\fG = \bbF_q^\times$.
\dated{25 May 2009}
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{2013a}{}  Structure in minor-closed classes of matroids.  In:  Simon R.\ Blackburn, Stefanie Gerke, and Mark Wildon, eds., \emph{Surveys in Combinatorics 2013}, Ch.\ 8, pp.\ 327--362.
	
\MR{3184116}.  \Zbl{1318.05015}.  
\codes{\CGG:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{2015a}{}  The highly connected matroids in minor-closed classes.  \emph{Ann.\ Combin.}\ 19 (2015), no.\ 1, 107--123.
\MR{3319863}.  \Zbl{1310.05044}.  \arXiv{1312.5012}.				
\annot{} Perturbations of representations of frame matroids are important.  
But see \xref{Grace and van Zwam}{2018a}{Kevin Grace \& Stefan H.M.\ van Zwam} for counterexamples.
\dated{27 Feb 2017, rev 15 Jul 2019} 
\codes{\Cgg:~\CMatrd:~\CInc}\codesctd{\Csg:~\CMatrd}%
\itemspace

\bsgitem{2018a}{}  Quasi-graphic matroids.  \emph{J.\ Graph Theory} 87 (2018), no.\ 2, 253--264.  
\MR{3742182}.  \Zbl{1387.05038}.  \arXiv{1512.03005}.				
\annot{} 	
\Cf\ \xref{Bowler--Funk--Slilaty}{2020a}{Nathan Bowler, Daryl Funk, \& Daniel Slilaty}.
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{2018b}{} Retraction [of previous on-line version], \emph{ibid.}\ 87 (2018), no.\ 2, 265.
\annotctd{} The retraction is of a previous version.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{James F.\ Geelen \& Bertrand Guenin}
\authorspace
\bsgitem{2002a}{}  Packing odd circuits in Eulerian graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 86 (2002), no.\ 2, 280--295.
\MR{1933464} (2004g:05129).  \Zbl{1023.05091}.					
\annot{} Adds to \xref{Guenin's theorem}{2001a}{Bertrand Guenin}:  Thm.:  If $\Sigma$ has no $-K_5$ minor, then the dual linear program has a half-integral minimum (assuming $f$ has nonnegative coefficients).	
Dictionary:  ``odd'' = negative; the ``Eulerian graph'' is signed.
\codes{\CSG,~\CDual:~\CInc,~\CGeom,~\CStr}%
\itemspace

\author{Jim Geelen \& Tony Huynh}
\authorspace
\bsgitem{2006a}{}  Colouring graphs with no odd-$K_n$ minor.  Manuscript, 2002, 2006.  URL \url{http://www.math.uwaterloo.ca/~jfgeelen/publications/colour.pdf}
\par
\codes{\CSG,~\CCol}%
\itemspace

\author{Jim Geelen \& Kasper Kabell}
\authorspace
\bsgitem{2009a}{}  The Erd\"os--P\'osa property for matroid circuits.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), 407--419.
\MR{2482958} (2010c:05026).  \Zbl{1229.05071}.  
\annot{}  Bicircular matroid is an example of the property.  \S 6.2, ``Cleaning a nest''.  Lemma 6.1:  A sufficiently large ``nest'' without a large uniform minor has a frame matroid $\bfF(K_n\full,\cB)$ as a minor.  \S 7, ``Cliques'':  A sufficiently large $\bfF(K_n\full,\cB)$ has $\bfM(K_n)$ or $\bfF(K_n,\eset)$ as a minor.
Dictionary:  ``Dowling clique'' = full frame matroid of biased $K_n$, ``Dowling representation'' = the biased $K_n$.
\dated{12 Jul 2016} 
\codes{\Cgg:~\CMatrd,~\CBic}%
\itemspace

\author{James Geelen, James Oxley, Dirk Vertigan, \& Geoff Whittle}
\authorspace
\bsgitem{2002a}{}  Totally free expansions of matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 84 (2002), no.\ 1, 130--179.
\MR{1877906} (2002j:05035).  \Zbl{1048.05020}.  
\annot{} A rank-$r$ swirl is $\bfF(2C_r,\eset)$.  Free spikes and rank-$r$ swirls, also the latter with one unbalanced loop, are important.  
\Conjecture:  The 3-connected, rank-$k$ matroids, representable over $\GF(q)$ and having no $\bfL(2C_k,\eset)$ or $\bfF(2C_k,\eset)$ minor, have a bounded number of inequivalent $\GF(q)$-repre\-sen\-ta\-tions.
\dated{25 May 2009, 29 Apr 2012} 
\codes{\Cgg:~\CMatrd:~\CInc}%
\itemspace

\bsgitem{2004a}{}  A short proof of non-$\GF(5)$-representability of matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 91 (2004), 105--121.
\MR{2047534} (2005b:05055).  \Zbl{1050.05024}.					
\annot{} The ``free swirl'' $\Delta_r$ is $\bfF(2C_k,\eset)$.  The ``free spike'' $\Lambda_r$ is $\bfL(2C_k,\eset)$.  They play a main role re large totally free matroids (Thm.\ 3.4).
\dated{8 Mar 2011} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Jim Geelen \& Cynthia Rodriguez}
\authorspace
\bsgitem{2021a}{}  Disjoint non-balanced $A$-paths in biased graphs.  \emph{Adv.\ Appl.\ Math.}\ 126 (2021), art.\ 102014, 6 pp.
\MR{4224061}.  
\codes{\CGG:~\CPaths}%
\itemspace

\author{M.C.\ Geetha}						%GEETHA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\authorx{Laura Gellert}						%GELLERT
\author{Laura Gellert \& Raman Sanyal}
\authorspace
\bsgitem{2017a}{}  On degree sequences of undirected, directed, and bidirected graphs.  \emph{European J.\ Combin.}\ 64 (2017), 113--124.
\MR{3658823}.  \Zbl{1365.05050}.  \arXiv{1512.08448}.				
\annotctd{} [\Cf\ \xref{Chartrand--Gavlas--Harary--Schultz}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz}, 
\xref{Yan--Lih--Kuo--Chang}{1997a}{Jing-Ho Yan, Ko-Wei Lih, David Kuo, \& Gerard J.\ Chang}, 
\xref{Michael}{2002a}{T.S.\ Michael}, 
\xref{Hoffman--Jordon}{2006a}{Dean Hoffman \& Heather Jordon}, 
\xref{Pirzada--Naikoo--Dar}{2007b}{S.\ Pirzada, T.A.\ Naikoo, \& F.A.\ Dar}, 
\xref{Pirzada--Dar}{2007b}{S.\ Pirzada \& F.A.\ Dar}, 
\xref{Tam--Fan--Zhou}{2008a}{Bit-Shun Tam, Yi-Zheng Fan, \& Jun Zhou}, 
\xref{Jordon--McBride--Tipnis}{2009a}{Heather Jordon, Richard McBride, \& Shailesh Tipnis}, 
\xref{Pirzada}{2012a}{S.\ Pirzada}.]
\codes{\CSG,~\COri:~\CInvar}%
\itemspace

\author{[Joseph Ben Geloun]}						%GELOUN
\authorspace
\authref  See \hyperlink{Joseph Ben Geloun}{J.~Ben~Geloun} (under `B').
\itemspace

\authorx{Xianya Geng}						%GENG XY
\author{Xianya Geng, Shuchao Li, \& Slobodan K.\ Simi\'c}
\authorx{Xianya Geng, Shuchao Li, \& Slobodan K.\ Simic}
\authorspace
\bsgitem{2010a}{}  On the spectral radius of quasi-$k$-cyclic graphs.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 8-10, 1561--1572.
\MR{2718221} (2011f:05178).  \Zbl{1211.05074}.  
\annot{} \S2 mentions $\Lap(-\Gamma)$.  
Quasi-$k$-cyclic means $\exists\ v$ such that $\Gamma \setminus v$ has cyclomatic number $k$.  For $k\leq2$, Thm.\ 3.2 describes all $\Gamma$ maximizing the largest eigenvalue of $\Lap(-\Gamma)$.
\dated{21 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Claudio Gentile}						%GENTILE
\authorspace
\authref  See \hyperlink{Nicolo Cesa-Bianchi}{N.~Cesa-Bianchi}.
\itemspace

\author{A.M.H.\ [Bert] Gerards} 						%GERARDS
\authorx{A.M.H.\ Gerards}
\authorspace
\authref  See also \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}, \hyperlink{Michele Conforti}{M.~Conforti}, and \hyperlink{Jim Geelen}{J.~Geelen}. 
\authrefspace

\bsgitem{1988a}{}  Homomorphisms of graphs into odd cycles.  \emph{J.\ Graph Theory} 12 (1988), 73--83. 
\MR{0928737} (89h:05045).  \Zbl{691.05013}.					
\annot{} If an antibalanced, unbalanced signed graph has no homomorphism into its shortest negative circle, then it contains a subdivision of $-K_4$ or of a loose $\pm C_3$ (here called an ``odd $K_4$'' and an ``odd $K^2_3$'').  (A loose $\pm C_n$ consists of $n$ negative digons in circular order, each adjacent pair joined either at a common vertex or by a link.)  [\Question.  Do the theorem and proof carry over to any unbalanced signed graph?]  Other results about antibalanced signed graphs are corollaries.  Several interesting results about signed graphs are lemmas.
\codes{\CPar,~\CSG:~\CHom}%
\itemspace

\bsgitem{1989a}{}  A min-max relation for stable sets in graphs with no odd-$K_4$.  \emph{J.\ Combin.\ Theory Ser.\ B} 47 (1989), 330--348.  
\MR{1026068} (91c:05143).  \Zbl{691.05021}.					
\annot{} Let $\Sigma$ be antibalanced and without isolated vertices and contain no subdivision of $-K_4$.  Then max.\ stable set size = min.\ cost of a cover by edges and negative circles.  Also, min.\ vertex-cover size = max.\ profit of a packing of edges and negative circles.  Also, weighted analogs.  [\Question.  Do the theorem and proof extend to any $\Sigma$?]
\codes{\Cpar,~\Csg:~\CCircles}%
\itemspace

\bsgitem{1989b}{}  A short proof of Tutte's characterization of totally unimodular matrices.  \emph{Linear Algebra Appl.}\ 114/115 (1989), 207--212.  
\MR{0986875} (90b:05033).  \Zbl{676.05028}.					
\annot{} The proof of Lemma 3 uses a signed graph. 
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{1990a}{\dag\dag}  \emph{Graphs and Polyhedra:  Binary Spaces and Cutting Planes}.  CWI Tract 73.  Centrum voor Wiskunde en Informatica, Amsterdam, 1990.  
\MR{1106635} (92f:\-52027).  \Zbl{727.90044}.					
\annot{} [Very incomplete annotation.]  Thm.:  Given $\Sigma$, the set $\{x \in \bbR^n : d_1 \leq x \leq d_2, b_1 \leq \Eta(\Sigma)\trans x \leq b_2 \}$ has Chvatal rank $\leq 1$ for all integral vectors $d_1, d_2, b_1, b_2$, iff $\Sigma$ contains no subdivided $-K_4$.   
	
\par
\codes{\CSG:~\CInc,~\CGeom,~\CBal,~\CSw,~\CStr}%
\itemspace

\bsgitem{1992a}{}  On shortest $T$-joins and packing $T$-cuts.  \emph{J.\ Combin.\ Theory Ser.\ B} 55 (1992), 73--82.  
\MR{1159855} (93d:05093).  \Zbl{810.05056}.					
\codes{\CSG:~\CStr}%
\itemspace

\bsgitem{1992b}{}  Odd paths and circuits in planar graphs with two odd faces.  CWI Report BS-R9218, September 1992.
\codes{\CSG:~\CCircles,~\Ctop}%
\itemspace

\bsgitem{1994a}{}  An orientation theorem for graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 62 (1994), 199--212.  
\MR{1305048} (96d:05051).  \Zbl{807.05020}.					
\codes{\Cpar,~\Csg:~\CMatrd,~\COri}%
\itemspace

\bsgitem{1995a}{\dag}  On Tutte's characterization of graphic matroids---a graphic proof.  \emph{J.\ Graph Theory} 20 (1995), 351--359.  
\MR{1355434} (96h:05038).  \Zbl{836.05017}.					
\annot{} Signed graphs used to prove Tutte's theorem.  The signed-graph matroid employed is the extended lift matroid $\bfL_0(\Sigma)$ (``extended even cycle matroid'').  The main theorem (Thm.\ 2):  Let $\Sigma$ be a signed graph with no $-K_4$, $\pm K_3$, $-Pr_3$, or $\Sigma_4$ link minor; then $\Sigma$ can be converted by Whitney 2-isomorphism operations (``breaking'' = splitting a component in two at a cut vertex, ``glueing'' = reverse, ``switching'' = twisting across a vertex 2-separation) to a signed graph that has a balancing vertex (``blocknode'').  Here $\Sigma_4$ consists of $+K_4$ with a 2-edge matching doubled by negative edges and one other edge made negative.
\annotctd{} More translation:  His ``$\Sigma$'' is our $E^-$.  ``Even, odd'' = positive, negative (for edges and circles).  ``Bipartite'' = balanced; ``almost bipartite'' = has a balancing vertex.
\codes{\CSG:~\CMatrd,~\CStr,~\CInc}%
\itemspace

\bsgitem{1995b}{}  Matching.  In:  M.O.\ Ball, T.L.\ Magnanti, C.L.\ Monma, and G.L.\ Nemhauser, eds., \emph{Network Models}, Ch.\ 3, pp.\ 135--224.   Handbooks Oper.\ Res.\ Management Sci., Vol.\ 7.  North-Holland, Amsterdam, 1995.
\MR{1420868}.  \Zbl{839.90131}.							
\annot{} \S7.2.2, ``Network flows and bidirected graphs''.  Generalized matchings in bidirected graphs.
\dated{9 Jun 2011} 
\codes{\Csg:~\COri:~\CInc}%
\itemspace

\author{A.M.H.\ Gerards \& M.\ Laurent}
\authorspace
\bsgitem{1995a}{}  A characterization of box $\frac 1d$-integral binary clutters.  \emph{J.\ Combin.\ Theory Ser.\ B} 65 (1995), 186--207.  
\MR{1358985} (96k:90052).  \Zbl{835.05017}.						
\annot{} Thm.\ 5.1:  The collection of negative circles of $\Sigma$ is box $\frac 1d$-integral for some/any integer $d \geq 2$ iff it does not contain $-K_4$ as a link minor.
\par
\codes{\CSG:~\CCircles,~\CGeom}%
\itemspace

\author{A.M.H.\ Gerards, L.\ Lov\'asz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}
\authorx{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}
\authorspace
\bsgitem{1990a}{}  Manuscript to be prepared as of 1990.
\annot{} Extension of \xref{Gerards and Schrijver}{1986b}{A.M.H.\ Gerards \& A.\ Schrijver} [same comments apply.  The proliferating authorship is preventing this major contribution from ever being published as such---though one hopes not!  See \xref{Seymour}{1995a}{P.D.\ Seymour} for description of two main theorems].
\codes{\CSG:~\CStr,~\CMatrd,~\CTop}%
\itemspace

\author{A.M.H.\ Gerards \& A.\ Schrijver} 
\authorspace
\bsgitem{1986b}{\dag\dag}  Signed graph -- regular matroids -- grafts.  Research Memorandum, Faculteit der Economische Wetenschappen, Tilburg University, 1986, 57 pp.  
\\ \url{https://research.tilburguniversity.edu/en/publications/signed-graphs-regular-matroids-grafts}
\annot{} Essential, major theorems.  The (extended) lift matroid of a signed graph is one of the objects studied.  Some of this material is published in \xref{Gerards}{1990a}{A.M.H.\ Gerards}.  This paper is in the process of becoming \xref{Gerards, Lov\'asz, \etal\!}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper} [was in process; that is unlikely to be written.]
\dated{rev 18 May 2017}%
\codes{\CSG:~\CStr,~\CMatrd}%
\itemspace

\bsgitem{1986a}{}  Matrices with the Edmonds--Johnson property.  \emph{Combinatorica} 6 (1986), 365--379.  
\MR{0879340} (88g:05087).  \Zbl{641.05039}, (\Zbl{565.90048}).			
\annot{} A subsidiary result:  If $-\Gamma$ contains no subdivided $-K_4$, then $\Gamma$ is $t$-perfect.
\codes{\Csg:~\CPar:~\CGeom,~\CStr}%
\itemspace

\author{A.M.H.\ Gerards \& F.B.\ Shepherd}
\authorspace
\bsgitem{1998a}{}  Strong orientations without even directed circuits.  \emph{Discrete Math.}\ 188 (1998), 111--125.
\MR{1630434} (99i:05091).  \Zbl{957.05048}.					
\codes{\Csd:~\CPar:~\CCycles}%
\itemspace

\bsgitem{1998b}{}  The graphs with all subgraphs $t$-perfect.  \emph{SIAM J.\ Discrete Math.}\ 11 (1998), 524--545.  
\MR{1640924} (2000e:05074).  \Zbl{980.38493}.					
\annot{} Extension of \xref{Gerards}{1989a}{A.M.H.\ Gerards}.  An ``odd-$K_4$'' is a graph whose all-negative signing is a subdivided $-K_4$.  A ``bad-$K_4$'' is an odd-$K_4$ which does not consist of exactly two undivided $K_4$ edges that are nonadjacent while the other edges are replaced by even paths.  Thm.\ 1:  A graph that contains no bad-$K_4$ as a subgraph is $t$-perfect.  Thm.\ 2 characterizes the graphs that are subdivisions of 3-connected graphs and contain an odd-$K_4$ but no bad-$K_4$.  [The fact that `badness' is not strictly a parity property weighs against the possibility that \xref{Gerards}{1989a}{A.M.H.\ Gerards} extends well to signed graphs.]
\codes{\Cpar,~\Csg:~\CStr,~\CAlgor}%
\itemspace

\author{K.A.\ Germina}						%GERMINA
\authorspace
\authref  See also \hyperlink{Ashraf P K}{Ashraf~P~K}, \hyperlink{Shahul Hameed K}{S.~Hameed}, \hyperlink{Albin Mathew}{A.~Mathew}, \hyperlink{Roshni T Roy}{R.T.~Roy}, \hyperlink{Shijin T V}{T.V.~Shijin}, and \hyperlink{Naduvath K.\ Sudev}{N.K.~Sudev}.
\authrefspace

\author{K.A.\ Germina \& P.K.\ Ashraf}
\authorspace
\bsgitem{2013a}{}  On open domination and domination in signed graphs.  \emph{Int.\ Math.\ Forum} 8 (2013), no.\ 38, 1863--1872.
\MR{3152954} (no rev).  \Zbl{1301.05247}.  
\codes{\CSG:~\CDom}%
\itemspace

\authorx{K.A.\ Germina \& Shahul Hameed K}
\author{K.A.\ Germina \& Shahul Hameed K (as K.\ Shahul Hameed)}
\authorspace
\bsgitem{2010a}{}  On signed paths, signed cycles and their energies.  \emph{Appl.\ Math.\ Sci.\ (Ruse)} 4 (2010), no.\ 70, 3455--3466.
\MR{2769200} (no rev).  \Zbl{1237.05125}.  
\annot{} Eigenvalues and energies of $A(\Sigma)$ and Laplacian matrices $\Lap(\Sigma)$ of signed paths and circles; also recurrences for the characteristic polynomials.  Energy of $A$ $:= \sum |\lambda_i(A)|$; energy of $\Lap$ $:= \sum |\lambda_i(\Lap) - \bar d|$ where $\bar d :=$ average degree.  [\Cf\ \xref{Mathai and Zaslavsky}{2012a}{A.M.\ Mathai \& Thomas Zaslavsky}.]
\dated{14 Nov 2010}
\codes{\CSG:~\CEig:~\CPaths,~\CCircles}%
\itemspace

\author{K.A.\ Germina, Shahul Hameed K, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2011a}{}  On products and line graphs of signed graphs, their eigenvalues and energy.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2432--2450.
\MR{2811128} (2012j:05254).  \Zbl{1222.05223}.  \arXiv{1010.3884}.			
\annot{} Adjacency matrix $A$ and eigenvalues and energy for the general ``Cvetko\-vi\'c product'', NEPS$(\Sigma_1, \ldots, \Sigma_k;\cB)$, and for a line graph $\Lambda(\Sigma)$ (as in \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}).  
Laplacian matrix $\Lap(\Sigma)$; $\Lap(+\Gamma) =$ Laplacian of graph $\Gamma$; $\Lap(-\Gamma) =$ signless Laplacian; and its eigenvalues and energy for Cartesian product $\Sigma_1 \times \cdots \times \Sigma_r$.  Also, $A(\Lambda(\Sigma))$.  Thm.:  The Cartesian product is balanced iff all $\Sigma_i$ are balanced.  
Examples:  Planar, cylindrical, and toroidal grids with product signatures; line graphs of those grids and of $+K_n$ and $-K_n$.
\dated{19 Oct 2010}
\codes{\CSG:~\CBal,~\CAdj,~\CEig,~\CLG}%
\itemspace

\author{K.A.\ Germina \& Sahariya}
\authorspace
\bsgitem{2015a}{}  On 2-path invariant signed graphs.  \emph{Adv.\ Appl.\ Discrete Math.}\ 15 (2015), no.\ 1, 21--32.
\MR{3242810}.  \Zbl{1320.05056}.  
\annot{} The ``$k$-path graph'' $(\Sigma)_k$ = $k$-path graph $(|\Sigma|)_k$ of $|\Sigma|$ with $\sigma_k(vw) := \max\sigma(P_{vw})$ over $vw$-paths of length $k$.  
(Different from \xref{Gill and Patwardhan}{1986a}{M.K.\ Gill \& G.A.\ Patwardhan}.)  
Question:  $(\Sigma)_2 \sim \Sigma$?  By Escalante and Montejano (1974),  $|\Sigma| \cong K_n$ or $C_{2m+1}$.
Thm.\ 7:  $(C_{2m+1},\sigma)_2$ is balanced.  Thms.\ 11--13:  Sufficient conditions for $(K_n,\sigma)_2 \sim (K_n,\sigma)$.
\dated{4 Jan 2022} 
\par
\codes{\CSG,~\CSw}%
\itemspace

\author{Mehrdad Ghadiri}						%GHADIRI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{A.\ Ghafari}							%GHAFARI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Joobin Gharibshah}						%GHARIBSHAH
\authorspace
\authref  See \hyperlink{Mohsen Shahriari}{M.~Shahriari}.
\itemspace

\author{E.\ Ghasemian}						%GHASEMIAN
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}.
\authrefspace

\author{E.\ Ghasemian \& G.H.\ Fath-Tabar}
\authorspace
\bsgitem{2017a}{}  On signed graphs with two distinct eigenvalues.  \emph{Filomat} 31 (2017), no.\ 20, 6393--6400.
\MR{3746874}.  
\annot{} Classifies 3- and 4-regular, triangle-free $\Sigma$ with two eigenvalues.  Cor.\ 2.5 says for 3-regular, the unique example is the cube $Q_3$ with all $C_4$'s negative.  Thm.\ 2.7 states a 4-regular characterization [incomplete; \cf\ \xref{Hou, Tang, and Wang}{2019a}{Yaoping Hou, Zikai Tang, \& Dijian Wang}.  Succeeded by \xref{Stani\'c}{2020b}{Zoran Stanic}.]
\dated{29 May 2018} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Anna Maria Ghirlanda} 						%GHIRLANDA
\authorspace
\authref  See \hyperlink{Luigi Muracchini}{L.~Muracchini}.
\itemspace

\author{Ebrahim Ghorbani}						%GHORBANI E
\authorspace
\authref  See also \hyperlink{R.B.\ Bapat}{R.B.~Bapat} and \hyperlink{Saieed Akbari}{S.~Akbari}.
\authrefspace

\bsgitem{2017a}{}  On eigenvalues of Seidel matrices and Haemers' conjecture.  \emph{Designs Codes Cryptogr.}\ 84 (2017), no.\ 1-2, 189--195.
\MR{3654204}.  \Zbl{1367.05128}.  \arXiv{1301.0075}.				
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Ebrahim Ghorbani, Willem H.\ Haemers, Hamid Reza Maimani, \& Leila Parsaei Majd}
\authorspace
\bsgitem{2020a}{}  On sign-symmetric signed graphs.  \emph{Ars Math.\ Contemp.}\ 19 (2020), 83--93.
\MR{4173666}.  \Zbl{1465.05072}.  \arXiv{2003.09981}.  
\annot{} ``Sign-symmetric'' means $\Sigma \simeq -\Sigma$.  Nonbipartite examples are found.  Sign symmetry implies symmetric spectrum but not conversely; shown by examples.
\dated{23 Nov 2020, 23 May 2022} 
\codes{\CSG:~\CStr,~\CAdj(\CEig)}%
\itemspace

\author{Ebrahim Ghorbani \& Arezoo Majidi}
\authorspace
\bsgitem{2021a}{}  Signed graphs with maximal index.  \emph{Discrete Math.}\ 344 (2021), no.\ 8, art.\ 112463, 8 pp.
\MR{4262020}.  \Zbl{1466.05087}.  \arXiv{2101.01503}.  
\annot{} Given $(K_n,\sigma)$ and $m=\#E^- \leq n^2/4$ (about half the edges), finds $\max_\sigma \lambda_1(A(K_n,\sigma))$ and the cases of equality.  
[Supersedes \xref {Akbari, Dalvandi, \etal}{2019a}{Saieed Akbari, Soudabeh Dalvandi, Farideh Heydari, \& Mohammad Maghasedi}, \xref{Kafai, Heydari, \etal}{2021a}{N.\ Kafai, F.\ Heydari, N.\ Jafari Rad, \& M.\ Maghasedi}.]
\dated{27 May 2021} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Modjtaba Ghorbani}						%GHORBANI M
\authorspace
\authref  See also \hyperlink{Mardjan Hakimi-Nezhaad}{M.~Hakimi-Nezhaad}.
\authrefspace

\author{Modjtaba Ghorbani, Mardjan Hakimi-Nezhaad, \& Bo Zhou}
\authorspace
\bsgitem{2023a}{}  Graphs with at most four Seidel eigenvalues.  \emph{Kragujevac J.\ Math.}\ 47 (2023), no.\ 2, 173--186 (to appear).
\annot{} Seidel matrix $S(\Gamma) := A(K_\Gamma)$.  When does it have eigenvalue with multiplicity $\geq n-d$, $d \leq 3$?  Partial solution.
\dated{9 Nov 2020} 
\par
\codes{\Csg:~\CKG:~\Csw,~\CAdj:~\CEig}%
\itemspace

\authorx{Vahid Ghorbani}						%GHORBANI V
\author{Vahid Ghorbani, Ghodratollah Azadi, \& Habib Azanchiler}
\authorspace
\bsgitem{2020a}{}  On the structure of spikes.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 17 (2020), no.\ 3, 883--886.
\MR{4181588}.  \Zbl{1471.05018}.  
\annot{} Spike = extended lift matroid $\bfL_0(2C_r,\cB)$ where $\cB \subseteq \{C_r \subset 2C_r\}$.  Construction of spikes by matroid operations from binary spikes (in which $\cB$ is maximum).  \Question.  Are these biased-graph operations?
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Prantar Ghosh}						%GHOSH P
\authorspace
\authref  See \hyperlink{Sandip Das}{S.~Das}.
\itemspace

\author{A.\ Ghouila-Houri} 						%GHOUILA-HOURI*
\authorspace
\authref  See \hyperlink{Claude Berge}{C.~Berge}.
\itemspace

\author{H.\ Giacomini}						%GIACOMINI
\authorspace
\authref  See \hyperlink{Hung T.\ Diep}{H.T.~Diep}.
\itemspace

\author{Christos Giatsidis}						%GIATSIDIS
\authorspace
\authref  See also \hyperlink{Fragkiskos D.\ Malliaros}{F.D.~Malliaros}.
\authrefspace

\author{Christos Giatsidis, Bogdan Cautis, Silviu Maniu, Dimitrios M.\ Thilikos, \& Michalis Vazirgiannis}
\authorspace
\bsgitem{2014a}{}  Quantifying trust dynamics in signed graphs, the S-Cores approach.  In:  \emph{Proceedings of the 2014 SIAM International Conference on Data Mining} (Philadelphia, 2014).  SIAM, 2014.
\codes{\CSD:~\CStr}%
\itemspace

\author{Rick Giles} 							%GILES
\authorspace
\bsgitem{1982a}{}  Optimum matching forests.  I:  Special weights.  II: General weights.  III:  Facets of matching forest polyhedra.  \emph{Math.\ Programming} 22 (1982), 1--11, 12--38, 39--51.  
\MR{0637527}--0637529 (82m:05075a,b,c).  \Zbl{468.90053}, \Zbl{468.90054}, \Zbl{468.90055}.
\annot{} In the author's ``mixed'' graphs, the undirected edges are really extraverted bidirected edges.	
\codes{\Csg:~\Cori}%
\itemspace

\author{Mukhtiar Kaur Gill [Mukti Acharya]} 				%GILL MK
\authorx{Mukhtiar Kaur Gill}
\authorspace
\authref  See also \hyperlink{B.\ Devadas Acharya}{B.D.~Acharya} and \hyperlink{Mukti Acharya}{Mukti Acharya}.
\authrefspace

\bsgitem{1981a}{}  A graph theoretical recurrence formula for computing the characteristic polynomial of a matrix.  
In:  S.B.~Rao, ed., \emph{Combinatorics and Graph Theory}  (Proc.\ Sympos., Calcutta, 1980), pp.\ 261--265.  \LNM 885.  Springer-Verlag, Berlin, 1981.
\MR{0655622} (83f:05047).  \Zbl{479.05030}.
\annot{} Introduces ``quasicospectrality'' of graphs or digraphs, i.e., they have cospectral signatures.
See \xref{B.D.~Acharya, Gill, and Pathwardhan}{1984a}{B.D.\ Acharya, M.K.\ Gill, \& G.A.\ Patwardhan} and \xref{M.~Acharya}{2012a}{Mukti Acharya}.
\dated{3 Feb 2012}
\codes{\CSG,~\CSD:~\CEig}%
\itemspace

\bsgitem{1981b}{}  A note concerning Acharya's conjecture on a spectral measure of structural balance in a social system.  
In:  S.B.~Rao, ed., \emph{Combinatorics and Graph Theory} (Proc.\ Sympos., Calcutta, 1980), pp.\ 266--271.  \LNM 885.  Springer-Verlag, Berlin, 1981.  
\MR{0655623} (84d:05121).  \Zbl{476.05073}.
\annot{} Assume $|\Sigma_1| = |\Sigma_2|$.  If $\Sigma_1$ and $\Sigma_2$ have the same value of \xref{B.D.~Acharya's}{1980a}{B.\ Devadas Acharya} measure of imbalance, $A(\Sigma_1)$ and $A(\Sigma_2)$ may have different spectra.  [Not surprisingly.]
\codes{\CSG:~\CBal,~\CEig}%
\itemspace

\bsgitem{1982a}{}  \emph{Contributions to Some Topics in Graph Theory and Its Applications}.  Ph.D.\ thesis, Dept.\ of Mathematics, Indian Institute of Technology, Bombay, 1982.  
\annot{} Most of the results herein have been published separately.  See \xref{M.K.\ Gill}{1981a}{Mukhtiar Kaur Gill}, \xrefd{1981b}{Mukhtiar Kaur Gill}, \xref{Gill and Patwardhan}{1981a}{M.K.\ Gill \& G.A.\ Patwardhan}, \xrefd{1982a}{M.K.\ Gill \& G.A.\ Patwardhan}, \xrefd{1986a}{M.K.\ Gill \& G.A.\ Patwardhan}, \xref{M.\ Acharya}{2009a}{Mukti Acharya}.
\codes{\CSG,~\CSD:~\CBal,~\CLG,~\CAdj,~\CEig}%
\itemspace

\author{M.K.\ Gill \& B.D.\ Acharya} 
\authorspace
\bsgitem{1980a}{} A recurrence formula for computing the characteristic polynomial of a sigraph.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 5 (19 80), 68--72.  
\MR{0586322} (81m:05097).  \Zbl{448.05048}.
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{1980b}{}  A new property of two dimensional Sperner systems.  \emph{Bull.\ Calcutta Math.\ Soc.}\ 72 (1980), 165--168.  
\MR{0669580} (83m:05121).  \Zbl{531.05058}.
\par
\codes{\CSG:~\CBal,~\CGeom}%
\itemspace

\author{M.K.\ Gill \& G.A.\ Patwardhan} 
\authorspace
\bsgitem{1981a}{} A characterization of sigraphs which are switching equivalent to their line sigraphs.  \emph{J.\ Math.\ Phys.\ Sci.}\ 15 (1981), 567--571.  
\MR{0650430} (84h:05106).  \Zbl{488.05054}.
\annot{} The line graph is that of \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand}.	
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{1982a}{}  A characterization of sigraphs which are switching equivalent to their iterated line sigraphs.  \emph{J.\ Combin.\ Inform.\ System.\ Sci.}\ 7 (1982), 287--296.  
\MR{0724371} (86a:05103).  \Zbl{538.05060}.
\annot{} The line graph is that of \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand}.	
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{1986a}{}  Switching invariant two-path signed graphs.  \emph{Discrete Math.}\ 61 (1986), 189--196.  
\MR{0855324} (87j:05138).  \Zbl{594.05059}.
\annot{} The $k$-path signed graph of $\Sigma$ [I write $D_k(\Sigma)$] is the distance-$k$ graph on $V$ with signs $\sigma_k(uv) = -$ iff every length-$k$ path is all negative.  The equation $\Sigma \simeq D_2(\Sigma)$ is solved.
\dated{29 Apr 2009}
\codes{\CSG,~\CSw}%
\itemspace

\author{Robert Gill}							%GILL R
\authorspace
\bsgitem{1998a}{}  (as Robert Voorhees Gill)  \emph{A Generalization of the Partition Lattice: Combinatorial Properties and the Action of the Symmetric Group}.   Doctoral dissertation, University of Michigan, 1998.
\MR{2697157} (no rev).  
\par
\codes{\Cgg:~\Cmatrd:~\CGeom,~\CInvar,~\CAut}%
\itemspace

\authorspace
\bsgitem{1998b}{}  The number of elements in a generalized partition semilattice.  \emph{Discrete Math.}\ 186 (1998), 125--134.
\MR{1623892} (99e:52014).  \Zbl{956.52009}.					
\annot{} The semilattice is the intersection semilattice of an affinographic hyperplane arrangement representing $[-k,k]K_n$ [and is therefore isomorphic to the geometric semilattice of all $k$-composed partitions of a set; see, e.g., \xref{Zaslavsky}{2002a}{Thomas Zaslavsky}, Ex.\ 10.5].  The rank and the Whitney numbers of the first kind are calculated with the aid of species.  See \xref{Kerr}{1999a}{Julie Kerr} for homology.
\par
\codes{\Cgg:~\Cmatrd:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2000a}{}  The action of the symmetric group on a generalized partition semilattice.  \emph{Electronic J.\ Combin.}\ 7 (2000), Research Paper 23, 20 pp.
\MR{1755612} (2001g:\-05107).  \Zbl{947.06001}.					
\annot{} See \xrefd{1998a}{Robert Gill}.
\codes{\Cgg:~\Cmatrd:~\CGeom,~\CInvar,~\CAut}%
\itemspace

\author{Ernst D.\ Gilles}						%GILLES
\authorspace
\authref  See \hyperlink{Steffen Klamt}{S.~Klamt}.
\itemspace

\author{John Gimbel} 						%GIMBEL
\authorspace
\bsgitem{1988a}{}  Abelian group labels on graphs.  \emph{Ars Combin.}\ 25 (1988), 87--92.  
\MR{0944350} (89k:05046).  \Zbl{655.05034}.						
\annot{} The topic is ``induced'' edge labellings, that is, $w(e_{uv}) = f(u)f(v)$ for some $f: V \to \fA$.  The number of $f$ that induce a given induced labelling, the number of induced labellings, and a characterization of induced labellings.  All involve the $2$-torsion subgroup of $\fA$, unless $\Gamma$ is bipartite.  The inspiration is dualizing magic graphs.  [Somewhat dual to \xref{Edelman and Saks}{1979a}{Paul H.\ Edelman \& Michael Saks}.]
\codes{\Cpar:~\Cinc}\codesctd{\CVS(\CGen):~\CEnum}%
\itemspace

\authorx{Omer Gim\'enez}						%GIMENEZ
\authorx{Omer Gimenez}
\author{Omer Gim\'enez, Anna de Mier, \& Marc Noy}
\authorx{Omer Gimenez, Anna de Mier, \& Marc Noy}
\authorspace
\bsgitem{2005a}{}  On the number of bases of bicircular matroids.  \emph{Ann.\ Combin.}\ 9 (2005), no.\ 1, 35--45.
\MR{2135774} (2005m:05049).  \Zbl{1059.05030}.						
\annot{} The number of bases is bounded above by $C^n\cdot$(number of spanning trees) in a simple graph but not in a multigraph.  More precise results for $K_n$ and $K_{n,m}$.  [See \xref{Neudauer, Meyers, and Stevens}{2001a}{Nancy Ann Neudauer, Andrew M.\ Meyers, \& Brett Stevens} and \xref{Neudauer and Stevens}{2001a}{Nancy Ann Neudauer \& Brett Stevens}.]
\codes{\CBic:~\CInc}%
\itemspace

\author{Omer Gim\'enez \& Marc Noy}
\authorspace
\bsgitem{2006a}{}  On the complexity of computing the Tutte polynomial of bicircular matroids.  \emph{Combin.\ Probab.\ Comput.}\ 15 (2006), no.\ 3, 385--395.
\MR{2216475} (2007a:05029).  \Zbl{1094.05013}.
\annot{} Known NP-hardness results for transversal matroids apply to their proper subclass, bicircular matroids, with a few possible exceptions.
\par
\codes{\CBic:~\CInc:~\CAlgor}%
\itemspace

\author{Giulia Giordano}						%GIORDANO
\authorspace
\authref  See \hyperlink{Franco Blanchini}{F.~Blanchini}.
\itemspace

\authorx{Ioannis Giotis}						%GIOTIS
\author{Ioannis Giotis \& Venkatesan Guruswami}
\authorspace
\bsgitem{2006a}{}  Correlation clustering with a fixed number of clusters.  In:  \emph{Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms}, pp.\ 1167--1176.   ACM, New York, 2006.
\MR{2373844} (2009f:62098).  \Zbl{1194.62087}.					
\par
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\bsgitem{2006b}{}  Correlation clustering with a fixed number of clusters.  \emph{Theory Comput.}\ 2 (2006), 249--266.
\MR{2322880} (2009e:68118).  
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\author{Noriane Girard}						%GIRARD
\authorspace
\authref  See \hyperlink{Emanuele Delucchi}{E.~Delucchi}.
\itemspace

\authorx{Pierre-Louis Giscard}						%GISCARD
\author{Pierre-Louis Giscard, Paul Rochet, \& Richard C. Wilson}
\authorspace
\bsgitem{2017a}{}  Evaluating balance on social networks from their simple cycles.  \emph{J.\ Complex Networks} 5 (2017), no.\ 5, 750--775.
\MR{3801709}.  \arXiv{1606.03347}.  
\par
\codes{\CSG:~\CFr,~\CPsS}%
\itemspace

\author{Nicolas Glade}						%GLADE	
\authorspace
\authref  See \hyperlink{Loic Forest}{L.~Forest}.
\itemspace

\authorx{Roland Glantz}						%GLANTZ
\author{Roland Glantz \& Marcello Pelillo}
\authorspace
\bsgitem{2006a}{}  Graph polynomials from principal pivoting.  \emph{Discrete Math.}\ 306 (2006), no.\ 24, 3253--3266.
\MR{2279060} (2008d:05112) (\qv).  \Zbl{1125.05073}.					
\annot{} The polynomials arise from $A(\Phi)$ where $\Phi$ is an $F^+$-gain graph, $F$ a field.  
\codes{\CGG:~\CInvar,~\CAdj}%
\itemspace

\author{Terry C.\ Gleason} 						%GLEASON
\authorspace
\authref  See also \hyperlink{Dorwin Cartwright}{D.~Cartwright}.
\authrefspace

\author{Terry C.\ Gleason \& Dorwin Cartwright}
\authorspace
\bsgitem{1967a}{} A note on a matrix criterion for unique colorability of a signed graph.  \emph{Psychometrika} 32 (1967), 291--296.  
\MR{0214509} (35 \#5359).  \Zbl{184.49202} (184, p.\ 492b).				
\annot{} ``Colorable'' = clusterable.  The adjacency matrices of $\Sigma^+$ and $\Sigma^-$ are employed separately.  The arithmetic is mostly ``Boolean'', i.e., $1+1=0$.  A certain integral matrix $T$ shows whether or not $\Sigma$ is clusterable.
\dated{11 Nov 2008}
\codes{\CSG:~\CClu,~\CAdj}%
\itemspace

\author{Fred Glover} 							%GLOVER
\authorspace
\authref  See also \hyperlink{Joyce Elam}{J.~Elam}.
\authrefspace

\author{F.\ Glover, J.\ Hultz, D.\ Klingman, \& J.\ Stutz} 
\authorspace
\bsgitem{1978a}{}  Generalized networks:  A fundamental computer-based planning tool.  \emph{Management Sci.}\ 24 (1978), 1209--1220.
\codes{\CGN:~\CAlgor,~\CMatrd(\Cbases):~\CExp,~\CRef}%
\itemspace

\author{Fred Glover \& D.\ Klingman} 
\authorspace
\bsgitem{1973a}{}  On the equivalence of some generalized network problems to pure network problems.  \emph{Math.\ Programming} 4 (1973), 269--278.
\MR{0317845} (47 \#6393).  \Zbl{259.90012}.  
\par
\codes{\CGN:~\CBal,~\CInc}%
\itemspace

\bsgitem{1973b}{}  A note on computational simplifications in solving generalized transportation problems.  \emph{Transportation Sci.}\ 7 (1973), 351--361.  
\MR{0418463} (54 \#6502).  
\par
\codes{\CGN:~\CMatrd(\Cbases),~\Cgeom}%
\itemspace

\author{Fred Glover, Darwin Klingman, \& Nancy V.\ Phillips}
\authorspace
\bsgitem{1992a}{}  \emph{Network Models in Optimization and Their Applications in Practice}.  Wiley-\break Interscience, New York, 1992.  
\annot{} Textbook.  See especially Ch.\ 5:  ``Generalized networks.''
\par
\codes{\CGN:~\CAlgor:~\CExp}%
\itemspace

\author{F.\ Glover, D.\ Klingman, \& J.\ Stutz} 
\authorspace
\bsgitem{1973a}{}  Extensions of the augmented predecessor index method to generalized network problems.  \emph{Transportation Sci.}\ 7 (1973), 377--384.  
\codes{\CGN:~\CMatrd(\Cbases),~\Cmatrd}%
\itemspace


\author{Luis Goddyn}						%GODDYN
\authorspace
\authref  See also \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}.
\authrefspace

\author{Luis Goddyn, Winfried Hochst\"attler, \& Nancy Ann Neudauer}	
\authorspace
\bsgitem{2016a}{}  Bicircular matroids are 3-colorable.  \emph{Discrete Math.}\ 339 (2016), 1425--1429.
\MR{3475555}.  \Zbl{1333.05063}.  
\par
\codes{\CBic:~\CBic}%
\itemspace

\author{C.D.\ Godsil} 						%GODSIL
\authorspace
\authref  See also \hyperlink{John Brown}{J.~Brown} and \hyperlink{G.\ Coutinho}{G.~Coutinho}.
\authrefspace

\bsgitem{1985a}{}  Inverses of trees.  \emph{Combinatorica} 5 (1985), 33--39. 
\MR{0803237} (86k:05084).  \Zbl{578.05049}.					
\annot{} If $T$ is a tree with a perfect matching, then $A(T)\inv = A(\Sigma)$ where $\Sigma$ is balanced and $|\Sigma| \supseteq \Gamma$.  \Question.  When does $|\Sigma| = \Gamma$?  [Solved by \xref{Simion and Cao}{1989a}{R.\ Simion \& D.-S.\ Cao}.]  [\Cf\ \xref{Buckley, Doty, and Harary}{1988a}{Fred Buckley, Lynne L.\ Doty, \& Frank Harary}, \xref{Tifenbach \& Kirkland}{2009a}{R.M.\ Tifenbach \& S.J.\ Kirkland}, and for a generalization to rings \xref{Bapat and Ghorbani}{2014a}{R.B.\ Bapat \& E.\ Ghorbani}.  For a different notion, \xref{Greenberg, Lundgren, and Maybee}{1984b}{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}.]
\dated{$<$ 1988 \emph{et seq}}
\annotctd{} \Problem.  A bipartite graph with a unique perfect matching has an inverse signed multigraph: $A(\Gamma)\inv=A(\Sigma)$.  When is $\Sigma$ balanced (i.e., switches to an unsigned graph)?  Elegant solution by \xref{Yang and Ye}{2018a}{Yujun Yang \& Dong Ye}.
\dated{11 Dec 2018}
\codes{\Csg:~\CAdj,~\CBal,~\Csw}%
\itemspace

\bsgitem{1996a}{}  Covers of complete graphs.  In:  \emph{Progress in Algebraic Combinatorics} (Fukuoka, 1993, Kyoto, 1994--5), pp.\ 137--163.   Adv.\ Stud.\ Pure Math., Vol.\ 24.  Kinokuniya, Kyoto, 1996.
\MR{1414466} (97m:05201).  \Zbl{859.05076}.  
\annot{} Covering graphs of complete permutation gain graphs.  
Dictionary:  ``arc function'' = permutation gain function.
\dated{7 Nov 2020} 
\par
\codes{\CGG:~\CKG:~\CCov}%
\itemspace

\author{C.D.\ Godsil \& I.\ Gutman}
\authorspace
\bsgitem{1981a}{}  On the matching polynomial of a graph.  In:  L.\ Lov\'asz and Vera T.\ S\'os, eds.,  \emph{Algebraic Methods in Graph Theory} (Proc., Szeged, 1978), Vol.\ I, pp.\ 241--249.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, Vol.\ 25.  North-Holland, Amsterdam, 1981.
\MR{0642044} (83b:05101).  \Zbl{0476.05060}.  
\codes{\CSG:~\CAdj~\Ceig}%
\itemspace

\author{Chris Godsil \& Gordon Royle}
\authorspace
\bsgitem{2001a}{}  \emph{Algebraic Graph Theory}.  Graduate Texts in Math., Vol.\ 207.  Springer-Verlag, New York, 2001.
\MR{1829620} (2002f:05002).  \Zbl{968.05002}.
\annot{} Ch.\ 11, ``Two-graphs'':  Equiangular lines (\xref{van Lint and Seidel}{1966a}{J.H.\ van Lint \& J.J.\ Seidel}, \xref{Lemmens and Seidel}{1973a}{P.W.H.\ Lemmens \& J.J.\ Seidel}), graph switching (\xref{van Lint and Seidel}{1966a}{J.H.\ van Lint \& J.J.\ Seidel}, \xref{Seidel}{1976a}{J.J.\ Seidel}), regular two-graphs (\xref{Taylor}{1977a}{D.E.\ Taylor}).
\par
\codes{\CTG:~\CAdj,~\CEig,~\CGeom,~\CSw}%
\annotctd{} Ch.\ 12, ``Line graphs and eigenvalues'':  Based on \xref{Cameron, Goethals, Seidel, and Shult}{1976a}{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}.
\codes{\CLG:~\Csg:~\CEig,~\CGeom,~\CSw}%
\annotctd{} \S15.3, ``Signed matroids'':  Sign-colored matroids and graphs.  Rank generating polynomial (see \xref{Kauffman}{1989a}{Louis H.\ Kauffman}).  \S16.3, ``Signed plane graphs'', \S16.5, ``Reidemeister invariants'', \S16.6, ``The Kauffman brack\-et'', \S16.8, ``Connectivity'':  Properties of \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} ``signed-graph'' (really sign-colored graph) Tutte polynomial.  \S16.7, ``The Jones polynomial'' of a knot.
\codes{\CSc,~\CSGc:~\CAdj,~\CInc,~\CTop}%
\itemspace

\author{J.M.\ Goethals} 						%GOETHALS
\authorspace
\authref  See also \hyperlink{Peter J.\ Cameron}{P.J.~Cameron}.
\authrefspace

\author{J.M.\ Goethals \& J.J.\ Seidel}
\authorspace
\bsgitem{1970a}{}  Strongly regular graphs derived from combinatorial designs.
\emph{Canad.\ J.\  Math.}\ 22 (1970) 597--614.
\MR{0282872} (44 \#106).  \Zbl{198.29301}.					
\annot{} A symmetric Hadamard matrix $H$ with constant diagonal can be put in the form $A(K_n,\sigma) \pm I$ for some signed $K_n$ that represents a regular two-graph [see \xref{D.E.\ Taylor}{1977a}{D.E.\ Taylor}] of order $4s^2$ (Thm.\ 4.1).
\codes{\Ctg:~\CAdj}%
\itemspace

\author{Michael Goff}						%GOFF
\authorspace
\bsgitem{2003a}{}  Recovering networks with signed conductivities.  REU paper, University of Washington, 2003.  URL \url{http://www.math.washington.edu/~reu/papers/2003/goff/mgoff.pdf}
\annot{} Partial treatment of the problem in \xref{W.~Johnson}{2012a}{Will Johnson}.
\dated{26 Dec 2012} 
\codes{\Csg:~\CWG:~\CAdj}%
\itemspace

\author{G\"ulistan Kaya G\"ok}						%GOK
\authorx{Gulistan Kaya Gok}
\authorspace
\bsgitem{2019a}{}  Some bounds on the Seidel energy of graphs.  \emph{TWMS J.\ Appl.\ Engineering Math.}\ 9 (2019), no.\ 4, 949--956.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\authorx{Andrew V.\ Goldberg}						%GOLDBERG
\author{Andrew V.\ Goldberg \& Alexander V.\ Karzanov}
\authorspace
\bsgitem{1994a}{}  Path problems in skew-symmetric graphs.  In:  \emph{Proceedings of the 5th annual ACM-SIAM symposium on discrete algorithms} (Arlington, Va., 1994), pp.\ 526--535.  New York, Assoc.\ Comput.\ Machinery (ACM), 1994.
\MR{1285193} (95c:05074).  \Zbl{867.90118}.					
\codes{\Csd:~\CFlows,~\CCov}%
\itemspace

\bsgitem{1996a}{}  Path problems in skew-symmetric graphs.  \emph{Combinatorica} 16 (1996), no.\ 3, 353--382.
\MR{1417346} (97h:05099).  \Zbl{867.05037}.					
\codes{\Csd:~\CFlows,~\CCov}%
\itemspace

\bsgitem{2004a}{} Maximum skew-symmetric flows and matchings.  \emph{Math.\ Program., Ser.\ A} 100 (2004), no.\ 3, 537--568. 
\MR{2129927} (2005m:90142).  \Zbl{1070.90090}.					
\annot{} Techniques for digraph flows are extended to bidirected flows, treated via the double covering digraph (\cf\ \xref{Tutte}{1967a}{W.T.\ Tutte}).
\dated{9 Sept 2010} 
\codes{\Csg:~\COri:~\CFlows,~\CCov}%
\itemspace

\authorx{Felix Goldberg}						%GOLDBERG F
\author{Felix Goldberg \& Steve Kirkland}
\authorspace
\bsgitem{2014a}{}  On the sign patterns of the smallest signless Laplacian eigenvector.  \emph{Linear Algebra Appl.}\ 443 (2014), 66--85.
\MR{3148894}.  \Zbl{1282.05114}.  \arXiv{1307.7749}.				
\annot{} The sign pattern of an eigenvector of the smallest eigenvalue of $\Lap(-\Gamma)$ for a bipartite graph + some edges may be predictable.  [\Problem.  Generalize to signed graphs.]
\dated{23 Nov 2014} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Andrew V.\ Goldberg, \'Eva Tardos, \& Robert E.\ Tarjan}
\authorx{Andrew V.\ Goldberg, Eva Tardos, \& Robert E.\ Tarjan}
\authorspace
\bsgitem{1990a}{}  Network flow algorithms.  
In:  B.~Korte, L.~Lov\'asz, H.J.~Pr\"omel, and A.~Schrijver, eds., \emph{Paths, Flows, and VLSI-Layout}, pp.\ 101--164.  Algorithms and Combinatorics, Vol.\ 9.  Springer-Verlag, Berlin, 1990. 
\MR{1083378} (92i:90043).  \Zbl{728.90035}.  
\annot{} \S1.5, ``The generalized flow problem'':  Max flow, conservative except at the source, in networks with (real, positive) gains; generalized augmenting paths.  \S1.6, ``The restricted problem'':  Flows with gains, conservative except at source and sink, whose residual flow has no gainy cycles that avoid the source.  \S1.7, ``Decomposition theorems'' for flows with or without gains.  
\S6, ``The generalized flow problem'':  Combinatorial algorithms; connections between flow problems with and without gains
\dated{11 Jun 2012} 
\codes{\CGN:~\CAlgor}%
\itemspace

\authorx{Jay R.\ Goldman}						%GOLDMAN
\author{Jay R.\ Goldman \& Louis H.\ Kauffman}
\authorspace
\bsgitem{1993a}{} Knots, tangles, and electrical networks.  \emph{Adv.\ Appl.\ Math.}\ 14 (1993), 267--306.
\MR{1228742} (94m:57013).  \Zbl{806.57002}.						
Repr.\ in Louis H.\ Kauffman,  \emph{Knots and Physics}, 2nd edn., pp.\ 684--723.  Ser.\ Knots Everything, Vol.\ 1.  World Scientific, Singapore, 1993.
\MR{1306280} (95i:57010).  \Zbl{868.57001}.
\annot{} The parametrized Tutte polynomial [as in \xref{Zaslavsky}{1992b}{Thomas Zaslavsky} \etal] of an $\bbR^\times$-weighted graph is used to define a two-terminal ``conductance''.  Interpreting weights as crossing signs ($\pm1$) in a planar link diagram with two blocked regions yields invariants of tunnel links.  
[Also see \xref{Kauffman}{1997a}{Louis H.\ Kauffman}.]
\codes{\CSGw:~\CGen:~\CInvar,~Knot,~\CPhys}%
\itemspace

\author{Avraham Goldstein}						%GOLDSTEIN A
\authorspace
\authref  See \hyperlink{Yonah Cherniavsky}{Y.~Cherniavsky}.
\itemspace

\authorx{Richard Z.\ Goldstein}			 			%GOLDSTEIN RZ
\author{Richard Z.\ Goldstein \& Edward C.\ Turner}
\authorspace
\bsgitem{1979a}{} Applications of topological graph theory to group theory.  \emph{Math.\ Z.}\ 165 (1979), 1--10.  
\MR{0521516} (80g:20050).  \Zbl{377.20027}, (\Zbl{387.20034}).			
\codes{\CSG:~\CTop}%
\itemspace

\author{Eric Goles}							%GOLES
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Harry F.\ Gollub} 						%GOLLUB
\authorspace
\bsgitem{1974a}{} The subject-verb-object approach to social cognition.  \emph{Psychological Rev.}\ 81 (1974), 286--321.
\codes{\CPsS:~\Cvs}%
\itemspace

\author{Martin Charles Golumbic}					%GOLUMBIC
\authorspace
\bsgitem{1979a}{}  A generalization of Dirac's theorem on triangulated graphs.  
In:  Allan Gewirtz and Louis V.\ Quintas, eds., Second Int.\ Conf.\ on Combinatorial Mathematics (New York, 1978).  
\emph{Ann.\ New York Acad.\ Sci.}\ 319 (1979), 242--246.  
\MR{0556028} (81c:05077).  \Zbl{479.05055}.
\annot{} Further results on chordal bipartite graphs.  Their properties imply standard properties of ordinary chordal graphs.  [See \xrefd{1980a}{Martin Charles Golumbic} for more.]  (The ``only if'' portion of Thm.\ 4 is false, according to \xrefd{1980a}{Martin Charles Golumbic}, p.\ 267.)
\par
\codes{\Csg:~\Cbal,~\Ccov}%
\itemspace

\bsgitem{1980a}{}  \emph{Algorithmic Graph Theory and Perfect Graphs}.  Academic Press, New York, 1980.  
\MR{0562306} (81e:68081).  \Zbl{541.05054}.
\annot{} \S12.3: ``Perfect elimination bipartite graphs,'' and \S12.4: ``Chordal bipartite graphs,'' expound perfect elimination and chordality for bipartite graphs from \xref{Golumbic and Goss}{1978a}{Martin Charles Golumbic \& Clinton F.\ Goss} and \xref{Golumbic}{1979a}{Martin Charles Golumbic}.  In particular, Cor.\ 12.11:  A bipartite graph is chordal bipartite iff every induced subgraph has perfect edge elimination scheme.  [\Problem.  Guided by these results, find a signed-graph generalization of chordality that corresponds to supersolvability and perfect vertex elimination (\cf\ \xref{Zaslavsky}{2001a}{Thomas Zaslavsky}).]
\codes{\Csg:~\Cbal,~\Ccov}%
\itemspace

\author{Martin Charles Golumbic \& Clinton F.\ Goss}
\authorspace
\bsgitem{1978a}{}  Perfect elimination and chordal bipartite graphs.  \emph{J.\ Graph Theory} 2 (1978), 155--163.  
\MR{0493395} (80d:05037).  \Zbl{411.05060}.
\annot{} A perfect edge elimination scheme is a bipartite analog of a perfect vertex elimination scheme.  A chordal bipartite graph is a bipartite graph in which every circle longer than $4$ edges has a chord.  Analogs of properties of chordal graphs, e.g., Dirac's separator theorem, are proved.  In particular, a chordal bipartite graph has a perfect edge elimination scheme.  [See \xref{Golumbic}{1980a}{Martin Charles Golumbic} for more.]
\codes{\Csg:~\Cbal}%
\itemspace

\authorx{Sergio G\'omez}						%GOMEZ
\authorx{Sergio Gomez}
\author{Sergio G\'omez, Pablo Jensen, \& Alex Arenas}
\authorx{Sergio Gomez, Pablo Jensen, \& Alex Arenas}
\authorspace
\bsgitem{2009a}{}  Analysis of community structure in networks of correlated data.  \emph{Phys.\ Rev.\ E} 80 (2009), no.\ 1, 016114.
\arXiv{0812.3030}.  
\annot{} \Cf\ \xref{Bansal, Blum, and Chawla}{2004a}{Nikhil Bansal, Avrim Blum, \& Shuchi Chawla}.
\codes{\CSG,~\CWG:~\CClu}%
\itemspace

\author{J.R.\ Gon{\c c}alves}						%GONCALVES
\authorx{J.R.\ Goncalves}
\authorspace
\authref  See also \hyperlink{J.A.\ Blackman}{J.A.~Blackman}.
\authrefspace

\author{J.R.\ Gon{\c c}alves, J.\ Poulter, \& J.A.\ Blackman}
\authorx{J.R.\ Goncalves, J.\ Poulter, \& J.A.\ Blackman}
\authorspace
\bsgitem{1995a}{}  $\pm J$ Ising model in 2D and of general composition.  \emph{J.\ Magnetism Magnetic Materials} 140--144 (1995), 1701--1702.
\codes{\CPhys:~\Csg:~\CFr,~\CEig}%
\itemspace

\bsgitem{1997a}{}  Bond and site defects in fully frustrated two-dimensional Ising systems.  \emph{J.\ Phys.\ A} 30 (1997), no.\ 9, 2947--2962.
\MR{1456894} (98d:82030).  \Zbl{920.60082}.  
\annot{} Signed square and triangular lattice graphs with all face circles (``plaquettes'') negative (``frustrated'').  Entropy change due to changing an edge sign (creating a ``bond defect''), or one or two vertex deletions (``site defects'').  Eigenstates correspond to negative plaquettes, as in \xref{Blackman}{1982a}{J.A.\ Blackman} and \xref{Blackman and Poulter}{1991a}{J.A.\ Blackman \& J.\ Poulter}.
\dated{17 May 2013} 
\par
\codes{\CPhys,~\CSG:~\CFr,~\CEig}%
\itemspace

\author{Chen Gong}							%GONG C
\authorspace
\authref  See \hyperlink{Weichan Liu}{W.-C.~Liu}.
\itemspace

\author{Maoguo Gong}						%GONG MG
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai} and \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\author{Shi-Cai Gong}						%GONG SC
\authorspace
\authref  See also \hyperlink{Yi Wang}{Y.~Wang}.
\authrefspace

\bsgitem{2011a}{}  The unicyclic graphs with extremal signless Laplacian spectral spread.  In:  \emph{Proceedings of 2011 World Congress on Engineering and Technology} (CET 2011), Vol.\ 1, pp.\ ?.  [This may not have been published.]
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shicai Gong, Hangen Duan, \& Yizheng Fan}
\authorspace
\bsgitem{2006a}{}  On eigenvalues distribution of mixed graphs.  \emph{J.\ Math.\ Study} 39 (2006), no.\ 2, 124--128.
\MR{2248100} (2007b:05136).  \Zbl{1104.05045}.					
\annot{} The ``mixed graphs'' are signed graphs.  ``[R]elations between the eigenvalues and matching number, diameter, and the number of quasi-pendant vertices of mixed graphs.''  (From the abstract.)
\dated{9 Jan 2013} 
\codes{\Csg:~\CEig}%
\itemspace

\author{Shi-Cai Gong \& Yi-Zheng Fan}
\authorspace
\bsgitem{2007a}{}  Nonsingular unicyclic mixed graphs with at most three eigenvalues greater than two.  \emph{Discuss.\ Math.\ Graph Theory} 27 (2007), no.\ 1, 69--82.
\MR{2321423} (2008b:\-05102).  \Zbl{1139.05033}.					
\annot{} Characterizes such signed (``mixed'') graphs, for $\Lap(\Sigma)$, for $n\geq9$.
\dated{23 Mar 2009}
\codes{\Csg:~\Cinc,~\CEig}%
\itemspace

\author{Shi-Cai Gong \& Guang-Hui Xu}
\authorspace
\bsgitem{2012a}{}  The characteristic polynomial and the matchings polynomial of a weighted oriented graph.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 9, 3597--3607.
\MR{2900738}.  \Zbl{1244.05120}.  
\annot{} A ``weighted oriented graph'' is an $\bbR^+$-gain graph.  The ``skew adjacency matrix'' is the gain-graphic adjacency matrix.
\dated{7 Feb 2012} 
\par
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Mauricio Gonz\'alez}						%GONZALEZ
\authorx{Mauricio Gonzalez}
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\authorx{Andrew Goodall}						%GOODALL
\author{Andrew Goodall, Bart Litjens, Guus Regts, \& Llu{\'\i}s Vena}
\authorx{Andrew Goodall, Bart Litjens, Guus Regts, \& Lluis Vena}
\authorspace
\bsgitem{2017a}{}  A Tutte polynomial for non-orientable maps.  Ninth European Conf.\ Combinatorics, Graph Theory Appl.\ (EuroComb 2017, Vienna, 2017).  \emph{Electronic Notes Discrete Math.}\ 61 (2017), 513--519.
\Zbl{1378.05094}.  
\annot{} Extended abstract of \xrefd{2020a}{Andrew Goodall, Bart Litjens, Guus Regts, \& Lluis Vena}.
\codes{\CSG:\CTop,~\CInvar}%
\itemspace

\bsgitem{2019a}{}  The canonical Tutte polynomial for signed graphs.  \emph{Acta Math.\ Univ.\ Comenianae (N.S.)}\ 88 (2019), no.\ 3, 749--754.
\MR{4012875}.  
\annot{} Early, shorter version of \xrefd{2021a}{Andrew Goodall, Bart Litjens, Guus Regts, \& Lluis Vena}.
\codes{\CSG:~\CInvar,~\CMatrd}%
\itemspace

\bsgitem{2020a}{}  A Tutte polynomial for maps II: the non-orientable case.  \emph{European J.\ Combin.}\ 86 (2020), art.\ 103095, 32 pp.
\MR{078935}.  \Zbl{1437.05108}.  \arXiv{1804.01496}.  
\codes{\CSG:~\CTop,~\CInvar,~\CMatrd}%
\itemspace

\bsgitem{2021a}{}  Tutte's dichromate for signed graphs.  \emph{Discrete Appl.\ Math.}\ 289 (2021), 153--184.
\MR{4164543}.  \Zbl{1454.05045}.  \arXiv{1903.07548}.				
\codes{\CSG:~\CInvar,~\CMatrd}%
\itemspace

\author{Gary Gordon}						%GORDON G
\authorspace
\authref  See also \hyperlink{Lori Fern}{L.~Fern}.
\authrefspace

\bsgitem{1997a}{}  Hyperplane arrangements, hypercubes and mixed graphs.  Proc.\ Twenty-eighth Southeastern Int.\ Conf.\ on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1997).  \emph{Congressus Numer.}\ 126 (1997), 65--72.
\MR{1604975} (98j:\-05038).  \Zbl{901.05055}.					
\annot{} An explicit bijection between the regions of the real hyperplane arrangement corresponding to $\pm K_n^{\circ}$ and the set of ``good signed [complete] mixed graphs'' $G_{\text{\bf a}}$ of order $n$.  The latter are a notational variant of the acyclic orientations $\tau$ of $\pm K_n^{\circ}$ [and are therefore in bijective correspondence with the regions, by \xref{Zaslavsky}{1991b}{Thomas Zaslavsky}, Thm.\ 4.4].  Dictionary: a directed edge in $G_{\text{\bf a}}$ is an oriented positive edge in $\tau$, while a positive or negative undirected edge in $G_{\text{\bf a}}$ is an introverted or extroverted negative edge of $\tau$.  
The main result, Thm.\ 1, is an interesting and significant explicit description of the acyclic orientations of $\pm K_n^{\circ}$.  Namely, one orders the vertices and directs all positive edges upward; then one steps inward randomly from both ends of the ordered vertex set, one vertex at a time, at each new vertex orienting all previously unoriented negative edges to be introverted if the vertex was approached from below, extroverted if from above in the vertex ordering.  [This clearly guarantees acyclicity.]  [\Problem.  Generalize to arbitrary signed graphs.]
\annotctd{} Lemma 2, ``a standard exercise'', is that an orientation of $\pm K_n^{\circ}$ (with the loops replaced by half edges) is acyclic iff the magnitudes of its net degrees are a permutation of $\{1, 3, \ldots, 2n-1]\}$.  [Similarly, an orientation of $\pm K_n^{\circ}$ is acyclic iff its net degree vector is a signed permutation of $\{2, 4, \ldots, 2n\}$ (\xref{Zaslavsky}{1991b}{Thomas Zaslavsky}, p.\ 369, but possibly known beforehand in other terminology).  Both follow easily from \xref{Zaslavsky}{1991b}{Thomas Zaslavsky}, Cor.\ 5.3: an acyclic orientation has a vertex that is a source or sink.] 
\par
\codes{\CSG:~\Cori:~\Cinc,~\CGeom}%
\itemspace

\bsgitem{1999a}{}  The answer is $2^n \cdot n!$  What's the question?  \emph{Amer.\ Math.\ Monthly} 106 (Aug.--Sept., 1999), no.\ 7, 636--645.
\MR{1720459} (2000j:05050).  \Zbl{982.05052}.					
\annot{} \S5 presents the (signed-graph) question: an appealing presentation of material from \xrefd{1997a}{Gary Gordon}.
\codes{\CSG:~\Cori,~\CInc,~\CGeom,~N:~\CExp}%
\itemspace

\author{Gary Gordon, Jennifer McNulty, \& Nancy Ann Neudauer}
\authorspace
\bsgitem{2016a}{}  Fixing numbers for matroids.  \emph{Graphs Combin.}\ 32 (2016), 133--146.
\MR{3436955}.  \Zbl{1332.05029}.  \arXiv{1307.7460}.				
\annot{} \S4, ``Cycle and bicircular matroids'':  Thm.\ 4.3:  If $\Gamma$ is 3-connected and $\#V\geq5$, then $\bfM(\Gamma)$ and $\bfF(\Gamma,\eset)$ have the same fixing number.  Thm.\ 4.4:  If $\Gamma$ is 2-connected, $\#V\geq5$, min degree $\geq3$, then $\Aut \bfF(\Gamma,\eset) \cong \Aut\Gamma$.  Cor.\ 4.6.2:  Fixing number of $\bfF(K_n,\eset)$ is 5 for $n=4$, $\lfloor 2n/3 \rfloor$ if $n>4$.
\dated{8 Jan 2016} 
\codes{\CGG:~\CBic:~\CAut}%
\itemspace

\authorx{Y.\ Gordon}			 				%GORDON Y
\author{Y.\ Gordon \& H.S.\ Witsenhausen}
\authorspace
\bsgitem{1972a}{}  On extensions of the Gale--Berlekamp switching problem and constants of $l_p$-spaces.  \emph{Israel J.\ Math.}\ 11 (1972), 216--229.  
\MR{0304078} (46 \#3213).  \Zbl{238.46009}.
\annot{} Asymptotic estimates of $l(K_{r,s})$, the maximum frustration index of signatures of $K_{r,s}$, improving the bounds of \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}.
\par
\codes{\Csg:~\CFr}%
\itemspace

\author{Clinton F.\ Goss}						%GOSS
\authorspace
\authref  See \hyperlink{Martin Charles Golumbic}{M.C.~Golumbic}.
\itemspace

\author{Eric Gottlieb}							%GOTTLIEB
\authorspace
\bsgitem{1998a}{}  (as Eric Inness Gottlieb)  \emph{Cohomology of Dowling Lattices and Lie Superalgebras}.  Ph.D.\ thesis, University of Miami, 1998.
\MR{2698113}.  
\annot{} See \xref{Gottlieb and Wachs}{2000a}{Eric Gottlieb \& Michelle L.\ Wachs}.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2003a}{}  On the homology of the $h,k$-equal Dowling lattice.  \emph{SIAM J.\ Discrete Math.}\ 17 (2003), no.\ 1, 50--71. 
\MR{2033305} (2004k:05209).  \Zbl{1033.05098}.  
\annot{} The lattice is the subposet of $\Lat \bfF(\fG K_n)$ consisting of the flats whose nontrivial balanced components have order $\geq k$ and whose unbalanced component, if any, has order $\geq h$.  If $\#\fG=2$ and $h\leq k$ we have the lattice of \xref{Bj\"orner and Sagan}{1996a}{Anders Bjorner \& Bruce E.\ Sagan}.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2003b}{}  On EL-shelling for the nondecreasing partition lattice.  Proc.\ Thirty-Fourth Southeastern Int.\ Conf.\ Combinatorics, Graph Theory and Computing.  \emph{Congressus Numer.}\ 162 (2003), 119--127. 
\MR{2050544} (2005e:05159).  \Zbl{1056.05147}.  
\codes{\Cgg,~\Csg:~\CMatrd}%
\itemspace

\author{Eric Gottlieb \& Michelle L.\ Wachs}
\authorspace
\bsgitem{2000a}{}  Cohomology of Dowling lattices and Lie (super)algebras.  \emph{Adv.\ Appl.\ Math.}\ 24 (2000), no.\ 4, 301--336.
\MR{1761776} (2001i:05161).  \Zbl{1026.05104}.						
\annot{} Two monomorphisms of the cohomology of the order complex of the lattice of flats of $Q_n(\fG)$, upon which $\fS_n\wr\fG$ acts as operators, into enveloping algebras of certain Lie algebras and Lie superalgebras.
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Samira Goudarzi}							%GOUDZARI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\authorx{Ian P.\ Goulden}						%GOULDEN
\author{Ian P.\ Goulden, Jin Ho Kwak, \& Jaeun Lee}
\authorspace
\bsgitem{2005a}{}  Enumerating branched orientable surface coverings over a non-orientable surface.  \emph{Discrete Math.}\ 303 (2005), 42--55.
\MR{2181041} (2006i:05089).  \Zbl{1079.05025}.					
\codes{\CSG:~\CCov,~\CTop,~\Cgg}%
\itemspace

\author{Antoine Gournay}						%GOURNAY
\authorspace
\bsgitem{2016a}{}  An isoperimetric constant for signed graphs.  \emph{Expositiones Math.}\ 34 (2016), no.\ 3, 339--351.
\MR{3521482}.  \Zbl{1342.05085}.  \arXiv{1410.5995}.  
\codes{\CSG:~\CInvar,~\CCov}%
\itemspace

\author{Jean-Luc Gouz\'e}						%GOUZE
\authorx{Jean-Luc Gouze}
\authorspace
\bsgitem{1998a}{}  Positive and negative circuits in dynamical systems.  \emph{J.\ Biol.\ Systems} 6 (1998), 11--15.
\Zbl{1058.37537}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Kevin Grace}							%GRACE
\authorspace
\bsgitem{2021a}{}  The templates for some classes of quaternary matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 146 (2021), 286--363.
\MR{4155284}.  \Zbl{1478.05019}.  \arXiv{1902.07136}.  
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Kevin Grace \& Stefan H.M.\ van Zwam}
\authorspace
\bsgitem{2018a}{}  On perturbations of highly connected dyadic matroids.  \emph{Ann.\ Combin.}\ 22 (2018), 513--542.
\MR{3845746}.  \Zbl{1396.05021}.  \arXiv{1712.07702}.				
\annot{} Counterexamples to \xref{Geelen, Gerards, and Whittle}{2015a}{Jim Geelen, Bert Gerards, \& Geoff Whittle}.
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{2019a}{}  The highly connected even-cycle and even-cut matroids.  \emph{SIAM J.\ Discrete Math.}\ 33 (2019), no.\ 1, 26--67.
\MR{3896635}.  \Zbl{1402.05030}.  \arXiv{1610.01106}.  
\codes{\CSG:~\CMatrd,~\CDual}%
\itemspace

\author{Jaros{\l}aw Grytczuk}						%GRYTCZUK
\authorx{Jaroslaw Grytczuk}
\authorspace
\authref  See \hyperlink{Marcin Anholcer}{M.~Anholcer}.
\itemspace

\author{R.L.\ Graham}		 				%GRAHAM
\authorspace
\bsgitem{1968a}{}  On finite 0-simple semigroups and graph theory.  \emph{Math.\ Systems Theory} 2 (1968), 325--339.
\MR{0240228} (39 \#1580).  \Zbl{177.03103} (177, 31c).				
\annot{} Bipartite gain graphs are essential in the representation theory of 0-simple semigroups.
\Cf\ \xref{Rees}{1940a}{David Rees}, \xref{Houghton}{1977a}{C.H.\ Houghton}, \xref{Rhodes and Steinberg}{2009a}{John Rhodes \& Benjamin Steinberg}.
\dated{19 Aug 2019} 
\codes{\Cgg:~\Csw,~\CAut:~\CAlgeb}%
\itemspace

\author{R.L.\ Graham \& N.J.A.\ Sloane}
\authorspace
\bsgitem{1985a}{}  On the covering radius of codes.  \emph{IEEE Trans.\ Inform.\ Theory} IT-31 (1985), 385--401.  
\MR{0794436} (87c:94048).  \Zbl{585.94012}.					
\annot{} See Example b, p.\ 396 (the Gale--Berlekamp code).  
\codes{\Csg:~\CFr}%
\itemspace
 
\authorx{M.J.\ Grannell}						%GRANNELL
\author{M.J.\ Grannell \& T.S.\ Griggs}
\authorspace
\bsgitem{2009a}{}  Embeddings and designs.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 13, pp.\ 268--288.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581550} (no rev).  \Zbl{1225.05034}.  
\annot{} The voltage graph (i.e., gain graph) construction is used to generate embeddings of combinatorial designs.
\dated{12 Jun 2013} 
\par
\codes{\CTop:~\CGG,~\CCov:~\CExp}%
\itemspace

\author{Douglas D.\ Grant}						%GRANT
\authorspace
\authref  See \hyperlink{Lars Dovling Andersen}{L.D.~Andersen}.
\itemspace

\authorx{Ante Graovac}						%GRAOVAC*
\author{Ante Graovac, Ivan Gutman, \& Nenad Trinajsti\'c}
\authorx{Ante Graovac, Ivan Gutman, \& Nenad Trinajstic}
\authorspace
\bsgitem{1977a}{}  \emph{Topological Approach to the Chemistry of Conjugated Molecules}.  Lect.\ Notes in Chem., Vol.\ 4.  Springer-Verlag, Berlin, 1977.  
\Zbl{385.05032}.  
\annot{} \S2.7.  ``Extension of graph-theoretical considerations to Mobius systems.''
\codes{\CSG:~\CAdj,~\CEig,~\CChem}%
\itemspace

\author{A.\ Graovac \& N.\ Trinajsti\'c} 
\authorx{A.\ Graovac \& N.\ Trinajstic} 
\authorspace
\bsgitem{1975a}{}  Mobius molecules and graphs.  \emph{Croatica Chemica Acta (Zagreb)} 47 (1975), 95--104.  
\codes{\CSG:~\CAdj,~\CEig,~\CChem}%
\itemspace

\bsgitem{1976a}{}  Graphical description of M\"obius molecules.  \emph{J.\ Molecular Structure} 30 (1976), 416--420.  
\annot{} The ``M\"obius graph'' (i.e., signed graph of a suitably twisted ring hydrocarbon) is introduced with examples of the adjacency matrix and characteristic polynomial.
\codes{\CChem:~\CSG:~\CAdj,~\CEig}%
\itemspace

\author{Timothy Graves}						%GRAVES
\authorspace
\authref  See \xref{Brewster and Graves}{2009a}{Richard C.\ Brewster \& Timothy Graves}.
\itemspace

\authorx{Gary Greaves}						%GREAVES
\author{Gary Greaves, Jack Koolen, Akihiro Munemasa, Yoshio Sano, \& Tetsuji Tani\-guchi}
\authorx{Gary Greaves, Jack Koolen, Akihiro Munemasa, Yoshio Sano, \& Tetsuji Taniguchi}
\authorspace
\bsgitem{2015a}{}  Edge-signed graphs with smallest eigenvalue greater than $-2$.  \emph{J.\ Combin.\ Theory Ser.\ B} 110 (2015), 90--111.
\MR{3279389}.  \Zbl{1302.05074}.  
\par
\codes{\CSG:~\CAdj:~\CEig:~\CStr}%
\itemspace

\author{Gary Greaves, Jacobus H.\ Koolen, Akihiro Munemasa, \& Ferenc Sz\"oll\H{o}si}
\authorx{Gary Greaves, Jacobus H.\ Koolen, Akihiro Munemasa, \& Ferenc Szollosi}
\authorspace
\bsgitem{2016a}{}  Equiangular lines in Euclidean spaces.  \emph{J.\ Combin.\ Theory Ser.\ A} 138 (2016), 208--235.
\MR{3423477}.  \Zbl{1330.51006}.  
\annot{} \S3, ``Some structural results on Seidel matrices''.  Remarkable congruence results.  Let $\Sigma := (K_n,\sigma)$.  
Thm.\ 3.5:  $\det A(\Sigma) \equiv (-1)^n(1-n) + 4n(\#E)\ (\mod 8)$.  Cor.\ 3.6:  $\det A(\Sigma) \equiv 1-n\ (\mod 4)$.  Thm.\ 3.7 gives $\per A(\Sigma)\ (\mod 8)$.  Prop.\ 3.8:  $\Sigma \not\simeq -\Sigma$ if $n \equiv 3\ (\mod 4)$.  
\annotctd{} \S4, ``Small sets of equiangular lines'', finds all $[\Sigma]$ for $n\leq12$, thus proving \xref{Haemers'}{2012a}{Willem H.\ Haemers} conjecture for $n\leq12$.  Application to equiangular lines.
\dated{10 Dec 2020} 
\codes{\CSG:~\CKG:~\CGeom,~\Csw,~\CAdj:~\CEig}%
\itemspace

\author{Gary Greaves, Bojan Mohar, \& Suil O}
\authorspace
\bsgitem{2019a}{}  Interlacing families and the Hermitian spectral norm of digraphs.  \emph{Linear Algebra Appl.}\ 564 (2019), 201--208.
\MR{3885710}.  \Zbl{1405.05069}.  
\annot{} Gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Gary R.W.\ Greaves \& Zoran Stani\'c}
\authorspace
\bsgitem{2022a}{}  Signed $(0,2)$-graphs with few eigenvalues and a symmetric spectrum.  \emph{J.\ Combin.\ Designs} 30 (2022), no.\ 5, 332--353.
\MR{4403834}.  
\arXiv{2107.11556}.  
\annot{} $(0,2)$-graph: all $\# [N(u)\cap N(v)] = 0,2$.
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{[John G.\ del Greco]}						%GRECO
\authorspace
\authref  See \hyperlink{John G.\ del Greco}{J.G.~del~Greco} (under `D').
\itemspace

\author{F.\ Green}							%GREEN
\authorspace
\bsgitem{1987a}{}  More about NP-completeness in the frustration model.  \emph{OR Spektrum} 9 (1987), 161--165.  
\MR{0908232} (88m:90053).  \Zbl{625.90070}.						
\annot{} Proves polynomial time for the reduction employed in \xref{Bachas}{1984a}{Constantin P.\ Bachas} and improves the theorem to:  The frustration-index decision problem on signed (3-dimensional) cubic lattice graphs with 9 layers is NP-complete.  [2 layers, in \xref{Barahona}{1982a}{Francisco Barahona}.]
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{Jan Green-Kr\'otki} 						%GREEN-KROTKI
\authorspace
\authref  See \hyperlink{Julian Araoz}{J.~Ar\'aoz}.
\itemspace

\authorx{Harvey J.\ Greenberg}						%GREENBERG
\author{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}
\authorspace
\bsgitem{1983a}{}  Rectangular matrices and signed graphs.  \emph{SIAM J.\ Algebraic Discrete Methods} 4 (1983), 50--61.  
\MR{0689865} (84m:05052).  \Zbl{525.05045}.						
\annot{} From a matrix $B$, with row set $R$ and column set $C$, form the ``signed bipartite graph'' $BG^+$ with vertex set $R \cup C$ and an edge $r_ic_k$ signed $\sgn b_{ik}$ whenever $b_{ik} \neq 0$.  The ``signed row graph'' $RG^+$ is the two-step signed graph of $BG^+$ on vertex set $R$: that is, $r_ir_j$ is an edge if $\dist^{BG^+}(r_i, r_j) = 2$ and its sign is the sign of any shortest $r_ir_j$-path.  If some edge has ill-defined sign, $RG^+$ is undefined.  The ``signed column graph'' $CG^+$ is similar.  The paper develops simple criteria for existence and balance of these graphs and the connection to matrix properties.  It examines simple special forms of $B$.
\codes{\CQM:~\CSG,~\CBal,~\CAppl}%
\itemspace

\bsgitem{1984a}{}  Signed graphs of netforms.  Proc.\ Fifteenth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing.  \emph{Congressus Numer.}\ 44 (1984), 105--115.  
\MR{0777533} (87c:05085).  \Zbl{557.05048}. 						
\annot{} Application of \xrefd{1983a}{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}, \xrefd{1984b}{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}.  ``Netform'' = incidence matrix of a positive real gain graph (neglecting a minor technicality).  Thm.\ 1: $B$ is a netform iff $RG^+(B)$ exists and is all negative.  (Then $CG^+(B)$ also exists.)  Thm.\ 2:  If the row set partitions so that all negative elements are in some rows and all positives are in the other rows, then $RG^+(B)$ is all negative and balanced.  Thm.\ 3:  If $\Sigma$ is all negative and balanced, then $B$ exists as in Thm.\ 2 with $RG^+(B) = \Sigma$.  [Equivalent to theorem of \xref{Hoffman and Gale}{1956a}{[A.J.\ Hoffman \& D.\ Gale]}.]  $B$ is an ``inverse'' of $\Sigma$.  Thm.\ 4 concerns ``inverting'' $-\Gamma$ in a minimal way.  Then $B$ will be (essentially) the incidence matrix of $+\Gamma$.
\codes{\CSG,~\Cgg:~\Cinc,~\CBal,~\CVS,~\CExp,~\CAppl}%
\itemspace

\bsgitem{1984b}{}  Inverting signed graphs.  \emph{SIAM J.\ Algebraic Discrete Methods} 5 (1984), 216--223.  
\MR{0745440} (86d:05085).  \Zbl{581.05052}.						
\annot{} See \xrefd{1983a}{Harvey J.\ Greenberg, J.\ Richard Lundgren, \& John S.\ Maybee}.  ``Inversion'' means, given a signed graph $\Sigma_R$, or $\Sigma_R$ and $\Sigma_C$, finding a matrix $B$ such that $\Sigma_R = RG^+(B)$, or $\Sigma_R = RG^+(B)$ and $\Sigma_C = CG^+(B)$.  The elementary solution is in terms of coverings of $\Sigma_R$ by balanced cliques.  It may be desirable to minimize the size of the balanced clique cover; this difficult problem is not tackled.
\codes{\CQM:~\CSG,~\CVS,~\CBal}%
\itemspace

\author{Harvey J.\ Greenberg \& John S.\ Maybee, eds.}
\authorspace
\bsgitem{1981a}{}  \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ First Sympos., Univ.\ of Colorado, Boulder, Col., 1980).  Academic Press, New York, 1981.
\MR{0617930} (82g:00016).  \Zbl{495.93001}.
\annot{} Several articles relevant to signed (di)graphs. 
\codes{\CQM}\codesctd{\CSD,~\CSG:~\CBal}%
\itemspace

\authorx{Curtis Greene}			 			%GREENE
\author{Curtis Greene \& Thomas Zaslavsky}
\authorspace
\bsgitem{1983a}{} On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 280 (1983), 97--126.
\MR{0712251} (84k:05032).  \Zbl{539.05024}.					
\annot{} \S9: ``Acyclic orientations of signed graphs.''  Continuation of \xref{Zaslavsky}{1991b}{Thomas Zaslavsky}, counting acyclic orientations with specified unique source; also, with edge $e$ having specified orientation and with no termini except at the ends of $e$.  The proof is geometric.
\codes{\CSG:~\CMatrd,~\COri,~\CGeom,~\CInvar}%
\itemspace

\author{David A.\ Gregory}						%GREGORY*
\authorspace
\authref  See also \hyperlink{Michael S.\ Cavers}{M.~Cavers}.
\authrefspace

\bsgitem{2012a}{}  Spectra of signed adjacency matrices.  Presented to Queen's-R.M.C. Discrete Mathematics Seminar.  Ms., 2012.  \url{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.225.7817}
\annot{} $\rho=$ spectral radius.  \Conjecture\ 2:  Given simple $\Gamma$, for some $\sigma$ we have $\rho(A(\Gamma,\sigma)) \leq 2\sqrt{\Delta-1}$.  Thus, focus is on small $\rho$---via matching polynomial of $\Gamma$.  
Lemma 3:  $\rho(\Sigma) \geq \surd{\bar{d}(\Gamma)}$; $=$ iff $A(\Sigma)^2 = \bar{d} I_n$.  \S6, ``2-covers and Ramanujan families'':  ``2-cover'' $= \tilde\Sigma$.  Lemma 6 rediscovers \xref{Fowler's}{2002a}{Patrick W.\ Fowler} theorem (\qv).
\dated{13 Jun 2019} 
\codes{\CSG:~\CAdj:~\CEig,~\Csw,~\CKG}%
\itemspace

\author{David A.\ Gregory, Kevin N.\ Vandermeulen, \& Bryan L.\ Shader}
\authorspace
\bsgitem{1996a}{}  Rank decompositions and signed bigraphs.  \emph{Linear Multilinear Algebra} 40 (1996), 283--301.
\MR{1384648} (97a:05147).  \Zbl{866.05042}.					
\annot{} For bipartite $\Sigma$, $\cM :=$ class of matrices with weak sign pattern $\Sigma$.  Every $A \in \cM$ is the sum of $\rk A$ rank-1 matrices in $\cM$ iff (*) $\sigma(C) = -(-1)^{\#C/2}$ for every circle with $\#C\geq6$.  Thm.\ 3.2:  $\Sigma$ has (*) for every circle iff it is a spanning subgraph of a signed 4-cockade.  Thm.\ 3.7.  $\Sigma$ has (*) for circles with $\#C\geq6$ iff, after switching, it is obtained by three constructions from a negative $C_4$, a subgraph of $+K_{3,n}$, or a signed graph $R_n$.
\dated{6 Mar 2011} 
\codes{\CSG:~\CQM,~\CCircles}%
\itemspace

\author{Gary S.\ Grest}						%GREST
\authorspace
\authref  See also \hyperlink{Daniel Blankschtein}{D.~Blankschtein}.
\authrefspace

\bsgitem{1985a}{}  Fully and partially frustrated simple cubic Ising models: a Monte Carlo study.  \emph{J.\ Phys.\ C} 18 (1985), 6239--6246.
\annot{} Simulation of the cubic signed graph of \xref{Blankschtein, Ma, and Nihat Berker}{1984a}{Daniel Blankschtein, M.\ Ma, \& A.\ Nihat Berker}.
\dated{18 Jun 2012} 
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\author{T.S.\ Griggs}							%GRIGGS
\authorspace
\authref  See \hyperlink{M.J.\ Grannell}{M.J.~Grannell}.
\itemspace

\authorx{Will Grilliette}						%GRILLIETTE
\author{Will Grilliette, Josephine Reynes, \& Lucas J.\ Rusnak}
\authorspace
\bsgitem{2022a}{}  Incidence hypergraphs: injectivity, uniformity, and matrix-tree theorems.  \emph{Linear Algebra Appl.}\ 634 (2022), 77--105.
\MR{4339605}.  \Zbl{1479.05206}.  \arXiv{1910.02305}.				
\codes{\CSH:~\CInc}%
\itemspace

\author{Will Grilliette \& Lucas J.\ Rusnak}
\authorspace
\bsgitem{20xxa}{}  Incidence hypergraphs: Box products \& the Laplacian.  Submitted.  
\arXiv{2007.01842}.						
\codes{\CSH:~\CAlgeb}%
\itemspace

\author{G.\ Grimmett}						%GRIMMETT
\authorspace
\bsgitem{1994a}{}  The random-cluster model.  
In: F.P.~Kelly, ed., \emph{Probability, Statistics and Optimisation}, Ch.\ 3, pp.\ 49--63.  Wiley, Chichester, 1994.
\MR{1320741} (96d:60154).  \Zbl{858.60093}.						
\annot{} Reviews \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn} and subsequent developments esp.\ in multidimensional lattices.  The viewpoint is mainly probabilistic and asymptotic.  \S3.7, ``Historical observations,'' reports Kasteleyn's account of the origin of the model.
\codes{\Csgc:~\CGen:~\CInvar,~\CPhys:~\CExp}%
\itemspace

\authorx{Ya.R.\ Grinberg}						%GRINBERG
\author{Ya.R.\ Grinberg \& A.M.\ Rappoport}
\authorspace
\bsgitem{2011a}{}  Configuration and minimal coloring of unbalanced graphs.  (In Russian.)  \emph{Doklady Akad.\ Nauk} 439 (2011), no.\ 6, 743--745.
\MR{2883804} (no rev).  \Zbl{1238.05118}.  
\annot{} See \xrefd{2011b}{Ya.R.\ Grinberg \& A.M.\ Rappoport}.
\codes{\CSG:~\CFr,~\CStr,~\CClu}%
\itemspace

\bsgitem{2011b}{}  Configuration and minimal coloring of disbalanced graphs.  \emph{Doklady Math.}\ 84 (2011), no.\ 1, 579--581.
\MR{2883804} (no rev).  \Zbl{1238.05118}.  
\annot{} Thm.\ 1:  The contrabalanced signed graphs are the cactus forests (Husimi forests) in which every circle is negative.  Dictionary:  ``disbalance'' = contrabalance, ``junction'' = cutpoint, ``cyclically splittable'' = every block is a circle, ``$p$-groupable'' = $p$-clusterable.  
\dated{9 Jun 2012, 22 Jan 2015} 
\codes{\CSG:~\CFr,~\CStr,~\CClu}%
\itemspace

\author{Richard C.\ Grinold} 						%GRINOLD
\authorspace
\bsgitem{1973a}{}  Calculating maximal flows in a network with positive gains.  \emph{Operations Res.}\ 21 (1973), 528--541.  
\MR{0351412} (50 \#3900).  \Zbl{304.90043}.
\annot{} Objective:  to find the maximum output for given input.  Basic solutions correspond to bases of $\bfF(\Phi')$, $\Phi'$ being the underlying gain graph $\Phi$ together with an unbalanced loop adjoined to the sink.  \xref{Onaga}{1967a}{Kenji Onaga} also treats this problem.  
\codes{\CGN:~\CMatrd(\Cbases),~\CAlgor}%
\itemspace

\authorx{Heinz Gr\"oflin}			 			%GROFLIN
\authorx{Heinz Groflin}
\author{Heinz Gr\"oflin \& Thomas M.\ Liebling}
\authorx{Heinz Groflin \& Thomas M.\ Liebling}
\authorspace
\bsgitem{1981a}{}  Connected and alternating vectors:  polyhedra and algorithms.  \emph{Math.\ Programming} 20 (1981), 233--244.  
\MR{0607409} (83k:90061).  \Zbl{448.90035}.
\codes{\Csg,~\CGeom}%
\itemspace

\author{Piotr Gronek}						%GRONEK
\authorspace
\authref  See \hyperlink{Premiyslaw Gawronski}{P.~Gawro\'nski} and \hyperlink{Krzysztof Kulakowski}{K.~Ku{\l}akowski}.
\itemspace

\author{Jonathan L.\ Gross}  						%GROSS
\authorspace
\authref  See also \hyperlink{Jianer Chen}{J.~Chen}.
\authrefspace

\bsgitem{1974a}{} Voltage graphs.  \emph{Discrete Math.}\ 9 (1974), 239--246.  
\MR{0347651} (50 \#153).  \Zbl{286.05106}.
\codes{\CGG:~\CTop,~\CCov}%
\itemspace

\bsgitem{2009a}{}  Distribution of embeddings.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 3, pp.\ 45--61.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581540} (no rev).  \Zbl{1197.05032}.  
\annot{} \S3, ``Total embedding distributions'':  ``Twist'' (= edge signature) is used to construct nonorientable embeddings, which increase the count of embeddings.
\dated{12 Jun 2013} 
\codes{\CTop:~\Csg:~\CExp}%
\itemspace

\author{Jonathan L.\ Gross, Toufik Mansour, \& Thomas W.\ Tucker}
\authorspace
\bsgitem{2020a}{}  Partial-duality for ribbon graphs, I:  Distributions.  \emph{European J.\ Combin.}\ 86 (2020), art.\ 103084, 20 pp.
\MR{4056111}.  \Zbl{1437.05060}.  
\par
\codes{\Csg:~\CTop:~\CDual:~\CInvar}%
\itemspace

\bsgitem{2021a}{}  Partial-duality for ribbon graphs, II:  Partial-twuality polynomials and monodromy computations.  \emph{European J.\ Combin}\ 95 (2021), art.\ 103329, 28 pp.
\MR{4248388}.  \Zbl{1466.05050}.  
\annot{} \Cf\ \xref{Chen and Chen}{20xxa}{Qiyao Chen \& Yichao Chen}.
\codes{\Csg:~\CTop:~\CDual:~\CInvar}%
\itemspace

\bsgitem{2021b}{}  Partial-duality for ribbon graphs, III:  a Gray code algorithm for enumeration.  \emph{J.\ Algebraic Combin.}\ 54 (2021), 1119--1135.
\MR{4348919}.  \Zbl{1479.05076}.  
\par
\codes{\Csg:~\CTop:~\CDual:~\CInvar:~\CAlgor}%
\itemspace

\author{Jonathan L.\ Gross \& Thomas W.\ Tucker}
\authorspace
\bsgitem{1977a}{} Generating all graph coverings by permutation voltage assignments.  \emph{Discrete Math.}\ 18 (1977), 273--283.  
\MR{0465917} (57 \#5803).  \Zbl{375.55001}.
\par
\codes{\CGG:~\CTop,~\CCov}%
\itemspace

\bsgitem{1979a}{}  Fast computations in voltage graph theory.  
In:  Allan Gewirtz and Louis V.\ Quintas, eds., Second Int.\ Conf.\ on Combinatorial Mathematics (New York, 1978).  
\emph{Ann.\ New York Acad.\ Sci.}\ 319 (1979), 247--253.  
\MR{0556029} (80m:94111).  \Zbl{486.05027}.
\codes{\CGG:~\CTop,~\CCov,~\CSw}%
\itemspace

\bsgitem{1987a}{}  \emph{Topological Graph Theory}.  Wiley, New York, 1987.
\MR{0898434} (88h:05034).  \Zbl{621.05013}.
Repr.\ with minor additions: Dover Publications, Mineola, N.Y., 2001.
\MR{1855951}.  \Zbl{991.05001}.
\annot{} Ch.\ 2:  ``Voltage graphs and covering spaces.''  Ch.\ 4:  ``Imbedded voltage graphs and current graphs.'' 
\codes{\CGG:~\CTop,~\CCov}%
\annotctd{} \S3.2.2:  ``Orientability.'' \S3.2.3:  ``Rotation systems.''  \S4.4.5: ``Nonorientable current graphs'', discusses how to deduce, from the signs on a current graph, the signs of the ``derived'' graph of the dual voltage graph.  [The same rule gives the signs on the surface dual of any orientation-embedded signed graph.]  (The sign group here is $\bbZ_2$.)
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{2009a}{}  Embedding graphs on surfaces.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 1, pp.\ 18--33.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581538} (no rev).  \Zbl{1197.05033}.  
\annot{} \S 4, ``Rotation systems'':  Mentions signed graphs for orientation embedding.  \S 5, ``Covering spaces and voltage graphs'':  Voltage graphs for surface embedding of graphs.  \S 6, ``Enumeration'':  See \xref{Kwak and Lee}{2009a}{Jin Ho Kwak \& Jaeun Lee}.  Dictionary:  ``voltage graph'' = gain graph; edge sign $-$ is ``twisted'', $+$ is ``flat''.
\dated{12 Jun 2013} 
\par
\codes{\CTop:~\CSG,~\Csw,~\CGG,~\CCov:~\CExp}%
\itemspace

\author{Jerrold W.\ Grossman} 						%GROSSMAN
\authorspace
\authref  See also \hyperlink{R.B.\ Bapat}{R.B.~Bapat}.
\authrefspace

\author{Jerrold W.\ Grossman \& Roland H\"aggkvist}
\authorx{Jerrold W.\ Grossman \& Roland Haggkvist}
\authorspace
\bsgitem{1983a}{}  Alternating cycles in edge-partitioned graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 34 (1983), 77--81.  
\MR{0701173} (84h:05044).  \Zbl{491.05039}, (\Zbl{506.05040}).
\annot{} They prove the special case in which $\Beta$ is all negative of the following generalization, which is an immediate consequence of their result.  [\emph{Theorem}.  If $\Beta$ is a bidirected graph such that for each vertex $v$ there is a block of $\Beta$ in which $v$ is neither a source nor a sink, then $\Beta$ contains a coherent circle.  (``Coherent'' means that at each vertex, one edge is directed inward and the other outward.)]
\codes{\Cpar:~\Cori}%
\itemspace

\author{Jerrold W.\ Grossman, Devadatta M.\ Kulkarni, \& Irwin E.\ Schochetman}
\authorspace
\bsgitem{1994a}{}  Algebraic graph theory without orientation.  \emph{Linear Algebra Appl.}\ 212/213 (1994), 289--307.
\MR{1306983} (96b:05111).  \Zbl{817.05047}.						
\annot{} Topics:  The unoriented incidence matrix of $\Gamma$ [i.e., the incidence matrix $\incid(-\Gamma)$], the Laplacian matrix $\Lap(-\Gamma)$, the even-cycle (``even circuit'') matroid $\bfF(-\Gamma)$, a partial all-minors matrix-tree theorem [completed in \xref{Bapat, Grossman, and Kulkarni}{1999a}{Ravindra B.\ Bapat, Jerrold W.\ Grossman, \& Devadatta M.\ Kulkarni}].  [This part is not new.  See \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen} for rank($\incid(-\Gamma)$); \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, \S8 for both matrices; \xref{Tutte}{1981a}{W.T.\ Tutte}, \xref{Doob}{1973a}{Michael Doob}, and \xref{Sim\~oes-Pereira}{1973a}{J.M.S.\ Simoes-Pereira} for the matroid; \xref{Chaiken}{1982a}{Seth Chaiken} for the whole matrix-tree theorem.]
\annotctd{} \S\S4, 5:  Vector spaces associated with $\bfF(-\Gamma)$ and its dual, expressed both combinatorially in terms of vectors associated with matroid circuits and cocircuits (of two kinds) and as null and row spaces of $\incid(-\Gamma)$ and $\incid(-\Gamma)\trans$.  E.g., in \S5 is the all-negative case of:  A basis for $\Nul \incid(\Sigma)\trans$ consists of one switching function positivizing each balanced component of $\Sigma$.  [The viewpoint, going from matroids to vector spaces over fields, usually with characteristic $\neq 2$, contrasts sharply with that of \xref{Tutte}{1981a}{W.T.\ Tutte}, who starts with integral chain groups ($\bbZ$-modules) and ends with chain-group properties and matroids.  This is the only thorough development I know of vector spaces of a signed graph before \xref{Chen and Wang}{2009a}{Beifang Chen \& Jue Wang}, despite some aspects' having appeared e.g.\ in \xref{Bolker}{1977a}{Ethan D.\ Bolker}, \xrefd{1979a}{Ethan D.\ Bolker}, and \xref{Tutte}{1981a}{W.T.\ Tutte}.  It will be still more valuable if it is extended to $\bbR^\times$-gain graphs and to $F^\times$-gain graphs for any field $F$.]
\annotctd{} Dictionary:  $M = \incid(-\Gamma)$; ``$k$-reduced spanning substructure'' $\cong$ independent set of rank $n-k$ in $\bfF(-\Gamma)$;  ``quasi edge cut'' = balancing set;  ``quasibond'' = minimal balancing set;  ``even circuit'' = positive closed walk;  ``bowtie'' = contrabalanced handcuff; ``marimba stick'' = half edge.
\par
\codes{\CECyc,~\Cpar:~\CInc,~\CBal,~\CDual}%
\itemspace

\bsgitem{1995a}{}  On the minors of an incidence matrix and its Smith normal form.  \emph{Linear Algebra Appl.}\ 218 (1995), 213--224.
\MR{1324059} (95m:15020).  \Zbl{819.05043}.					
\annot{} Rank of unoriented incidence matrix of $\Gamma$ (i.e., $\incid(-\Gamma)$) [as in \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}].  Thm.\ 2.2: the nonzero minors of $\incid(-\Gamma)$ are $\pm2^k$ for all $0\leq k \leq \tau_o(\Gamma)$, where $\tau_o :=$ max \# disjoint odd circles.  [Improves \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Lemma 8A.2, for $\Sigma=-\Gamma$.  The latter implies that the nonzero minors of $\incid(\Sigma)$ are $\pm2^k$ for all $0\leq k \leq l_0(\Sigma)$, $l_0 :=$ frustration number = max \# disjoint negative circles.] 
\annotctd{} Consequences are Smith normal form of $\incid(-\Gamma)$ (\S3) and total integrality of some integer programs with $\incid(-\Gamma)$ as coefficient matrix.
\dated{rev.\ 11 Mar 2022}
\codes{\Cpar:~\CIncid,~\Cecyc,~\CGeom}%
\itemspace

\author{Martin Gr\"otschel} 						%GROTSCHEL
\authorx{Martin Grotschel}
\authorspace
\authref  See also \hyperlink{Francisco Barahona}{F.~Barahona}.
\authrefspace

\author{M.\ Gr\"otschel, M.\ J\"unger, \& G.\ Reinelt}
\authorx{M.\ Grotschel, M.\ Junger, \& G.\ Reinelt}
\authorspace
\bsgitem{1987a}{}  Calculating exact ground states of spin glasses:  a polyhedral approach.  
In:  J.L.~van Hemmen and I.~Morgenstern, eds., \emph{Heidelberg Colloquium on Glassy Dynamics} (Proc., 1986), pp.\ 325--353.  Lect.\ Notes in Physics, Vol.\ 275.  Springer-Verlag, Berlin, 1987.
\MR{0916885} (no rev).  
\annot{} \S2, ``The spin glass model'': finding the weighted frustration index in a weighted signed graph $(\Sigma, w)$, or finding a ground state in the corresponding Ising model, is equivalent to the weighted max-cut problem in $(-\Sigma, w)$.  This article concerns finding the exact weighted frustration index.  \S3, ``Complexity'', describes previous results on NP-completeness and polynomial-time solvability.  \S4, ``Exact methods'', discusses previous solution methods.  \S5, ``Polyhedral combinatorics'', shows that finding weighted frustration index is a linear program on the cut polytope; also expounds related work.  The remainder of the paper concerns a specific cutting-plane method suggested by the polyhedral combinatorics.
\par
\codes{\Csg:~\Cfr(\Cgen),~\CState:~\CAlgor,~\CGeom,~\CRef}\codesctd{\CPhys,~\CRef:~\CExp}%
\itemspace

\author{Martin Gr\"otschel, L\'aszl\'o Lov\'asz, \& Alexander Schrijver}
\authorx{Martin Grotschel, Laszlo Lovasz, \& Alexander Schrijver}
\authorspace
\bsgitem{1988a}{}  \emph{Geometric Algorithms and Combinatorial Optimization}.  Algorithms and Combin., Vol.\ 2.  Springer-Verlag, Berlin, 1988.
\MR{0936633} (89m:90135).  \Zbl{634.05001}.
\annot{} Ch.\ 8, ``Combinatorial optimization: A tour d'horizon'':  Topics mentioned include odd circles, maximum-gain flow, odd cuts.
\par
\codes{\Cpar,~\Cgg:~\CCircles,~\CAlgor}%
\itemspace

\bsgitem{1993a}{}  \emph{Geometric Algorithms and Combinatorial Optimization}.  Second corrected ed.  Algorithms and Combin., Vol.\ 2.  Springer-Verlag, Berlin, 1993.
\MR{1261419} (95e:90001).  \Zbl{837.05001}.
\annot{} Essentially the same as \xrefd{1988a}{Martin Grotschel, Laszlo Lovasz, \& Alexander Schrijver}.
\codes{\Cpar,~\Cgg:~\CCircles,~\CAlgor}%
\itemspace

\author{M.\ Gr\"otschel \& W.R.\ Pulleyblank}
\authorx{M.\ Grotschel \& W.R.\ Pulleyblank}
\authorspace
\bsgitem{1981a}{}  Weakly bipartite graphs and the max-cut problem.  \emph{Operations Res.\ Lett.}\ 1 (1981/82), 23--27.  
\MR{0643056} (83e:05048).  \Zbl{478.05039}, \Zbl{494.90078}.				
\annot{} Includes a polynomial-time algorithm, which they attribute to ``Waterloo folklore'', for shortest (more generally, min-weight) even or odd path, hence (in an obvious way) odd or even circle.
[Attributed by \xref{Thomassen}{1985a}{Carsten Thomassen} to Edmonds (unpublished).  Adapts to signed graphs by the negative subdivision trick:  Subdivide each positive edge of $\Sigma$ into two negative edges, each with half the weight.  The min-weight algorithm applied to the subdivision finds a min-weight (e.g., a shortest) negative circle of $\Sigma$.]
[This paper is very easy to understand.  It is one of the best written I know.]
[Weakly bipartite graphs are certain signed graphs.  Further work: \xref{Barahona, Gr\"otschel, and Mahjoub}{1985a}{Francisco Barahona, Martin Grotschel, \& Ali Ridha Mahjoub}, \xref{Poljak and Tuza}{1995a}{Svatopluk Poljak \& Zsolt Tuza}, and esp.\ \xref{Guenin}{1998a}{Bertrand Guenin}, \xrefd{2001a}{Bertrand Guenin}.]
\par
\codes{\Cpar:~\CAlgor,~\CGeom,~\CPaths,~\CCircles}\codesctd{\Csg:~\CGeom}%
\itemspace

\authorx{D.A.\ Grundy}						%GRUNDY
\author{D.A.\ Grundy, D.D.\ Olesky, \& P.\ van den Driessche}
\authorspace
\bsgitem{2012a}{}  Constructions for potentially stable sign patterns.  \emph{Linear Algebra Appl.}\ 436 (2012), 4473--4488.
\MR{2917424}.  \Zbl{1245.15030}.  
\codes{\CQM:~\CSD}%
\itemspace

\author{Xiangbai Gu}							%GU XB
\authorspace
\authref  See \hyperlink{Di Peng}{D.~Peng}.
\itemspace

\author{Huanhuan Guan}						%GUAN HH
\authorspace
\authref  See \hyperlink{Chao Wen}{C.~Wen}.
\itemspace

\author{Zhi-Hong Guan}						%GUAN ZH
\authorspace
\authref  See \hyperlink{Bin Hu}{B.~Hu}.
\itemspace

\authorx{Victor Guba}						%GUBA
\author{Victor Guba \& Mark Sapir}
\authorspace
\bsgitem{1997a}{}  \emph{Diagram Groups}.  Mem.\ Amer.\ Math.\ Soc., vol.\ 130 (1997), no.\ 620. 
\MR{1396957} (98f:20013).  \Zbl{930.20033}.
\annot{} The ``labelled oriented graph'' (pp.\ 12--13) is a gain graph with a gain semigroup (instead of group) which is the semigroup generated by an alphabet and its inverse.
\codes{\Cgg(\CGen)}%
\itemspace

\author{James E.\ Gubernatis}						%GUBERNATIS
\authorspace
\authref  See \hyperlink{Naomichi Hatano}{N.~Hatano}.
\itemspace

\author{Bertrand Guenin}						%GUENIN
\authorspace
\authref  See also \hyperlink{Ahmad Abdi}{A.~Abdi}, \hyperlink{Gerard Cornuejols}{G.~Cornu\'ejols}, \hyperlink{Zouhaier Ferchiou}{Z.~Ferchiou}, \hyperlink{Jim Geelen}{J.F.~Geelen}, and \hyperlink{Cheolwon Heo}{C.~Heo}.
\authrefspace

\bsgitem{1998a}{}  \emph{On Packing and Covering Polyhedra}.  Ph.D.\ dissertation, Grad.\ Sch.\ Industrial Engin., Carnegie--Mellon University, 1998.
\codes{\CSG:~\CGeom}\codesctd{\CS(\CMatrd):~\CGeom}
\itemspace

\bsgitem{1998b}{}  A characterization of weakly bipartite graphs.   
In: Robert E.\ Bixby, E.~Andrew Boyd, and Roger Z.\ R\'{\i}os-Mercado, eds., \emph{Integer Programming and Combinatorial Optimization} (6th Int.\ IPCO Conf., Houston, 1998), pp.\ 9--22.  \LNCS 1412.  Springer, Berlin, 1998.
\MR{1726332} (2000i:05158).  \Zbl{909.90264}.					
\annot{} Extended abstract of \xrefd{2001a}{Bertrand Guenin}.
\codes{\CSG:~\CGeom}
\itemspace

\bsgitem{2001a}{\dag}  A characterization of weakly bipartite graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 83 (2001), 112--168.
\MR{1855799} (2002h:05145).  \Zbl{1030.05103}.					
\annot{} $\Sigma$ is ``weakly bipartite'' (\xref{Gr\"otschel and Pulleyblank}{1981a}{M.\ Grotschel \& W.R.\ Pulleyblank}) if its clutter of negative circles is ideal (i.e., has the ``weak MFMC'' property of \xref{Seymour}{1977a}{P.D.\ Seymour}).  [This is a polyhedral property that can be equivalently stated:  Define a ``negative circle cover'' to be an edge multiset that intersects every negative circle, and a ``weighted negative circle cover'' to be an edge weighting by nonnegative real numbers such that the total weight of each negative circle is at least $1$.  Weak biparticity means that, for every linear functional $f: E\to\bbR$, the minimum value over all weighted negative circle covers is attained by a negative circle cover.]
Thm.:  $\Sigma$ is weakly bipartite iff it has no $-K_5$ minor.  This proves part of \xref{Seymour's}{1981a}{P.D.\ Seymour} conjecture (see \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}).  [Short proof:  \xref{Schrijver}{2002a}{Alexander Schrijver}.]
Dictionary: ``odd'' = negative, ``even'' = positive.
\par
\codes{\CSG:~\CGeom,~\CCircles,~\CStr}%
\itemspace

\bsgitem{2001b}{}  Integral polyhedra related to even cycle and even cut matroids.  
In:  Karen Aardal and Bert Gerards, eds., \emph{Integer Programming and Combinatorial Optimization} (8th Int.\ IPCO Conf., Utrecht, 2001).  \LNCS 2081, 196--209.  Springer, Berlin, 2001.
\MR{1939172} (2003j:90090).  \Zbl{1010.90088}.					
\annot{} See \xrefd{2002a}{Bertrand Guenin}.
\codes{\Csg:~\CPar:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{2002a}{}  Integral polyhedra related to even-cycle and even-cut matroids.  \emph{Math.\ Operations Res.}\ 27 (2002), no.\ 4, 693--710.
\MR{1939172} (2003j:90090).  \Zbl{1082.90584}.					
\annot{} In $\Sigma$ distinguish a negative link $e_{st}$.  An ``unbalanced port'' is $C \setminus e_{st}$ where $C$ is an unbalanced circuit of $\bfL(\Sigma)$ that contains $e_{st}$.  Replace ''negative circle'' by ''negative port'' in the definition of \xrefd{2001a}{Bertrand Guenin}.  Thm.:  The minimum value over all weighted unbalanced port covers is attained by an unbalanced port cover, iff $\Sigma$ has no $-K_5$ minor and $\bfL(\Sigma)$ has no $F_7^*$ minor.  
[The latter can be replaced by:  $\Sigma$ has no $(\pm C_4 \setminus \text{edge})$ minor, by \xref{Zaslavsky}{1990a}{Thomas Zaslavsky}.]
Dictionary:  ``even-cycle matroid'' = lift matroid $\bfL(\Sigma)$, not the even-cycle matroid $\bfF(-\Gamma)$ in \xref{W.T.\ Tutte}{1981a}{W.T.\ Tutte}, \xref{M.\ Doob}{1973a}{Michael Doob}; ``odd $st$-walk'' = unbalanced port.
\codes{\CSG:~\CGeom,~\CMatrd,~\CStr}%
\itemspace

\bsgitem{2002b}{}  A short proof of Seymour's characterization of the matroids with the max-flow min-cut property.  \emph{J\ Combin.\ Theory Ser.\ B} 86 (2002), no.\ 2, 273--279.
\MR{1933463}.  \Zbl{1023.05029}.  
\annot{} The excluded minor is $\bfL(-K_4)$.
\codes{\CSG,~\CSgnd(\CMatrd):~\CFlows}%
\itemspace

\bsgitem{2002c}{}  A short proof of Seymour's characterization of the matroids with the max-flow min-cut property.  In:  William J.\ Cook and Andreas S.\ Schulz, eds., \emph{Integer Programming and Combinatorial Optimization} (9th Int.\ IPCO, Cambridge, Mass., 2002), pp.\ 1--12.  \LNCS 2337.  Springer, Berlin, 2002.
\MR{2061054}.  \Zbl{1049.90529}.  
\annot{} See \xrefd{2002b}{Bertrand Guenin}.
\codes{\CSG,~\CSgnd(\CMatrd):~\CFlows}%
\itemspace

\bsgitem{2016a}{}  A survey on flows in graphs and matroids.  \emph{Discrete Appl.\ Math.}\ 209 (2016), 122--132.
\MR{3510436}.  \Zbl{1339.05165}.  
\codes{\CSG:~\CFlows,~\CStr}%
\itemspace

\author{Bertrand Guenin, Cheolwon Heo, \& Irene Pivotto}
\authorspace
\bsgitem{20xxa}{}  Signed graphs with the same even cycles.  Preprint.  \emph{}\ 
\codes{\CSG:~\CMatrd,~\CDual,~\CSw}%
\itemspace

\author{Bertrand Guenin, Irene Pivotto, \& Paul Wollan}
\authorspace
\bsgitem{2011a}{}  Isomorphism for even cycle matroids - I.  Manuscript, 2011.  
\arXiv{1109.2978}.  
\annotctd{} The lift matroids $\bfL(\Sigma)$ of signed graphs (not \xref{Doob's}{1973a}{Michael Doob} even-cycle matroids $\bfF(-\Gamma)$).  To appear, augmented, as several papers.  Dictionary:  See \xrefd{2016b}{Bertrand Guenin, Irene Pivotto, \& Paul Wollan}.
\dated{7 Aug 2019} 
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{2013a}{}  Relationships between pairs of representations of signed binary matroids.  \emph{SIAM J.\ Discrete Math.}\ 27 (2013), no.\ 1, 329--341.
\MR{3032922}.  \Zbl{1268.05042}.  
\codes{\CS(\CMatrd)}%
\itemspace

\bsgitem{2016a}{}  Displaying blocking pairs in signed graphs.  \emph{European J.\ Combin.}\ 51 (2016), 135--164.
\MR{3398845}.  \Zbl{1321.05102}.  
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{2016b}{}  Stabilizer theorems for even cycle matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 118 (2016), 44--75.
\MR{3471844}.  \Zbl{1332.05030}.  
\annotctd{} Dictionary:  ``Even'' = positive, ``odd'' = negative, ``cycle'' = binary cycle = even subgraph, ``even cycle matroid'' = lift matroid $\bfL(\Sigma)$, not the even-cycle matroid defined in \xref{Doob}{1973a}{Michael Doob} (\cf\ \xref{Tutte}{1981a}{W.T.\ Tutte}).
\par
\codes{\Csg:~\CMatrd:~\CStr}%
\itemspace

\author{Bertrand Guenin \& Leanne Stuive}
\authorspace
\bsgitem{2013a}{}  Single commodity-flow algorithms for lifts of graphic and cographic matroids.  In:  Michel Goemans and Jos\'e Correa, eds., \emph{Integer Programming and Combinatorial Optimization} (16th Int.\ IPCO, Valpara\'{\i}so, Chile, 2013), pp.\ 193--204.  \LNCS 7801.  Springer, Heidelberg, 2013.
\MR{3085546}.  \Zbl{1372.05030}.  
\annot{} Extended abstract of \xrefd{2016a}{Bertrand Guenin \& Leanne Stuive}.
\codes{\Cgg,~\CDual:~\CFlows}%
\itemspace

\bsgitem{2016a}{}  Single commodity-flow algorithms for lifts of graphic and cographic matroids.  \emph{SIAM J.\ Discrete Math.}\ 30 (2016), no.\ 3, 1775--1797.
\MR{3544848}.  \Zbl{1344.05040}.  
\codes{\Cgg,~\CDual:~\CFlows}%
\itemspace

\author{Sylvain Guillemot}						%GUILLEMOT
\authorspace
\bsgitem{2008a}{}  FPT algorithms for path-transversals and cycle-transversals problems in graphs.  In:  Martin Grohe and Rolf Niedermeier, eds., \emph{Parameterized and Exact Computation} (Proc.\ 3rd Int.\ Workshop, IWPEC 2008, Victoria, B.C., 2008), pp.\ 129-140.  \LNCS 5018.  Springer, Berlin, 2008.
\MR{2503559} (2010e:68087).  \Zbl{1142.68599}.  
\codes{\Csg,~\CGG:~\Cfr:~\CAlgor}%
\itemspace

\bsgitem{2011a}{}  FPT algorithms for path-transversal and cycle-transversal problems.  \emph{Discrete Optimization}\ 8 (2011), 61--71.
\MR{2772561} (2012h:68112).  \Zbl{1248.90072}.  
\codes{\Csg,~\CGG:~\Cfr:~\CAlgor}%
\itemspace

\authorx{Basak Guler}						%GULER
\author{Basak Guler, Burak Varan, Kaya Tutuncuoglu, Mohamed Nafea, Ahmed A.\ Zewail, Aylin Yener, \& Damien Octeau}
\authorspace
\bsgitem{2014a}{}  Optimal strategies for targeted influence in signed networks.  In:  \emph{2014 IEEE--ACM International Conference on Advances in Social Networks Analysis and Mining} (ASONAM 2014, Beijing), pp.\ 906--911.  IEEE Conf.\ Publications, 2014.
\annot{} Extending ideas of \xref{Li, Chen, Wang, \& Zhang}{2013a}{Yanhua Li, Wei Chen, Yajun Wang, \& Zhi-Li Zhang}.
\codes{\CSG:~\CPsS}%
\itemspace

\authorx{N.\ G\"ulpinar}						%GULPINAR
\authorx{N.\ Gulpinar}
\author{N.\ G\"ulpinar, G.\ Gutin, G.\ Mitra, \& A.\ Zverovitch}
\authorx{N.\ Gulpinar, G.\ Gutin, G.\ Mitra, \& A.\ Zverovitch}
\authorspace
\bsgitem{2004a}{}  Extracting pure network submatrices in linear programs using signed graphs.  \emph{Discrete Appl.\ Math.}\ 137 (2004), no.\ 3, 359--372.
\MR{2036638} (2004k:90145).  \Zbl{1095.90112}.					
\annot{} Problem:  Finding a largest embedded network matrix (up to ``reflection'' = row negation).  Given a $0,\pm1$-matrix $AS$, let $\Sigma$ have for vertices the rows of $A$, with an edge $\epsilon e_{ij}$ iff $\sgn(a_ {ik}a_{jk}) = -\epsilon$ in the $k$-th column for some $k$.  Let $\alpha :=$ maximum size of a stable set in a graph.  
Thm.:  The maximum height of a reflected network submatrix of $A$ equals $\max_X \alpha((\Sigma^X)^-)$ over all switchings $\Sigma^X$.  This implies a heuristic algorithm for finding a large embedded network matrix.
\dated{30 Sept 2009} 
\codes{\CSG:~\Cinc:~\CBal,~\CAlgor}%
\itemspace

\author{Mahadevappa M.\ Gundloor}					%GUNDLOOR
\authorspace
\authref  See \hyperlink{Harishchandra S.\ Ramane}{H.S.~Ramane}.
\itemspace

\author{Hangtian Guo}						%GUO HT
\authorspace
\authref  See also \hyperlink{Huiqiu Lin}{H.Q.~Lin}.
\authrefspace

\author{Hangtian Guo, Huiqiu Lin, \& Yanhua Zhao}
\authorspace
\bsgitem{2021a}{}  A spectral condition for the existence of a pentagon in non-bipartite graphs.  \emph{Linear Algebra Appl.}\ 627 (2021), 140--149.
\MR{4278142}.  \Zbl{1468.05153}.  
\annot{} [\Problem.  Generalize to negative pentagons in unbalanced signed graphs.]
\dated{6 Jul 2022} 
\codes{\Cpar:~\CAdj:~\CEig:~\Cbal}%
\itemspace

\authorx{Heng Guo}							%GUO H
\author{Heng Guo \& Mark Jerrum}
\authorspace
\bsgitem{2021a}{}  Approximately counting bases of bicircular matroids.  \emph{Combin.\ Probab.\ Comput.}\ 30 (2021), no.\ 1, 124--135.
\MR{4205662}.  
\arXiv{1808.09548}.  
\codes{\Cgg:~\CMatrd:~\CBic:~\CAlgor}%
\itemspace

\author{Ji Ming Guo}							%GUO JM
\authorspace
\authref  See also \hyperlink{Lin Feng}{L.~Feng}, \hyperlink{Jianxi Li}{J.X.~Li}, \hyperlink{Shang Wang Tan}{S.W.~Tan}, \hyperlink{Xiao Li Wu}{X.L.~Wu}, and \hyperlink{Jingming Zhang}{J.M.~Zhang}.
\authrefspace

\author{Ji-Ming Guo, Jianxi Li, \& Wai Chee Shiu}
\authorspace
\bsgitem{2013a}{}  On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph.  \emph{Czechoslovak Math.\ J.}\ 63 (2013), no.\ 3, 701--720.
\MR{3125650}.  \Zbl{1299.05216}.  
\annot{} Operations on the characteristic polynomial of $\Lap(-\Gamma)$ are applied to construct cospectral graphs.
\dated{23 Nov 2014} 
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\authorx{Jiong Guo}							%GUO JO
\author{Jiong Guo, Hannes Moser, \& Rolf Niedermeier}
\authorspace
\bsgitem{2009a}{}  Iterative compression for exactly solving NP-hard minimization problems.   
In:  J.\ Lerner, D.\ Wagner, and K.A.\ Zweig, eds., \emph{Algorithmics of Large and Complex Networks: Design, Analysis, and Simulation}, pp.\ 65--80.  \LNCS 5515.  Springer, Berlin, 2009.
\Zbl{1248.68380}.  
\annot{} Iterative compression results in vast speed-up for, e.g., Graph Bipartization and Signed Graph Balancing (\S3.1).  \Cf\ \xref{H\"uffner, Betzler, and Niedermeier}{2007a}{Falk Huffner, Nadja Betzler, \& Rolf Niedermeier}.
\dated{6 Feb 2011} 
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\author{Krystal Guo}							%GUO K
\authorspace
\bsgitem{2015a}{}  \emph{Simple Eigenvalues of Graphs and Digraphs}.  Doctoral thesis, Simon Fraser University, Burnaby, British Columbia, 2015.
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Krystal Guo \& Bojan Mohar}
\authorspace
\bsgitem{2017a}{}  Hermitian adjacency matrix of digraphs and mixed graphs.  \emph{J.\ Graph Theory}\ 85 (2017), no.\ 1, 217--248.
\MR{3634484}.  \Zbl{1365.05173}.  \arXiv{1505.01321}.				
\annot{} ``Digraph'' = ``mixed graph'' = $\{1,\pm\mathrm{i}\}$-gain graph $\Phi$.  ``Hermitian adjacency matrix'' = $A(\Phi)$.  Studies eigenvalue properties in terms of gain-graph properties.  \S8, ``Cospectrality'':  Reversing arcs in a cut without undirected edges, replacing undirected edges in a cut without arcs by arcs in the same direction, and ``four-way switching'' retain the spectrum (because they are gain switching operations).  So does reversing all arcs (it is conjugation).  
\Cf\ \xref{Liu and Li}{2015a}{Jianxi Liu \& Xueliang Li} and \xref{Mohar}{2016a}{Bojan Mohar}.
\dated{4 Feb 2018, 23 Dec 2020} 
\codes{\Cgg:~\Csw,~\CAdj:~\CEig}%
\itemspace

\bsgitem{2017b}{}  Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths.  \emph{Discrete Math.}\ 340 (2017), no.\ 11, 2616--2632.
\MR{3689910}.  \Zbl{1369.05092}.  
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\authorx{Qiao Guo}							%GUO Q
\author{Qiao Guo, Yaoping Hou, \& Deqiong Li}
\authorspace
\bsgitem{2020a}{}  The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $K$ pendant vertices.  \emph{Electronic J.\ Linear Algebra} 36 (2020), 390--399.
\MR{4118164}. \Zbl{1444.05091}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace
 
\author{Qiumin Guo}							%GUO QM
\authorspace
\authref  See also \hyperlink{Weili Guo}{W.L.~Guo}.
\authrefspace

\author{Qiumin Guo, Weili Guo, Wentao Hu, \& Guangfeng Jiang}
\authorspace
\bsgitem{2017a}{}  The global invariant of signed graphic hyperplane arrangements.  \emph{Graphs Combin.}\ 33 (2017), no.\ 3, 527--535.
\MR{3654136}.  \Zbl{1370.52070}.  
\annot{} For the complex arrangement $\cH[\Sigma]$, the third quotient of the lower central series of $\pi(\bbC^n\setm\bigcup\cH[\Sigma]$ has a combinatorial interpretation in terms of $\Sigma$.
\dated{1 Nov 2014} 
\codes{\CSG:~\CGeom,~\CAlgeb}%
\itemspace

\author{Shuguang Guo}						%GUO SG
\authorspace
\authref  See \hyperlink{Guanglong Yu}{G.L.~Yu}.
\itemspace

\author{Weili Guo}							%GUO WL
\authorspace
\authref  See also \hyperlink{Qiumin Guo}{Q.-M.~Guo}.
\authrefspace

\author{Weili Guo, Qiumin Guo, \& Guangfeng Jiang}
\authorspace
\bsgitem{2018a}{}  Falk invariants of signed graphic arrangements.  \emph{Graphs Combin.}\ 34 (2018), no.\ 6, 1247--1258.
\MR{3881264}.  \Zbl{1406.52041}.  
\codes{\CSG:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\author{Weili Guo \& Michele Torielli}
\authorspace
\bsgitem{2018a}{}   On the Falk invariant of signed graphic arrangements.  \emph{Graphs Combin.}\ 34 (2018), no.\ 3, 466--488.
\MR{3784929}.  \Zbl{1392.52012}.  \arXiv{1703.09402}.				
\par
\codes{\CSG:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\bsgitem{2020a}{}   On the Falk invariant of hyperplane arrangements attached to gain graphs.  \emph{Australasian J.\ Combin.}\ 77 (2020), no.\ 3, 301--317.
\MR{4224497}.  \Zbl{1444.05149}.  \arXiv{1707.08449}.				
\codes{\CGG:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\bsgitem{2021a}{}  On the Falk invariant of Shi and Linial arrangements.  \emph{Discrete Comput.\ Geom.}\ 66 (2021), no.\ 2, 751--768. 
\MR{4292763}.  \Zbl{1471.52021}.  \arXiv{2101.04517}.				
\par
\codes{\CGG:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\author{Yihao Guo}							%GUO YH
\authorspace
\authref  See \hyperlink{Min Zhu}{M.~Zhu}.
\itemspace

\author{Anika Gupta}							%GUPTA A
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{G.\ Gupta}							%GUPTA G
\authorspace
\authref  See \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{Neha Gupta}							%GUPTA N
\authorspace
\authref  See \hyperlink{Moses Charikar}{M.~Charikar} and \hyperlink{Vaggos Chatziafratis}{V.~Chatziafratis}.
\itemspace

\author{Razvan Gurau}						%GURAU
\authorspace
\bsgitem{2010a}{}  Topological graph polynomials in colored group field theory.  \emph{Ann.\ Henri Poin\-car\'e}\ 11 (2010), 565--584.
\MR{2677739} (2011k:81098).  \Zbl{1208.81153}.  \arXiv{0911.1945}.
\arXiv{0911.1945}.							
\codes{\Csg:~\CTop:~\CInvar}%
\itemspace

\author{Venkatesan Guruswami}					%GURUSWAMI
\authorspace
\authref  See \hyperlink{Moses Charikar}{M.~Charikar} and \hyperlink{Ioannis Giotis}{I.~Giotis}.
\itemspace

\author{Vladimir A.\ Gurvich}						%GURVICH
\authorspace
\authref  See \hyperlink{Endre Boros}{E.~Boros}.
\itemspace

\author{Karen Guthrie}						%GUTHRIE
\authorspace
\authref  See \hyperlink{Alvaro Carbonero}{A.~Carbonero}.
\itemspace

\author{Gregory Gutin}						%GUTIN
\authorspace
\authref  See also \hyperlink{Jorgen Bang-Jensen}{J.~Bang-Jensen}, \hyperlink{R.\ Crowston, G.\ Gutin, M.\ Jones, \& G.\ Muciaccia}{R.~Crowston}, and \hyperlink{N.\ Gulpinar}{N.~G\"ulpinar}.
\authrefspace

\author{G.\ Gutin \& D.\ Karapetyan}
\authorspace
\bsgitem{2009a}{}  A selection of useful theoretical tools for the design and analysis of optimization heuristics.  \emph{Memetic Computing} 1 (2009), 25--34.
\annot{} \S2.1, ``Preprocessing in linear programming'':  Exposition of \xref{G\"ulpinar, Gutin, Mitra, and Zverovitch}{2004a}{N.\ Gulpinar, G.\ Gutin, G.\ Mitra, \& A.\ Zverovitch}.
\dated{30 Sept 2009} 
\par
\codes{\CSG:~\Cinc,~\CBal,~\CAlgor:~\CExp}%
\itemspace

\author{Gregory Gutin, Daniel Karapetyan, \& Igor Razgon}
\authorspace
\bsgitem{2009a}{}  Fixed-parameter algorithms in analysis of heuristics for extracting networks in linear programs.  
In: J.\ Chen and F.V.\ Fomin, eds., \emph{Parameterized and Exact Computation} (4th Int.\ Workshop, IWPEC 2009, Copenhagen), pp.\ 222--233.  \LNCS 5917.  Springer, Berlin, 2009.
\MR{2773945} (no rev).  \Zbl{1273.68175}.  \arXiv{0906.1359}.				
\annot{} Algorithmics of finding a balanced signed-graph incidence matrix (``reflected network matrix'') in a given matrix.
\dated{26 Dec 2012} 
\par
\codes{\Csg:~\CBal:~\CSw,~\CAlgor}%
\itemspace

\author{Gregory Gutin, Benjamin Sudakov, \& Anders Yeo}
\authorspace
\bsgitem{1998a}{}  Note on alternating directed cycles.  \emph{Discrete Math.}\ 191 (1998), 101--107.
\MR{1644876} (99d:05050).  \Zbl{956.05060}.					
\annot{} Existence of a coherent circle with alternating colors in a digraph with an edge $2$-coloring is NP-complete.  However, if the minimum in- and out-degrees of both colors are sufficiently large, such a cycle exists.  
[This problem generalizes the undirected, edge-$2$-colored alternating-circle problem, which is a special case of the existence of a bidirected coherent circle---see \xref{Bang-Jensen and Gutin}{1997a}{Jorgen Bang-Jensen \& Gregory Gutin}.  \Question.  Is this alternating cycle problem also signed-graphic?]
\codes{\Cpar:~\Cori:~\CCircles:~\CGen}%
\itemspace

\author{Gregory Gutin \& Alexei Zverovitch}
\authorspace
\bsgitem{2003a}{}  Extracting pure network sub-matrices in linear programs using signed graphs, part II.  \emph{Commun.\ Dependability Quality Management} 5 (2003), no.\ 1, 58--65.
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Ivan Gutman} 						%GUTMAN
\authorspace
\authref  See also \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~de~Abreu}, \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}, \hyperlink{Ante Graovac}{A.~Graovac}, \hyperlink{Shyi-Long Lee}{S.-L.~Lee}, and \hyperlink{Harishchandra S.\ Ramane}{H.S.~Ramane}.
\authrefspace

\bsgitem{1978a}{}  Electronic properties of M\"obius systems.  \emph{Z.\ Naturforsch.}\ 33a (1978), 214--216.  
\MR{0489372} (58 \#8800).  
\codes{\CSG:~\CAdj,~\CEig,~\CChem}%
\itemspace

\bsgitem{1988a}{}  Topological analysis of eigenvalues of the adjacency matrices in graph theory:  A difficulty with the concept of internal connectivity.  \emph{Chem.\ Phys.\ Lett.}\ 148 (1988), 93--94.
\annot{} Points out an ambiguity in the definitions of \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu} in the case of multiple eigenvalues.  [See \xref{Lee and Gutman}{1989a}{Shyi-Long Lee \& Ivan Gutman} for the repair.]
\codes{\CVS,~\CSGw}%
\itemspace

\author{Ivan Gutman, Dariush Kiani, Maryam Mirzakhah, \& Bo Zhou}
\authorspace
\bsgitem{2009a}{}  On incidence energy of a graph.  \emph{Linear Algebra Appl.}\ 431 (2009), no.\ 8, 1223--1233.
\MR{2547906} (2010k:05174).  \Zbl{1175.05084}.					
\codes{\Cpar:~\CInc~\CEig}%
\itemspace

\author{Ivan Gutman, Shyi-Long Lee, Yeung-Long Luo, \& Yeong-Nan Yeh}
\authorspace
\bsgitem{1994a}{}  Net signs of molecular graphs:  dependence of molecular structure.  \emph{Int.\ J.\ Quantum Chem.}\ 49 (1994), 87--95.  
\annot{} How to compute the balanced signing of $\Gamma$ that corresponds to eigenvalue $\lambda_i$ (see \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}), without computing the eigenvector $X_i$.  Theorem:  If $v_rv_s \in E$, then $X_{ir}X_{is} = \sum_P f(P; \lambda_i)$, where $f(P; \lambda) := \phi(G-V(P); \lambda)/\phi'(G; \lambda)$, $\phi(G; \lambda)$ is the characteristic polynomial, and the sum is over all paths connecting $v_r$ and $v_s$.  Hence $\sigma_i(v_rv_s) = \sgn(X_{ir}X_{is})$ is determined.  [An interesting theorem.  \Questions.  Does it generalize if one replaces $\Gamma$ by a signed graph, this being the balanced (all-positive) case?  In such a generalization, if any, how will $\sigma$ enter in---by restricting the sum to positive paths, perhaps?  What about graphs with real gains, or weights?]
\codes{\CVS,~\CSGw}%
\itemspace

\author{Ivan Gutman, Shyi-Long Lee, Jeng-Horng Sheu, \& Chiuping Li}
\authorspace
\bsgitem{1995a}{}  Predicting the nodal properties of molecular orbitals by means of signed graphs.  \emph{Bull.\ Inst.\ Chem., Academica Sinica} No.\ 42 (1995), 25--31.
\annot{} Points out some difficulties with the method of \xref{Lee and Li}{1994a}{Shyi-Long Lee \& Chiuping Li}.  ``Net sign'' of $\tau: V \to \{\pm1\}$ means $\sum_v \tau(v)$ [= $\#V-2$(cut size)].  
\par
\codes{\CVS,~\CChem}%
\itemspace

\author{Ivan Gutman, Shyi-Long Lee, \& Yeong-Nan Yeh}
\authorspace
\bsgitem{1992a}{}  Net signs and eigenvalues of molecular graphs: some analogies.  \emph{Chem.\ Phys.\ Lett.}\ 191 (1992), 87--91.  
\annot{} A connected graph $\Gamma$ has $n$ eigenvalues and $n$ corresponding balanced signings (see \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}).  Let $S_1 \geq S_2 \geq \cdots \geq S_n$ be the net signs of these signings and $m = \#E$.  The net signs satisfy analogs of properties of eigenvalues.  (A) If $\Delta \subset \Gamma$, then $S_1(\Delta) < S_1$.  (B) $S_1 = m \geq S_2 + 2$.  (C, D) For bipartite $\Gamma$, $S_n = -m$.  Otherwise, $S_n \geq -m+2$.  From (B, C, D) we have $\#S_i \leq m-2$ for all $i \neq 1$ and, if $\Gamma$ is bipartite, $i \neq n$.  (E, F) If $\Gamma$ is bipartite, then $S_i = -S_{n+1-i}$ and at least $a-b$ net signs equal $0$, where $a \geq b$ are the numbers of vertices in the two color classes.  The analogy is imperfect, since $S_1 + S_2 + \cdots + S_n \geq 0$, while equality holds for eigenvalues.  [\Questions.  Some of these conclusions require $\Gamma$ to be bipartite.  Does that mean that they will generalize to an arbitrary balanced signed graph $\Sigma$ in place of the bipartite $\Gamma$, the eigenvectors being those of $\Sigma$?  Will the other results generalize with $\Gamma$ replaced by any signed graph?  How about real gains, or weights?]
\codes{\CVS,~\CSGw}%
\itemspace

\author{Ivan Gutman, Mar{\'\i}a Robbiano, Enide Andrade Martins, Domingos M.\ Cardoso, Luis Medina, \& Oscar Rojo}
\authorx{Ivan Gutman, Maria Robbiano, Enide Andrade Martins, Domingos M.\ Cardoso, Luis Medina, \& Oscar Rojo}
\authorspace
\bsgitem{2010a}{}  Energy of line graphs.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 7, 1312--1323.
\MR{2680258} (2012a:05188).  \Zbl{1194.05137}.					
\codes{\CPar:~\CEig,~\CInc,~\CLG}%
\itemspace

\author{Ivan Gutman \& Oskar E.\ Polansky}
\authorspace
\bsgitem{1986a}{}  \emph{Mathematical Concepts in Organic Chemistry}.  Springer-Verlag, Berlin, 1986.
\MR{0861119} (87m:92102).  \Zbl{657.92024}.					
\annot{} See pp.\ 54--55 for eigenvalues of adjacency matrices of positive and negative circles.
\dated{4 Nov 2010} 
\codes{\CChem:~\CExp:~\CSG:~\CEig}%
\itemspace

\authorx{Pavol Gvozdjak}						%GVOZDJAK
\author{Pavol Gvozdjak \& Jozef \v{S}ir\'a\v{n}}
\authorx{Pavol Gvozdjak \& Jozef Siran}
\authorspace
\bsgitem{1993a}{}  Regular maps from voltage assignments.  In: Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 441--454.  
Contemp.\ Math., Vol.\ 147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.  
\MR{1224722} (94j:05047).  \Zbl{791.05025}.  
\annot{} \S3,  ``Voltage assignments and derived maps'', defines gain graph and covering graph (and map).  \S4,  ``Lifting map automorphisms'':  A map automorphism lifts iff it preserves the class of identity-gain walks.  
[Initiates method developed in \xref{Nedela and \v{S}koviera}{1997b}{Roman Nedela \& Martin Skoviera}, \xref{Malni\v{c}, Nedela, \& \v{S}koviera}{2000a}{Aleksander Malnic, Roman Nedela, \& Martin Skoviera}, \xrefd{2002a}{Aleksander Malnic, Roman Nedela, \& Martin Skoviera}, \etal]  
Dictionary:  ``voltage'' = gain, ``derived graph'' = gain covering graph, ``map'' = combinatorial definition of embedded graph, ``local group'' (at a vertex) = fundamental group (at the vertex).
\dated{11 Jun 2012} 
\codes{\CGG:~\CAut,~\CCov,~\CTop}%
\itemspace

\author{A.\ Gy\'arf\'as}						%GYARFAS
\authorx{A.\ Gyarfas}
\authorspace
\authref  See \hyperlink{Bela Bollobas}{B.~Bollob\'as} and \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\itemspace

\authorx{Eszter Gyimesi}						%GYIMESI
\author{Eszter Gyimesi \& G\'abor Nyul}
\authorx{Eszter Gyimesi \& Gabor Nyul}
\authorspace
\bsgitem{2018a}{}  A comprehensive study of $r$-Dowling polynomials.  \emph{Aequationes Math.}\ 92 (2018), 515--527.
\MR{3805869}.  \Zbl{1420.11048}.  
\annot{} Combinatorial interpretation of the $r$-Dowling numbers, generalizing \xref{Belbachir and Bousbaa}{2013a}{Hacene Belbachir \& Imad Eddine Bousbaa}.  $D_{n,m,r} =$ \# of ways to partition $[n+r]$ into $k$ blocks so $n+1,\ldots,n+r$ are in different blocks, then color non-mininal elements of each block $B \subseteq [n]$ with $m$ colors.
\annotctd{} [Let group $\fG$ have order $m$; use color set $\fG$, least elements are colored $\epsilon$; then $W_m(n,k,1) = W_k(Q_n(\fG)) =$ Whitney number of Dowling lattice (= lattice of frame matroid $\bfF(\fG K_n\full)$).  Proved by \xref{Dowling}{1973b}{Thomas A.\ Dowling}.  \emph{Improvements}:  (1)  $W_m(n,k,r) =$ \# colorings by $\fG$ as above, but each minimal element has color $\epsilon$ (a Dowling-like special case of Gyimesi--Nyul's theorem).  (2)  \Conjecture.  Replacing partitions by permutations and blocks by cycles, $|w_m(n,k,r)| =$ \# colorings as above.  See note at \xref{Belbachir and Bousbaa}{2013a}{Hacene Belbachir \& Imad Eddine Bousbaa} for similar interpretations of $|w_m(n,k,0)|$ and $W_m(n,k,0)$.]
\dated{28 May 2018} 
\annotctd{} [Independently, Conjecture (2) is proved in \xrefd{2019a}{Eszter Gyimesi \& Gabor Nyul}.]
\dated{22 Sept 2018} 
\codes{\Cgg:~\Cmatrd:~\CInvar(\CGen)}%
\itemspace

\bsgitem{2019a}{}  New combinatorial interpretations of $r$-Whitney and $r$-Whitney-Lah numbers.  \emph{Discrete Appl.\ Math.}\ 255 (2019), 222--233.
\MR{3926342}.  \Zbl{1459.11071}.  
\annot{} [``Whitney'' should be ``Dowling'' or ``Dowling--Whitney''; Whitney numbers are more general.]  
\annotctd{} Definitions in terms of permutations [= partitions into circularly ordered blocks] ($r$-Dowling of first kind), partitions ($r$-Dowling of second kind), and partitions into linearly ordered blocks ($r$-Dowling--Lah), all with $r$ distinguished elements appearing in distinct blocks and an $m$-coloring rule for most elements of the other blocks.  [\Cf\ \xrefd{2018a}{Eszter Gyimesi \& Gabor Nyul} for Dowling-style reinterpretations.]  Identities, formulas, with combinatorial proofs.
\annotctd{} [\Cf\ notes on \xrefd{2018a}{Eszter Gyimesi \& Gabor Nyul}.  This $m$-coloring is equivalent to \xref{Dowling's}{1973b}{Thomas A.\ Dowling} homogeneously $\bbZ_m$-labelled partitions (or coloring by any $\fG$ of order $m$), applied also to circularly and linearly ordered blocks and generalized to multiple distinguished elements (Dowling being $r=1$).  Each such type forms a lattice in the natural partial ordering of homogeneously $\fG$-colored partitions \etal\ (ignoring distinguished elements).  Partitions generalize Dowling lattices (geometric lattice); permutations generalize regions of a hyperplane representation (Eulerian lattice; ).  
(3)  \Question.  Is there similar geometry for $r>1$?  (4)  \Question.  With $r=1$, is there a geometry for linearly ordered blocks?  
(5)  \Problem.  Determine the properties of these lattices and how they relate to each other.  E.g., ranked, geometric, Eulerian?  Effect of the natural homomorphisms?]
\dated{22 Sept 2018} 
\codes{\Cgg:~\Cmatrd:~\CInvar(\CGen)}%
\itemspace

\author{Ervin Gy\H{o}ri}						%GYORI
\authorx{Ervin Gyori}
\authorspace
\authref  See also \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\authrefspace

\author{Ervin Gy\"ori, Alexandr V.\ Kostochka, \& Tomasz \L uczak}
\authorx{Ervin Gyori, Alexandr V.\ Kostochka, \& Tomasz Luczak}
\authorspace
\bsgitem{1997a}{}  Graphs without short odd cycles are nearly bipartite.  \emph{Discrete Math.}\ 163 (1997), 279--284.  
\MR{1428582} (97g:05103).  \Zbl{871.05040}.						
\annot{} Given $\Sigma=-\Gamma$ and positive $\rho$, suppose every negative circle has length $\geq n/\rho$.  Then $\Sigma$ has frustration index $\leq 200\rho^2(\ln(10\rho))^2$ (best possible up to a constant factor) and vertex frustration number $\leq 15\rho\ln(10\rho)$ (best possible up to a logarithmic factor).  The proof is based on an interesting, refining lemma.  [\Problem.  Generalize to arbitrary $\Sigma$.]
\codes{\Csg:~\CPar:~\CFr}%
\itemspace


\label{LETTER H} \hypertarget{LETTER H}{}	% LETTER  H


\authorx{M.\ Hachimori}						%HACHIMORI
\author{M.\ Hachimori \& M.\ Nakamura}	
\authorspace
\bsgitem{2007a}{}  A factorization theorem of characteristic polynomials of convex geometries.  \emph{Ann.\ Combin.}\ 11 (2007), 39--46.
\MR{2311929} (2008b:52001).  \Zbl{1110.06006}.					
\annot{} Signed graph coloring is mentioned as an example.
\dated{10 Mar 2011} 
\codes{\CSG:~\CInvar:~\CExp}%
\itemspace

\author{Willem H.\ Haemers}						%HAEMERS
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Andries E.\ Brouwer}{A.E.~Brouwer}, \hyperlink{Michael S.\ Cavers}{M.~Cavers}, \hyperlink{Ebrahim Ghorbani}{E.~Ghorbani}, and \hyperlink{Edwin R.\ van Dam}{E.R.~van Dam}.
\authrefspace

\bsgitem{2012a}{}  Seidel switching and graph energy.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 68 (2012), 653--659.
\MR{3052170}.  \Zbl{1289.05290}.  
\annot{} Graphs whose energy is not altered by switching.  \S3, ``Seidel matrix'':  Energy $\cE(K_\Delta)$ of signed complete graphs $K_\Delta$ (``Seidel energy of $\Delta$'').  Thm.\ 3.1:  $\cE(K_\Delta) \leq n\sqrt{n-1}$, and $=$ iff $\Delta$ is a conference graph.  Thm.\ 3.3:  $\cE(K_\Delta) > \sqrt{2n(n-1)}$ if $n>2$.  Fact:  $\cE(+K_n) = 2(n-1)$.  \Conjecture.  $\min_{\sigma} \cE(K_n,\sigma) = 2(n-1)$.  
[Proved in \xref{Akbari, Einollahzadeh, \etal\!}{2020a}{S.\ Akbari, M.\ Einollahzadeh, M.M.\ Karkhaneei, \& M.A.\ Nematollahi}, previously for $n\leq12$ in \xref{Greaves \etal\!}{2016a}{Gary Greaves, Jacobus H.\ Koolen, Akihiro Munemasa, \& Ferenc Szollosi}.]
\dated{13 Jan 2015, 10 Dec 2020} 
\codes{\CSG(\CKG):~\CSw,~\CAdj:~\CEig}%
\itemspace

\author{W.H.\ Haemers \& G.R.\ Omidi}
\authorspace
\bsgitem{2011a}{}  Universal adjacency matrices with two eigenvalues.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2520--2529.
\MR{2811135} (2012e:05230).  \Zbl{1221.05233}.					
\par
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Willem H.\ Haemers \& Edward Spence}
\authorspace
\bsgitem{2004a}{}  Enumeration of cospectral graphs.  \emph{European J.\ Combin.}\ 25 (2004), 199--211.
\MR{2070541} (2005d:05102).  \Zbl{1033.05070}.					
\annot{} ``Sign-less Laplacian'' $Q(\Gamma) :=$ Laplacian matrix $\Lap(-\Gamma) = D(\Gamma) + A(\Gamma)$.  $\Lap(-\Gamma)$ seems ($n\leq11$) to allow fewer cospectral graphs than does $A(\Gamma)$ or $\Lap(\Gamma)$.
\dated{Sept 2010}
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Willem H.\ Haemers \& Hatice Topcu}
\authorspace
\bsgitem{20xxa}{}  On signed graphs with at most two eigenvalues unequal to $\pm1$.  Submitted.  \emph{}\ 
\arXiv{2109.02522}.  
\annot{} Disconnected, bipartite, and complete underlying graphs.  Examples that are neither balanced nor antibalanced.
\dated{23 Jan 2022} 
\codes{\CSG:~\CAdj:~\CEig,~\CSw}%
\itemspace

\authorx{Sumaira Hafeez}						%HAFEEZ
\author{Sumaira Hafeez \& Rashid Farooq}
\authorspace
\bsgitem{2020a}{}  On energy ordering of vertex-disjoint bicyclic sidigraphs.  \emph{AIMS Math.}\ 5 (2020), no.\ 6, 6693--6713.
\MR{4148974}.  \Zbl{1485.05084}.  \arXiv{2004.04032}.  
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{Sumaira Hafeez, Rashid Farooq, \& Mehtab Khan}
\authorspace
\bsgitem{2019a}{}  Bicyclic signed digraphs with maximal energy.  \emph{Appl.\ Math.\ Comput.}\ 347 (2019), 702--711. 
\MR{3881711} (no rev).  \Zbl{1428.05192}.  
\codes{\CSD,~\CWD:~\CAdj:~\CEig}%
\itemspace

\author{Sumaira Hafeez \& Mehtab Khan}
\authorspace
\bsgitem{2018a}{}  Iota energy of weighted digraphs.  \emph{Trans.\ Combin.}\ 7 (2018), no.\ 3, 55--73.
\MR{3841401}.  \Zbl{1463.05335}.  
\annot{} Iota energy $:= \sum |\Im\lambda_i|$, $\lambda_i$ = adjacency eigenvalues.  Weights from $\bbR^\times$.  Results on cyclic and unicyclic weighted digraphs; $\sgn($weight product) affects the results.  Thm.\ 3.5:  General upper bound.  \S4, ``Equienergetic weighted digraphs'':  Examples.
\dated{3 Jul 2018} 
\par
\codes{\CSD,~\CWD:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  Computing extremal energy of a class of bicyclic weighted digraphs.  \emph{Asian-European J.\ Math.}\ 13 (2020), no.\ 5, art.\ 2050090, 22 pp. 
\MR{4134198}.  \Zbl{1458.05121}.  
\codes{\CSD,~\CWD:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Total energy of signed digraphs.  \emph{Computer Sci.\ J.\ Moldova}\ 29 (2021), no.\ 1(85), 105--134.
\MR{4256598}.  \Zbl{1469.05087}.  
\annot{} \Cf\ \xref{Hafeez and Khan}{2018a}{Sumaira Hafeez \& Mehtab Khan}.
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{Jurriaan Hage}						%HAGE J
\authorspace
\authref  See also \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender} and \hyperlink{Andrzej Ehrenfeucht}{A.~Ehrenfeucht}.
\authrefspace

\bsgitem{1999a}{}  The membership problem for switching classes with skew gains.  \emph{Fund.\ Inform.}\ 39 (1999), 375--387.
\MR{1823982} (2002b:05071).  \Zbl{944.68144}.					
\annot{} An algorithm to decide whether two skew gain graphs are switching equivalent.
\codes{\CGG(\CGen):~\CSw,~\CAlgor}%
\itemspace

\bsgitem{2001a}{}  \emph{Structural Aspects of Switching Classes}.  Doctoral dissertation, Universiteit Leiden, 2001.  IPA Dissertation Ser., UL.2001-8.  [Instituut voor Programmatuurkunde en Algoritmiek, 2001.]
\annot{} Contains the material of the following papers, along with updates and improved results: \xref{Ehrenfeucht, Hage, Harju, and Rozenberg}{2000a}{Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, \& Grzegorz Rozenberg}, \xrefd{2000b}{Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, \& Grzegorz Rozenberg}, \xrefd{2006a}{Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, \& Grzegorz Rozenberg}, \xref{Hage}{1999a}{Jurriaan Hage}, and \xref{Hage and Harju}{1998a}{Jurriaan Hage \& Tero Harju}, \xrefd{2000a}{Jurriaan Hage \& Tero Harju}, \xrefd{2004a}{Jurriaan Hage \& Tero Harju}.
\annotctd{} Errata and a corrected version at URL (1/2002) \url{http://www.cs.uu.nl/people/jur/2s.html}
\codes{\CTG:~\CSw,~\CAlgor}\codesctd{\CGG(\CGen):~\CSw,~\CAlgor}%
\itemspace

\bsgitem{2003a}{}  Enumerating submultisets of multisets.  \emph{Inform.\ Processing Lett.}\ 85 (2003), no.\ 4, 221--226.
\MR{1950498} (2003m:05008).  \Zbl{1173.68511}.  
\annot{} \S3, ``The computation of the switches of a graph'':  Using a Hamiltonian path of vertex switchings in a graph switching class seems to speed up generating the class.
\dated{15 Jun 2022} 
\codes{\CTG:~\CSw:~\CAlgor}%
\itemspace

\bsgitem{2012a}{}  Subgroup switching of skew gain graphs.  \emph{Fund.\ Inform.}\ 116 (2012), 111--122.
\MR{2977837} (no rev).  \Zbl{1243.05205}.  
\annot{} Skew gains reverse by an involutory antiautomorphism of the gain group (\xref{Hage and Harju}{2000a}{Jurriaan Hage \& Tero Harju}).  Here switching is restricted by prescribing for each vertex a subgroup from which the switching value may be taken.  Properties of ordinary switching generalize, or become more complicated, or become too difficult.  Further research is needed.
\dated{17 Dec, 5 Jan 2011--12} 
\codes{\CGG(\CGen):~\CSw:~\CGen}%
\itemspace

\author{Jurriaan Hage \& Tero Harju}
\authorspace
\bsgitem{1998a}{}  Acyclicity of switching classes.  \emph{European J.\ Combin.}\ 19 (1998), 321--327.
\MR{1621017} (99d:05051).  \Zbl{905.05057}.					
\annot{} Classifies the switching-equivalent pairs of forests.  Thm.\ 2.2:  In a Seidel switching class of graphs there is at most one isomorphism type of tree; and there is at most one tree, with exceptions that are completely classified.  Thms.\ 3.1 and 4.1:  In a switching class that contains a disconnected forest there are at most 3 forests (not necessarily isomorphic); the cases in which there are 2 or 3 forests are completely classified.  (Almost all are trees plus isolated vertices.)  [\Question.  Regarding these results as concerning the negative subgraphs of switchings of signed complete graphs, to what extent do they generalize to switchings of arbitrary signed simple graphs?]  [\xref{B.D.~Acharya}{1981a}{B.\ Devadas Acharya} asked which simple graphs switch to forests, with partial results.]
\codes{\CTG:~\CSw}%
\itemspace

\bsgitem{2000a}{}  The size of switching classes with skew gains.  \emph{Discrete Math.}\ 215 (2000), 81--92.
\MR{1746450} (2001d:05074).  \Zbl{949.05039}.					
\annot{} Introducing ``skew gain graphs'', which generalize gain graphs (see \xref{Zaslavsky}{1989a}{Thomas Zaslavsky}) to incorporate dynamic labelled 2-structures (see Ehrenfeucht and Rozenberg).  Inversion is replaced by a gain-group antiautomorphism $\delta$ of period at most 2.  Thus $\phi(e^{-1}) = \delta(\phi(e))$, while in switching by $\tau$, one defines $\phi^{\tau}(e; v,w) = \delta(\tau(v))\phi(e; v,w)\tau(w)$.  The authors find the size of a switching class $[\phi]$ in terms of the centralizers and/or $\delta$-centralizers of various parts of the image of $\phi_T$, that is, $\phi$ switched to be the identity on a spanning tree $T$.  The exact formulas depend on whether $\Gamma$ is complete, or bipartite, or general, and on the choice of $T$ (the case where $T \cong K_{1,n-1}$ being simplest).
\codes{\CGG(\CGen):~\CSw}%
\itemspace

\bsgitem{2004a}{}  A characterization of acyclic switching classes of graphs using forbidden subgraphs.  \emph{SIAM J.\ Discrete Math.}\ 18 (2004), no.\ 1, 159--176.
\MR{2112496} (2005k:05205).  \Zbl{1071.05063}.					
\annot{} Solves the problem raised by \xref{B.D.~Acharya}{1981a}{B.\ Devadas Acharya}.
\codes{\CTG:~\CSw}%
\itemspace

\bsgitem{2007a}{}  Towards a characterization of bipartite switching classes by means of forbidden subgraphs.  \emph{Discuss.\ Math.\ Graph Theory} 27 (2007), no.\ 3, 471--483.
\MR{2412359} (2009b:05126).  \Zbl{1142.05042}.  
\annot{} Partial results on the forbidden induced subgraphs for graph switching classes with no bipartite member.
\dated{9 Sept 2010} 
\codes{\CTG:~\CSw}%
\itemspace

\bsgitem{2009a}{}  On involutions arising from graphs.  In:  Anne Condon \etal, eds., \emph{Algorithmic Bioprocesses}, pp.\ 623--630.  Springer, Berlin, 2007.
\MR{2762065} (2012j:20157).  \Zbl{1183.68407} (book).  
\annot{} Algebra related to the skewness, i.e., involutory anti-automorphisms, of skew gains on a graph.
\dated{12 Sept 2017} 
\codes{\Cgg(\CGen):~\CAlgeb}%
\itemspace

\author{Jurriaan Hage, Tero Harju, \& Elmo Welzl}
\authorspace
\bsgitem{2002a}{}  Euler graphs, triangle-free graphs and bipartite graphs in switching classes.  In:  \emph{Graph Transformation} (Proc.\ First Int.\ Conf., Rome, 2002), pp.\ 148--160.  \LNCS 2505.  Springer-Verlag, London, 2002. 
\MR{2049362} (no rev).  \Zbl{1028.68101}.						
\annot{} Preliminary version of \xrefd{2003a}{Jurriaan Hage, Tero Harju, \& Elmo Welzl}.
\dated{9 Sept 2010} 
\codes{\CTG:~\CSw}%
\itemspace

\bsgitem{2003a}{}  Euler graphs, triangle-free graphs and bipartite graphs in switching classes.  Special issue on ICGT 2002.  \emph{Fund.\ Inform.}\ 58 (2003), no.\ 1, 23--37.
\MR{2056589} (2005b:05206).  \Zbl{1054.05092}.					
\annot{} Polynomial-time algorithms for whether a graph switching class contains a triangle-free, or bipartite, or Eulerian, member.
\codes{\CTG:~\CSw}%
\itemspace

\author{Per Hage}			 				%HAGE P
\authorspace
\bsgitem{1979a}{}  Graph theory as a structural model in cultural anthropology.  \emph{Annual Rev.\ Anthropology} 8 (1979), 115--136.
\annot{} ``Signed graphs'', pp.\ 120--124.  ``Structural duality'', pp.\ 132--133.  Other examples.
\dated{2 Aug 2010} 
\codes{\CSG,~\CPsS:~\CBal,~\CFr,~\CClu~\CExp}%
\itemspace


\author{Per Hage \& Frank Harary}
\authorspace
\bsgitem{1983a}{}  \emph{Structural Models in Anthropology}.  Cambridge Univ.\ Press, Cambridge, Eng., 1983.  
\MR{0738630} (86e:92002).  
\annot{} Signed graphs are treated in Ch.\ 3 and 6, marked (i.e., vertex-signed) graphs in Ch.~6.
[Reviewed in \xref{Doreian}{1985a}{Patrick Doreian}.]
\par
\codes{\CSG,~\CPsS:~\CBal:~\CExp}\codesctd{\CVS:~\CExp}%
\itemspace

\bsgitem{1986a}{}  Some genuine graph models in anthropology.  \emph{J.\ Graph Theory} 10 (1986), no.\ 3, 353--361.  
\MR{0856121} (87i:92061).  \Zbl{605.05042}.						
\codes{\CPsS,~\CSG:~\CExp}%
\itemspace

\bsgitem{1987a}{}  \emph{Exchange in Oceania}.  Routledge and Kegan Paul, London, 1987. 
\codes{\CSG:~\CAppl}%
\itemspace

\author{Roland H\"aggkvist} 						%HAGGKVIST
\authorx{Roland Haggkvist}
\authorspace
\authref  See \hyperlink{Jerrold W.\ Grossman}{J.W.~Grossman}.
\itemspace

\author{Mohammad Hassan Shirdareh Haghighi}				%HAGHIGHI
\authorspace
\authref  See \hyperlink{Fatemeh Motialah}{F.~Motialah}.
\itemspace

\author{MohammadTaghi Hajiaghayi}					%HAJIAGHAYI
\authorspace
\authref  See \hyperlink{Erik D.\ Demaine}{E.D.~Demaine}.
\itemspace

\author{Mardjan Hakimi-Nezhaad}					%HAKIMI-NEZHAAD
\authorspace
\authref  See also \hyperlink{Modjtaba Ghorbani}{M.~Ghorbani}.
\authrefspace

\author{Mardjan Hakimi-Nezhaad \& Modjtaba Ghorbani}
\authorspace
\bsgitem{2020a}{}  On the Estrada index of Seidel matrix.  \emph{Math.\ Interdisciplinary Res.}\ 5 (2020), no.\ 1, 43--54.
\annot{} \Cf\ \xref{Askari, Iranmanesh, Das}{2016a}{Jalal Askari, Ali Iranmanesh, \& Kinkar Ch Das}.  Bounds.  Lem.\ 3.1: Seidel matrix of composition of two graphs.  \Conjectures: $[-K_n]$ maximizes and $[+K_n]$ minimizes Estrada index of signed $K_n$.  $K_{P_n}$ (path) minimizes and $K_{S_n}$ (star) minimizes for $K_T$, $T$ a tree.
\dated{9 Nov 2020} 
\par
\codes{\Csg:~\CKG:~\Csw,~\CAdj:~\CEig}%
\itemspace

\author{F.D.M.\ Haldane}						%HALDANE
\authorspace
\authref  See \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\itemspace

\author{Frank J.\ Hall}						%HALL FJ
\authorspace
\authref  See also \hyperlink{Marina Arav, Frank J.\ Hall, Zhongshan Li, \& Hein van der Holst}{M.~Arav} and \hyperlink{Carolyn Eschenbach}{C.A.~Eschenbach}.
\authrefspace

\author{Frank J.\ Hall \& Zhongshan Li}
\authorspace
\bsgitem{2007a}{}  Sign pattern matrices.  In:  Leslie Hogben, ed., \emph{Handbook of Linear Algebra}, pp.\ 33-1--33-21.  
Discrete Math.\ Appl.  Chapman \& Hall/CRC Press, Boca Raton, 2007.
\MR{2279160} (2007j:15001) (book).  \Zbl{1122.15001}.				
\codes{\CQM:~\Csd}
\itemspace

\author{H.\ Tracy Hall}						%HALL HT
\authorspace
\authref  See \hyperlink{Steve Butler}{S.~Butler}.
\itemspace

\author{Peter Hall}							%HALL P
\authorspace
\authref  See \hyperlink{Bai Xiao}{B.~Xiao}.
\itemspace

\authorx{Joshua Hallam}						%HALLAM
\author{Joshua Hallam, Jeremy L.\ Martin, \& Bruce E. Sagan}
\authorspace
\bsgitem{2019a}{}  Increasing spanning forests in graphs and simplicial complexes.  \emph{European J.\ Combin.}\ 76 (2019), 178--198.
\MR{3886522}.  \Zbl{1402.05183}.  \arXiv{1610.05093}.  
\annot{} \S4, ``ISFs in multigraphs'':  The matroid is the Dowling geometry $\bfF(\bbC^*K_n\full)$ or a submatroid.  
Dictionary:  ``Labeled $n$-multigraph'' = simple $\bbC^*$-gain graph; ``0-edge'' = half edge;  ``ISF''' (increasing spanning forest) = increasing basis of $\bfF(\bbC^*K_n\full)$; ``perfectly labeled'' = closed in $\bfF(\bbC^*K_n\full)$.
\dated{19 Oct 2019} 
\codes{\CGG:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\authorx{Maureen Hallinan}						%HALLINAN
\author{Maureen Hallinan \& David D.\ McFarland}
\authorspace
\bsgitem{1975a}{}  Higher order stability conditions in mathematical models of sociometric or cognitive structure.  \emph{J.\ Math.\ Sociology} 4 (1975), 131--148.
\MR{0421740} (54 \#9734).  \Zbl{322.92024}.						
\annot{} \S\ ``Signed directed graphs'' (pp.\ 134--141):  Tendency towards ``transitivity''  (balance or clusterability) in signed digraphs.  The impetus for single-arc change (sign change, or deletion or introduction) cannot be determined by triangles alone (Props.\ 1.1--1.3).
\dated{23 Nov 2012} 
\par
\codes{\CSD:~\Cbal,~\Cclu,~\CPsS}%
\itemspace

\author{Mark D.\ Halsey}						%HALSEY
\authorspace
\bsgitem{1987a}{}  Line-closed combinatorial geometries.  \emph{Discrete Math.}\ 65 (1987), no.\ 3, 245--248. 
\MR{0897649} (88g:05043).  \Zbl{613.05014}.  
\annot{} Dowling lattices are line closed; thus line closure does not imply vector representability.
\dated{8 Apr 2016} 
\codes{\Cgg:~\CMatrd:~\CStr}%
\itemspace

\author{Ginji Hamano}						%HAMANO
\authorspace
\bsgitem{2016a}{}  Existence of a regular unimodular triangulation of the edge polytopes of finite graphs.  \emph{Comment. Math. Univ. St. Pauli} 65 (2016), no.\ 2, 85--96.
\MR{3675806}.  \Zbl{1379.52015}.  \arXiv{1405.3341}.  
\annot{} Disjoint odd circles are important for edge polytope of all-extraverted graph.  [\Question.  How does this apply to edge polytopes of general bidirected graphs?]
\dated{6 Jul 2022} 
\codes{\Cpar:~\CCircles,~\CGeom}%
\itemspace

\author{Saira Hameed}						%HAMEED
\authorspace
\authref  See \hyperlink{Uzma Ahmad}{U.~Ahmad}.
\itemspace

\author{Shahul Hameed K}						%HAMEED
\authorspace
\authref  See also \hyperlink{K.A.\ Germina}{K.A.~Germina}, \hyperlink{Albin Mathew}{A.~Mathew}, \hyperlink{Roshni T Roy}{R.T.~Roy}, and \hyperlink{Shijin T V}{T.V.~Shijin}.
\authrefspace

\author{Shahul Hameed K \& K.A.\ Germina}
\authorspace
\bsgitem{2012a}{\dag}  Balance in gain graphs -- A spectral analysis.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 5, 1114--1121.
\MR{2890908}.  \Zbl{1236.05096}.						
\annot{} Thm.\ 2.4:  For a field $F$, an $F^\times$-gain graph $\Phi$ is balanced iff $\Spec_F A(\Phi) = \Spec_F A(\|\Phi\|)$---generalizing \xref{B.D.~Acharya}{1980a}{B.\ Devadas Acharya}.  Thms.\ 2.2, 3.1: a formula and a recurrence (involving circles) for the characteristic polynomial.  Corollaries for signed graphs.
\dated{12 Jul 2019} 
\par
\codes{\CGG:~\CEig,~\CBal,~\CSG}%
\itemspace

\bsgitem{2012b}{}  On composition of signed graphs.  \emph{Discuss.\ Math.\ Graph Theory} 32 (2012), no.\ 3, 507--516.
\MR{2974034}.  \Zbl{1257.05056}.						
\annot{} Balance and eigenvalues in a definition of composition (lexicographical product).  Only the composition of gains of the type balanced[identity] is balanced.  [\Cf\ \xref{Brunetti, Cavaleri, and Donno}{2019a}{Maurizio Brunetti, Matteo Cavaleri, \& Alfredo Donno}.]
\dated{20 Apr 2019} 
\codes{\CSG:~\CBal,~\CEig:~\CAdj,~\CLap}%
\itemspace

\bsgitem{2012c}{}  Balance in certain gain graph products.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 205--216.
\Zbl{1301.05280}.  
\annot{} Definitions of Cartesian, lexicographic, Cvetkovi\'c (NEPS) products, with balance, adjacency matrices, eigenvalues.
In part, generalizes to gain graphs portions of \xrefd{2012b}{Shahul Hameed K \& K.A.\ Germina} and \xref{Germina, Hameed, and Zaslavsky}{2011a}{K.A.\ Germina, Shahul Hameed K, \& Thomas Zaslavsky}.  For balance of lex product see \xrefd{2012b}{Shahul Hameed K \& K.A.\ Germina}.  Cvetkovi\'c product NEPS$(\Phi_1,\ldots,\Phi_\nu)$ preserves balance.
\dated{20 Apr 2019, rev 12 Jul 2019} 
\codes{\CGG:~\CBal,~\CAdj,~\CEig}%
\itemspace

\bsgitem{20xxa}{}  On bounds for the eigenvalues and energy of signed graphs.  Submitted.  
\par
\codes{\CSG:~\CEig}%
\itemspace

\author{Shahul Hameed K, Albin Mathew, Germina K A, \& Thomas Zaslavsky}
\authorspace
\bsgitem{20xxa}{}  Vector valued switching in signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ (in press and online).
\arXiv{2208.00149}.  
\annot{} Vector-valued switching of $\Sigma$, or $k$-switching, means $\sigma(uv) \mapsto \sigma^\zeta(uv) = \sgn(\zeta(u)\cdot\zeta(v)),$ where $\zeta: V \to \{\pm1,0\}^k \setm \mathbf0$.  Main question:  ``balancing dimension'' $\bdim(\Sigma) := \min\{k: (\exists \zeta)\ \Sigma^\zeta \text{ is balanced}\}$.  Variants such as injective $\zeta$ and $\zeta: V \to \{\pm1\}^k$.  Solved for examples, e.g.\ $\bdim(C_n^-) = 2$ ($n>3$), $= 3$ ($n=3$), where $C_n^-$ = negative circle.
\dated{16 Jun 2023} 
\codes{\CSG:~\CSw(\CGen),~\CBal}%
\itemspace

\author{Shahul Hameed K., Viji Paul, \& K.A.\ Germina}
\authorspace
\bsgitem{2015a}{}  On co-regular signed graphs.  \emph{Australasian J.\ Combin.}\ 62(1) (2015), 8--17.
\MR{3337173}.  \Zbl{1321.05103}.  
\annot{} See \xref{Viji~P}{2012a}{Viji Paul} for definitions.	
\codes{\CSG}%
\itemspace

\author{Shahul Hameed Koombail \& Ramakrishnan K O}
\authorspace
\bsgitem{20xxa}{}  Balance theory: An extension to conjugate skew gain graphs.  \emph{Commun.\ Combin.\ Optim.}\ (in press).
\codes{\CGG(\CGen):~\CBal:~\CAdj,~\CLap}%
\itemspace

\author{Shahul Hameed K, Remna K P, Divya T, Biju K, Rajeevan P, Santhosh G O, \& Ramakrishnan K O}
\authorspace
\bsgitem{20xxa}{}  On the metric dimension of signed graphs.  Submitted.  \emph{}\ 
\arXiv{2106.12539}.  
\annot{} For distance-compatible $\Sigma$ (\cf\ \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}), let $\dim\Sigma =$ min size of signed-distance resolving set.  Solved for star, circle; partial results for $K_n$, wheels, trees.
\dated{13 Aug 2021} 
\codes{\CSG}%
\itemspace

\author{Shahul Hameed K, Roshni T Roy, Soorya P, \& Germina K A}
\authorspace
\bsgitem{20xxa}{}  On the characteristic polynomial of skew gain graphs.  \emph{Southeast Asian Bull.\ Math.}\ (to appear).
\arXiv{2009.08708}.  
\codes{\CGG(~\CGen):~\CAdj:~\CEig}%
\itemspace

\author{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2021a}{}  Signed distance in signed graphs.  \emph{Linear Algebra Appl.}\ 608 (2021), 236--247.
\MR{4143538}.  \Zbl{1458.05054}.  \arXiv{2005.06202}.				
\annot{} Set $\sigma_{max}(u,v) := \max\{\sigma(P): \text{shortest } uv\text{-path } P\}$, $\sigma_{\min}$ similarly.
Signed distance: $d_{max}(u,v) := \sigma_{\max}(u,v)d(u,v)$, $d_{\min}$.  Signed distance matrices $D^{\max}$, $D^{\min}$.  If $d_{\max} = d_{\min}$ (``distance compatibility''), $D^\pm := D^{\max}$.  
Associated signed complete graphs.  
Thm.\ 3.5:  $\Sigma$ balanced iff any $D(\Sigma)$ cospectral with $D(|\Sigma|)$.
\annotctd{} $\Sigma$ is distance compatible if geodetic, balanced, or antibalanced.  Thm.\ 5.1:  Bipartite $\Sigma$ is distance-compatible iff balanced.  Ex.\ 5.4:  Distance compatible but not geodetic, balanced, or antibalanced.
\annotctd{} [\Cf\ \xref{Roy, Germina \etal\!}{20xxa}{Roshni T Roy, K A Germina, K Shahul Hameed, and Thomas Zaslavsky}, \xref{Shijin, Soorya, \etal\!}{20xxa}{Shijin T V, Soorya P, Shahul Hameed K, \& Germina K A}.  Generalized by \xref{Samanta and Kannan}{2022a}{Aniruddha Samanta \& M.\ Rajesh Kannan}, \xref{Shijin and Germina}{20xxa}{Shijin T V \& Germina K A}.]
\dated{11 Oct 2020} 
\par
\codes{\CSG:~\CWG,~\CAdj(\CGen),~\CBal,~\CEig,~\CSw}%
\itemspace

\authorx{Hasti Hamidzade}						%HAMIDZADE
\author{Hasti Hamidzade \& Dariush Kiani}
\authorspace
\bsgitem{2010a}{}  Erratum to ``The lollipop graph is determined by its $Q$-spectrum''.  \emph{Discrete Math.}\ 310 (2010), no.\ 10-11, 1649.
\MR{2601277} (2011c:05194).  \Zbl{1210.05080}.					
\annot{} Corrected proof of \xref{Y.P.~Zhang, Liu, Zhang, and Yong}{2009a}{Yuanping Zhang, Xiaogang Liu, Bingyan Zhang, \& Xuerong Yong}, Thm.\ 3.3.
\dated{16 Oct 2011} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Peter L.\ Hammer} 						%HAMMER*
\authorspace
\authref  See also \hyperlink{Egon Balas}{E.~Balas}, \hyperlink{C.\ Benzaken}{C.~Benzaken}, \hyperlink{Endre Boros}{E.~Boros}, \hyperlink{Jean-Marie Bourjolly}{J.-M.~Bourjolly}, \hyperlink{Yves Crama}{Y.~Crama}, and \hyperlink{Aviezri S.\ Fraenkel}{A.~Fraenkel}.
\authrefspace

\bsgitem{1974a}{}  Boolean procedures for bivalent programming.  
In:  P.L.~Hammer and G.~Zoutendijk, eds., \emph{Mathematical Programming in Theory and Practice} (Proc.\ NATO Adv.\ Study Inst., Figueira da Foz, Portugal, 1972), pp.\ 311--363.  North-Holland, Amsterdam, and American Elsevier, New York, 1974.  
\MR{0479387} (57 \#18817).  \Zbl{335.90034} (book).
\codes{\Csg:~\Cori}
\itemspace

\bsgitem{1977a}{}  Pseudo-Boolean remarks on balanced graphs.  
In: L.~Collatz, G.~Meinardus, and W.~Wetterling, eds., \emph{Numerische Methoden bei Optimierungsaufgaben, Band 3:  Optimierung bei graphentheoretischen und ganzzahligen Problemen} (Tagung, Oberwolfach, 1976), pp.\ 69--78.  Int.\ Ser.\ Numer.\ Math., Vol.\ 36.  Birkh\"auser, Basel, 1977.  
\MR{0465947} (57 \#5833).  \Zbl{405.05054}.
\codes{\CSG:~\CBal}%
\itemspace

\author{P.L.\ Hammer, C.\ Benzaken, \& B.\ Simeone} 
\authorspace
\bsgitem{1980a}{}  Graphes de conflit des fonctions pseudo-bool\'eennes quadratiques.
In:  P.~Hansen and D.~de Werra, eds., \emph{Regards sur la Th\'eorie des Graphes} (Actes du Colloq., Cerisy, 1980), pp.\ 165--170.  Presses Polytechniques Romandes, Lausanne, Switz., 1980.  
\MR{0614299} (82d:05054) (book).  
\codes{\Csg}%
\itemspace

\author{P.L.\ Hammer, T.\ Ibaraki, \& U.\ Peled} 
\authorspace
\bsgitem{1980a}{}  Threshold numbers and threshold completions.  
In:  M.~Deza and I.G.~Rosenberg, eds., \emph{Combinatorics 79} (Proc.\ Colloq., Montreal, 1979), Part II.  \emph{Ann.\ Discrete Math.}\ 9 (1980), 103--106.
\MR{0597360} (81k:05092).  \Zbl{443.05064}.
\codes{\Cpar:~\Cori}%
\itemspace

\bsgitem{1981a}{}  Threshold numbers and threshold completions.  
In:  Pierre Hansen, ed., \emph{Studies on Graphs and Discrete Programming} (Proc.\ Workshop, Brussels, 1979), pp.\ 125--145.  
North-Holland Math.\ Studies, 59.  Ann.\ Discrete Math., 11.  North-Holland, Amsterdam, 1981.  
\MR{0653822} (83m:90062).  \Zbl{0482.05060}.
\annot{} See description of Thm.\ 8.5.2 in \xref{Mahadev and Peled}{1995a}{N.V.R.\ Mahadev \& U.N.\ Peled}.
\codes{\Cpar:~\Cori}%
\itemspace

\author{P.L.\ Hammer \& N.V.R.\ Mahadev} 
\authorspace
\bsgitem{1985a}{}  Bithreshold graphs.  \emph{SIAM J.\ Algebraic Discrete Methods} 6 (1985), 497--506.  
\MR{0791177} (86h:05093).  \Zbl{579.05052}.					
\annot{} An auxiliary signed graph on $E$ is a proof tool.  See description in \S8.3 of \xref{Mahadev and Peled}{1995a}{N.V.R.\ Mahadev \& U.N.\ Peled}. 
[Later work by \xref{Mahadev and Peled}{1988a}{N.V.R.\ Mahadev \& U.N.\ Peled}, \hyperlink{Peter L.\ Hammer, N.V.R.\ Mahadev, \& Uri N.\ Peled}{Hammer, Mahadev, and Peled}, \hyperlink{Rossella Petreschi}{R.\ Petreschi} and \hyperlink{Tiziana Calamoneri}{T.\ Calamoneri}.]
\dated{rev.\ 22 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal}%
\itemspace

\author{Peter L.\ Hammer, N.V.R.\ Mahadev, \& Uri N.\ Peled}
\authorspace
\bsgitem{1989a}{} Some properties of 2-threshold graphs.  \emph{Networks} 19 (1989), 17--23.  
\MR{0973562} (89m:05096).  \Zbl{671.05059}.					
\annot{} Uses the auxiliary signed graph of \xref{Hammer and Mahadev}{1985a}{P.L.\ Hammer \& N.V.R.\ Mahadev}.
\dated{22 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal}%
\itemspace

\bsgitem{1993a}{}  Bipartite bithreshold graphs.  \emph{Discrete Math.}\ 119 (1993), 79--96.
\MR{1234060} (94e:05234).  \Zbl{790.05082}.  
\annot{} Uses the auxiliary signed graph of \xref{Hammer and Mahadev}{1985a}{P.L.\ Hammer \& N.V.R.\ Mahadev}.
\dated{17 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal}%
\itemspace

\author{Peter L.\ Hammer \& Sang Nguyen} 
\authorspace
\bsgitem{1977a}{}  APOSS.  A partial order in the solution space of bivalent programs.
In:  J.\ Rose and C.\ Bilciu, eds., \emph{Modern Trends in Cybernetics and Systems} (Proc.\ Third Int.\ Congress, Bucharest, 1975), Vol.\ I, pp.\ 869--883.  Springer, Berlin, 1977.  
\MR{0475847} (57 \#15430).  \Zbl{414.90063}.
\codes{\Csg:~\Cori}%
\itemspace

\bsgitem{1979a}{}  A partial order in the solution space of bivalent programs.
In:  Nicos Christofides, Aristide Mingozzi, Paolo Toth, and Claudio Sandi, eds., \emph{Combinatorial Optimization}, Ch.\ 4, pp.\ 93--106.  Wiley, Chichester, 1979.  
\MR{0557004} (82a:90099) (book).  \Zbl{414.90063}.
\codes{\Csg:~\Cori}%
\itemspace

\author{J.\ Hammann}						%HAMMANN
\authorspace
\authref  See \hyperlink{E.\ Vincent}{E.~Vincent}.
\itemspace

\authorx{Prabhakar Ramrao Hampiholi}						%HAMPIHOLI
\author{P.R.\ Hampiholi, H.S.\ Ramane, Shailaja S.\ Shirkol, Meenal M.\ Kaliwal, \& Saroja R.\ Hebbar}
\authorspace
\bsgitem{2017a}{}  A note on signed semigraphs.  \emph{Int.\ J.\ Comput.\ Appl.\ Math.}\ 12 (2017), no.\ 3, 887--898.
\annot{} Semigraph $:= (V,E)$; edge $e = (v_1,v_2,\ldots,v_k) = (v_k,\ldots,v_2,v_1)$, i.e., symmetrically ordered subset of $V$ (due to \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}).  Sign $\sigma(e)=(-1)^k$.  Parity and sign are not distinguished.  Balance results appear to be immediate corollaries of known signed-graph facts.  Vertices may also be signed.
\dated{11 May 2018} 
\codes{\CS:~\CBal,~\CVS}%
\itemspace

\author{Miaomiao Han}						%HAN MM
\authorspace
\authref  See \hyperlink{Jianbing Liu}{J.-B.~Liu} and \hyperlink{Xiying Yuan}{X.Y.~Yuan}.
\itemspace

\author{Wei Han}							%HAN W
\authorspace
\authref  See \hyperlink{Shiying Wang}{S.Y.~Wang}.
\itemspace

\author{Phil Hanlon} 						%HANLON
\authorspace
\bsgitem{1984a}{}  The characters of the wreath product group acting on the homology groups of the Dowling lattices.  \emph{J.\ Algebra} 91 (1984), 430--463.
\MR{0769584} (86j:05046).  \Zbl{557.20009}.
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\bsgitem{1988a}{}  A combinatorial construction of posets that intertwine the independence matroids of $B_n$ and $D_n$.  Manuscript, 1988.  
\annot{} Computes the M\"obius functions of posets obtained from $\Lat \bfF(\pm K_n^{\circ})$ by discarding those flats with unbalanced vertex set in a given lower-hereditary list.  Examples include $\Lat \bfF(\pm K_n^{(k)})$, the exponent denoting the addition of $k$ negative loops.  Generalized and superseded by \xref{Hanlon and Zaslavsky}{1998a}{Phil Hanlon \& Thomas Zaslavsky}.
\codes{\Csg:~\CMatrd:~\CGen:~\CInvar}%
\itemspace

\bsgitem{1991a}{}  The generalized Dowling lattices.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 325 (1991), 1--37.  
\MR{1014249} (91h:06011).  \Zbl{748.05043}.
\annot{} The lattices are based on a rank, $n$, a group, and a meet sublattice of the lattice of subgroups of the group.  The Dowling lattices are a special case.
\codes{\Cgg:~\CMatrd:~\CGen:~\CInvar}%
\itemspace

\bsgitem{1996a}{}  A note on the homology of signed posets.  \emph{J.\ Algebraic Combin.}\ 5 (1996), 245--250.  
\MR{1394306} (97f:05194).  \Zbl{854.06004}.
\annot{1394306} Partial summary of \xref{Fischer}{1993a}{Steven D.\ Fischer}.  
\codes{\CS}%
\itemspace

\author{Phil Hanlon \& Thomas Zaslavsky} 
\authorspace
\bsgitem{1998a}{}  Tractable partially ordered sets derived from root systems and biased graphs.  \emph{Order} 14 (1997--98), 229--257.
\MR{1634902} (2000a:06016).  \Zbl{990.03811}.					
\annot{} Computes the characteristic polynomials (Thm.\ 4.1) and hence the M\"obius functions (Cor.\ 4.4) of posets obtained from $\Lat \bfF(\Omega)$, $\Omega$ a biased graph, by discarding those flats with unbalanced vertex set in a given lower-hereditary list.  Examples include $\Lat \bfF(\fG K_n^{(k)})$ where $\fG$ is a finite group, the exponent denoting the addition of $k$ unbalanced loops.  The interval structure, existence of a rank function, covering pairs, and other properties of these posets are investigated.  There are many open problems.  
Simplifies, then generalizes, \xref{Hanlon}{1988a}{Phil Hanlon}.
\par
\codes{\CGG:~\CMatrd,~\CGen:~\CInvar,~\CStr,~\CCol}%
\itemspace

\author{Pierre Hansen} 						%HANSEN
\authorspace
\authref  See also \hyperlink{Mustapha Aouchiche}{M.~Aouchiche}, \hyperlink{Vincent Devloo}{V.~Devloo}, and \hyperlink{Carla Silva Oliveira}{C.S.~Oliveira}.
\authrefspace

\bsgitem{1978a}{} Labelling algorithms for balance in signed graphs.  
In: \emph{Probl\`emes combinatoires et th\'eorie des graphes} (Colloq.\ Int., Orsay, 1976), pp.\ 215--217.  
Colloques Int.\ du CNRS, 260.  Editions du C.N.R.S., Paris, 1978.
\MR{0539978} (80m:68057).  \Zbl{413.05060}.					
\annot{} \S1:  Algorithm 1 labels vertices of a signed graph to detect imbalance and a negative circle if one exists.  [It is equivalent to switching a maximal forest to all positive and looking for negative edges.  Independently discovered by \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}.]  \S2:  Algorithm 2 is the unweighted case of the algorithm of \xrefd{1984a}{Pierre Hansen}.  Path balance in a signed digraph is discussed.  \S3:  The frustration index of a signed graph is bounded below by the negative-circle packing number, which can be crudely bounded by Alg.\ 1.
\codes{\CSG,~\CSD:~\CBal,~\CFr:~\CAlgor,~\Csw}%
\itemspace

\bsgitem{1979a}{}  Methods of nonlinear 0--1 programming.  
In:  P.L.~Hammer, E.L.~Johnson, and B.H.~Korte, eds., \emph{Discrete Optimization II} (Proc., Banff and Vancouver, 1977), pp.\ 53--70.  
Ann.\ Discrete Math., Vol.\ 5.  North-Holland, Amsterdam, 1979.  
\MR{0558567} (no rev).  \Zbl{426.90063}.						
\annot{} See pp.\ 58--59. 
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\bsgitem{1983a}{}  Recognizing sign solvable graphs.  \emph{Discrete Appl.\ Math.}\ 6 (1983), 237--241.  
\MR{0712924} (84i:68112).  \Zbl{524.05048}.					
\annot{} Improves the characterization by \xref{Maybee}{1981a}{John S.\ Maybee} of sign-solvable digraphs with an eye to more effective algorithmic recognition.  Thm.\ 2.2:  A signed digraph $D$ is sign solvable iff its positive subdigraph is acyclic and each strongly connected component has a vertex that is the terminus of no negative, simple directed path.  \S3: ``An algorithm for sign solvability'' in time $O(\#V\,\#E)$.
\codes{\CSD:~\CSol:~\CAlgor}%
\itemspace

\bsgitem{1984a}{}  Shortest paths in signed graphs.  
In:  A.~Burkard \etal, eds., \emph{Algebraic Methods in Operations Research}, pp.\ 201--214.
North-Holland Math.\ Stud., 95.  Ann.\ of Discrete Math., 19.  North-Holland, Amsterdam, 1984.  
\MR{0780022} (86i:05086).  \Zbl{567.05032}.					
\annot{} Algorithm to find shortest walks of each sign from vertex $x_1$ to each other vertex, in a signed digraph with positive integral(?) weights (i.e., lengths) on the edges.  Applied to digraphs with signed vertices and edges; $N$-balance in signed graphs; sign solvability.  The problem for (simple) paths is discussed [which is solvable by any min-weight parity path algorithm; see the notes on \xref{Gr\"otschel and Pulleyblank}{1981a}{M.\ Grotschel \& W.R.\ Pulleyblank}].
\par
\codes{\CSD,~\CWD:~\CPaths,~\CVS,~\CBal,~\CSol:~\CAlgor}%
\itemspace

\author{Pierre Hansen \& Claire Lucas}
\authorspace
\bsgitem{2009a}{}  An inequality for the signless Laplacian index of a graph using the chromatic number.  \emph{Graph Theory Notes N.Y.}\ 57 (2009), 39--42.
\MR{2666279} (2011c:\-05195).  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  Bounds and conjectures for the signless Laplacian index of graphs.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 12, 3319--3336.
\MR{2639286} (2011m:05173).  \Zbl{1214.05079}.					
\annot{} 
[See \xref{Liu and Lu}{2014a}{Huiqing Liu \& Mei Lu} for solution of a conjecture.]
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Pierre Hansen \& Bruno Simeone}
\authorspace
\bsgitem{1986a}{}  Unimodular functions.  \emph{Discrete Appl.\ Math.}\ 14 (1986), 269--281.  
\MR{0848659} (88a:90138).  \Zbl{597.90058}.						
\annot{} Three types of relatively easily maximizable pseudo-Boolean function (``unimodular'' and two others) are defined. For quadratic pseudo-Boolean functions $f$, the three types coincide; $f$ is unimodular iff an associated signed graph is balanced (Thm.\ 3).  Thus one can quickly recognize unimodular quadratic functions, although not unimodular functions in general.  If the graph is a tree, the function can be maximized in linear time.
\codes{\CSG:~\CBal,~\CAlgor}%
\itemspace

\author{Christopher R.H.\ Hanusa}					%HANUSA
\authorspace
\authref   See \xref{Chaiken, Hanusa, and Zaslavsky}{2019a}{Seth Chaiken, Christopher R.H.\ Hanusa, \& Thomas Zaslavsky}.
\itemspace

\authorx{Fei Hao}							%HAO F
\author{Fei Hao, Stephen S.\ Yau, Geyong Min, \& Laurence T.\ Yang}
\authorspace
\bsgitem{2014a}{}  Detecting $k$-balanced trusted cliques in signed social networks.  \emph{IEEE Internet Computing}\ 18 (2014), no.\ 2, 24--31.
\annot{} Dictionary:  ``$k$-balanced trusted clique'' = all-positive $k$-clique in $\Sigma$.  The problem:  detect all such cliques.
\dated{27 Nov 2018} 
\par
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Rong Xia Hao}						%HAO RX
\authorspace
\authref  See also \hyperlink{Shengjie He}{S.-J.~He}.
\authrefspace

\author{Rong Xia Hao \& Yan Pei Liu}
\authorspace
\bsgitem{2010a}{}  Auxiliary graphs of projective planar signed graphs.  (In Chinese.)  \emph{J.\ Systems Sci. Math.\ Sci.}\ 30 (2010), no.\ 9, 1251--1258.
\MR{2785248} (2012c:05093).  \Zbl{1240.05139}.  
\annot{} $\Sigma$ is projective planar iff an auxiliary graph is balanced.  [The auxiliary graph may be a tree.  It may have order linear in that of $\Sigma$.]
\dated{25 Apr 2012} 
\codes{\CSG:~\CTop}%
\itemspace

\authorx{Xiao Hui Hao}						%HAO XH
\author{Xiao Hui Hao \& Bao Feng Li}
\authorspace
\bsgitem{2008a}{}  The quasi-Laplacian spectral radius of a graph.  (In Chinese.)  \emph{Math.\ Practice Theory}\ 38 (2008), no.\ 4, 158--160.
\MR{2435555} (no rev).  \Zbl{1174.05438}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xiao Hui Hao \& Li Jun Zhang} %[MR:  Zhang, Li Jun 1]
\authorspace
\bsgitem{2009a}{}  The largest eigenvalue of the quasi-Laplacian matrix of a connected graph.  (In Chinese.)  \emph{Math.\ Pract.\ Theory}\ 39 (2009), no.\ 7, 178--181.
\MR{2553871} (no rev).  \Zbl{1212.05154}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Masaaki Harada}						%HARADA
\author{Masaaki Harada \& Akihiro Munemasa}
\authorspace
\bsgitem{2012a}{}  On the classification of weighing matrices and self?orthogonal codes.  \emph{J.\ Combin.\ Designs} 20 (2012), no.\ 1, 40--57. 
\MR{2864617}.  \Zbl{1252.05026}.  \arXiv{1011.5382}.  
\annot{} Signed bipartite graphs with $B(\Sigma)B\transpose(\Sigma) = 5I$.
\dated{14 Oct 2020} 
\codes{\Csg:~\CAdj}%
\itemspace

\author{Frank Harary} 						%HARARY*
\authorspace
\authref  See also \hyperlink{Lowell W.\ Beineke}{L.W.~Beineke}, \hyperlink{Andreas Blass}{A.~Blass}, \hyperlink{Fred Buckley}{F.~Buckley}, \hyperlink{Dorwin Cartwright}{D.~Cartwright}, \hyperlink{Gary Chartrand}{G.~Chartrand}, \hyperlink{Ove Frank}{O.~Frank}, and \hyperlink{Per Hage}{P.~Hage}.
\authrefspace

\bsgitem{1953a}{\dag\dag} On the notion of balance of a signed graph.  \emph{Michigan Math.\ J.}\ 2 (1953--1954), 143--146 and addendum preceding p.\ 1.  
\MR{0067468} (16, 733).  \Zbl{056.42103} (56, p.\ 421c).					
\annot{$\Sigma$} The main theorem (Thm.\ 3) characterizes balanced signings as those for which there is a bipartition of the vertex set such that an edge is positive iff it lies within a part [I call this a Harary bipartition].  Thm.\ 2: A signing of a simple [or a loop-free] graph is balanced iff, for each pair of vertices, every path joining them has the same sign.  The generating function for counting nonisomorphic signed simple graphs with $n$ vertices by numbers of positive and negative edges is $g_n(x+y)$ where $g_n(x)$ is the g.f.\ of nonisomorphic simple graphs.
\annotctd{} [The birth of signed graph theory.  Although Thm.\ 3 was anticipated by \xref{K\"onig}{1936a}{Denes Konig} (Thm.\ X.11, for finite and infinite graphs) without the terminology of signs, here is the first recognition of the crucial fact that labelling edges by elements of a group---specifically, the sign group---can lead to a general theory.]  
\dated{ca.\ 1977.  Rev.\ 20 Jan 2010}
[\Cf\ \xref{Whiteley}{1991a}{Walter Whiteley}.]  
\dated{12 Jun 2012}
\codes{\CSG:~\CBal,~\CEnum}%
\itemspace

\bsgitem{1955a}{} On local balance and $N$-balance in signed graphs.  \emph{Michigan Math.\ J.}\ 3 (1955--1956), 37--41.  
\MR{0073170} (17, 394).  \Zbl{070.18502} (70, p.\ 185b).						
\annot{} $\Sigma$ is (locally) balanced at a vertex $v$ if every circle on $v$ is positive; then Thm.\ 3$'$:  $\Sigma$ is balanced at $v$ iff every block containing $v$ is balanced.  $\Sigma$ is $N$-balanced if every circle of length $\leq N$ is positive; Thm.\ 2 concerns characterizing $N$-balance.  Lemma 3: For each circle basis, $\Sigma$ is balanced iff every circle in the basis is positive.  [This strengthens \xref{K\"onig}{1936a}{Denes Konig} Thm.\ 13 for finite graphs.]
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{1957a}{} Structural duality. \emph{Behavioral Sci.}\ 2 (1957), 255--265.  
\MR{0134799} (24 \#B851).  
\annot{} ``Antithetical duality'' (pp.\ 260--261) introduces antibalance.  Remarks on signed and vertex-signed graphs are scattered about the succeeding pages.
\codes{\CSG:~\CBal,~\CPar}%
\itemspace

\bsgitem{1958a}{} On the number of bi-colored graphs.  \emph{Pacific J.\ Math.}\ 8 (1958), 743--755.  
\MR{0103834} (21 \#2598).  \Zbl{084.19402} (84, p.\ 194b).				
\annot{} \S6:  ``Balanced signed graphs''. 
\codes{\CSG:~\CBal:~\CEnum}%
\itemspace

\bsgitem{1959a}{} Graph theoretic methods in the management sciences.  \emph{Management Sci.}\ 5 (1959), 387--403.  
\MR{0108387} (21 \#7103).  
Repr.\ in:  Samuel Leinhardt, ed., \emph{Social Networks:  A Developing Paradigm}, pp.\ 371--387.  Academic Press, New York, 1977.
\annot{} Pp.\ 400--401: List of characterizations of balance.
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\bsgitem{1959b}{} On the measurement of structural balance.  \emph{Behavioral Sci.}\ 4 (1959), 316--323.  
\MR{0112850} (22 \#3696).  
\annot{} Proposes to measure imbalance by 
(i) $\beta(\Sigma)$, the proportion of positive circles (``degree of balance'') from \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}, 
(ii) the frustration index $l(\Sigma)$ (here called ``line index of balance'') [\cf\ \xref{Abelson and Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg}], i.e., the smallest number of edges whose deletion or equivalently (Thm.\ 7) negation results in balance, and 
(iii) the frustration number $l_0(\Sigma)$ (``point index'').  
For $\beta$ restricted to unbalanced blocks with cyclomatic number $\xi$:  Thm.\ 4: $\min\beta \leq (\xi-1)/(\xi-1+2^{\xi-1})$.  Thm.\ 5: $\max\beta \geq 1-2/(\xi+1)$ (e.g., a ladder with $\xi+1$ rungs and one rung negative).  Cors.: $\min\beta \to 0$, $\max\beta \to 1$ as $\xi \to \infty$.  \Conjecture.  The bounds are best possible.  [I know of no work on this.]  Thm.\ 6 (contributed by J.\ Riordan): Asymptotically, $\beta(-K_n) - \frac12 \sim \frac12(-1/e)^n$.
\dated{$\leq$1980.  Rev.\ 20 Jan 2010}
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{1960a}{}  A matrix criterion for structural balance.  \emph{Naval Res.\ Logistics Quarterly} 7, No.\ 2 (June, 1960), 195--199.  
\Zbl{091.15904} (91, p.\ 159d).  
\annot{} First explicit appearance of the incidence matrix $\incid(\Sigma)$, called $J$.  Thm.\ 2 (\xref{Heller and Tompkins}{1956a}{I.\ Heller \& C.B.\ Tompkins}, \xref{Hoffman and Gale}{1956a}{[A.J.\ Hoffman \& D.\ Gale]}):  $\Sigma$ is balanced iff $\incid(\Sigma)$ is totally unimodular.  Cor.:  The unoriented incidence matrix of $\Gamma$ is totally unimodular iff $\Gamma$ is bipartite.  
\dated{10 Nov 2008}
\codes{\CSG:~\CBal,~\CInc:~\CExp}%
\annotctd{} Thm.\ 3:  A necessary and sufficient condition that a subdeterminant is $0$ in $\incid(\Sigma)$, provided $\Sigma$ is balanced.  [\xref{Zaslavsky}{1981a}{Thomas Zaslavsky} \S8A evaluates subdeterminants for any $\Sigma$.]
\dated{20 Jan 2010}
\codes{\CSG:~\CBal,~\CInc}%
\itemspace

\bsgitem{1970a}{} Graph theory as a structural model in the social sciences.  
In: Bernard Harris, ed., \emph{Graph Theory and Its Applications}, pp.\ 1--16.  Academic Press, New York, 1970.  
\MR{0263676} (41 \#8277).  \Zbl{224.05129}.
\itemspace

\bsgitem{1971a}{}  Demiarcs:  An atomistic approach to relational systems and group dynamics.  \emph{J.\ Math.\ Sociology} 1 (1971), 195--205.
\MR{0371738} (51 \#7955).  
\annot{} Signed, oriented half edges, applied to represent interpersonal relations.
\par
\codes{\CPsS:~\CSD}%
\itemspace

\bsgitem{1979a}{}  Independent discoveries in graph theory.  
In:  Frank Harary, ed., Topics in Graph Theory (Proc.\ Conf., New York, 1977).  \emph{Ann.\ New York Acad.\ Sci.}\ 328 (1979), 1--4.  
\MR{0557880} (81a:05001).  \Zbl{465.05026}.
\itemspace

\bsgitem{1980a}{} Some theorems about graphs from social sciences.  
In: \emph{Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing} (Arcata, Calif., 1979), pp.\ 41--47.  
Congressus Numerantium, XXVI.  Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1980.  
\MR{0593895} (81m:05118).  \Zbl{442.92027}.					
\par
\codes{\CSG:~\CBal:~{\bf History},~\CExp}%
\itemspace

\bsgitem{1980b}{}  Graph theoretic models.  \emph{Theor.\ Computer Sci.}\ 11 (1980), 117--121.
\MR{0572211} (81g:68093).  \Zbl{0452.05053}.					
\annot{} Mentions the balance-detection algorithm of \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}.
\dated{15 Aug 2017} 
\codes{\CSG:~\CBal:~\CAlgor:~\CExp}%
\itemspace

\bsgitem{1981a}{}  Structural models and graph theory.  
In:  Harvey J.\ Greenberg and John S.\ Maybee, eds., \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ Sympos., Boulder, Col., 1980), pp.\ 31--58.  Discussion, pp.\ 103--111.  Academic Press, New York, 1981.  
\MR{0617930} (82g:00016) (book).  \Zbl{495.93001} (book).
\annot{} See remarks of Bixby (p.\ 111). 
\codes{\CSG,~\CVS,~\CSD:~\CBal,~\CAlgor:~\CExp}%
\itemspace

\bsgitem{1983a}{} Consistency theory is alive and well.  \emph{Personality and Social Psychology Bull.}\ 9 (1983), 60--64.  
\annot{} Historical remarks.  E.g., it was \xref{Osgood and Tannenbaum}{1955a}{Charles E.\ Osgood \& Percy H.\ Tannenbaum} that inspired Harary to study vertex signings (\xref{Beineke and Harary}{1978a}{Lowell W.\ Beineke \& Frank Harary}, \xrefd{1978b}{Lowell W.\ Beineke \& Frank Harary}).
\codes{\CPsS,~\CSG:~\CExp}%
\itemspace

\bsgitem{1985a}{}  The reconstruction conjecture for balanced signed graphs.
In: B.R.~Alspach and C.D.~Godsil, eds., \emph{Cycles in Graphs}, pp.\ 439--442. Ann.\ Discrete Math., Vol.\ 27.  
North-Holland Math.\ Stud., Vol.\ 115.  North-Holland, Amsterdam, 1985.  
\MR{0821544} (87d:05122).  \Zbl{572.05048}.
\annot{} Reconstruction from the multiset of vertex-deleted subgraphs.  $\Sigma^+$ is reconstructible if $\Sigma$ is connected and balanced and not all positive or all negative.
\codes{\CSG:~\CBal}%
\itemspace

\author{F.\ Harary \& G.\ Gupta} 					%HARARY COAUTH 
\authorspace
\bsgitem{1997a}{}  Dynamic graph models.  \emph{Math.\ Computer Modelling} 25 (1997), no.\ 7, 79--87.  
\MR{1452306} (98b:05092).  \Zbl{879.68085}.  					
\annot{} \S3.9, ``Signed graphs'', mentions that deletion index = negation index (\xref{Harary}{1959b}{Frank Harary}).
\codes{\CSG:~\CFr:~\CExp}%
\itemspace

\author{Frank Harary \& Jerald A.\ Kabell}
\authorspace
\bsgitem{1980a}{} A simple algorithm to detect balance in signed graphs.  \emph{Math.\ Social Sci.}\ 1 (1980/81), 131--136.  
\MR{0590724} (81j:05098).  \Zbl{497.05056}.
\annot{} Equivalent to switching so a spanning tree is all positive, then searching for a negative edge.  [Independently discovered by \xref{Hansen}{1978a}{Pierre Hansen}---and rather obvious by switching.]
\codes{\CSG:~\CBal,~\CAlgor}%
\itemspace

\bsgitem{1981a}{} Counting balanced signed graphs using marked graphs.  \emph{Proc.\ Edinburgh Math.\ Soc.}\ (2) 24 (1981), 99--104.  
\MR{0625282} (83a:05072).  \Zbl{476.05043}.					
\annot{} Generating functions and partially explicit formulas for connected and all isomorphism types of vertex-signed, and balanced signed, graphs of order $n$.
\dated{15 Aug 2017}
\codes{\CSG,~\CVS:~\CEnum}%
\itemspace

\author{Frank Harary \& Helene J.\ Kommel} 
\authorspace
\bsgitem{1978a}{} Matrix measures for transitivity and balance.  \emph{J.\ Math.\ Sociology} 6 (1978/79), 199--210.
\MR{0539378} (81a:05056).  \Zbl{408.05028}.					
\annot{} \S2, ``Balance measures in signed graphs'':  Balance in $\Sigma$ is measured by the proportion of positive triangles, or quadrilaterals, as computed from small powers of $A$ and $\#A$.  
\S3, ``Balance in signed digraphs'':  Similar measures for a signed digraph using digons or triangles.  
\S4, ``Example'':  A signed digraph of order 5.
\dated{10 Nov 2008, 2 Sept 2013}
\par
\codes{\CSG:~\CFr,~\CAdj}%
\itemspace

\bsgitem{1979a}{} The graphs with only self-dual signings.  \emph{Discrete Math.}\ 26 (1979), 235--241.  
\MR{0535250} (80h:05047).  \Zbl{408.05045}.						
\codes{\CSG,~\CVS:~\CAut}%
\itemspace

\author{Frank Harary, Meng-Hiot Lim, Amit Agarwal, \& Donald C.\ Wunsch}
\authorspace
\bsgitem{2004a}{}  Algorithms for derivation of structurally stable Hamiltonian signed graphs.  \emph{Int.\ J.\ Computer Math.}\ 81 (2004), no.\ 11, 1349--1356.
\MR{2172923} (no rev).  \Zbl{1065.05049}.						
\annot{} Thm.\ 1: The sizes of cuts in $K_n$.  Thm.\ 2: A subgraph of a balanced signed graph is balanced.
\dated{12 Sept 2009} 
\codes{\CSG:~\CBal}%
\itemspace

\author{Frank Harary, Meng-Hiot Lim, \& Donald C.\ Wunsch}
\authorspace
\bsgitem{2002a}{}  Signed graphs for portfolio analysis in risk management.  \emph{IMA J.\ Management Math.}\ 13 (2002), no.\ 3, 201--210.
\Zbl{1065.91025}.  
\annot{} ``Assets'' (vertices) have positive or negative correlation (edges of $K_n$).  Switching is a means of hedging risk, which is highest with all edges positive.  Imbalance indicates unpredictability; measured by the proportion of positive triangles.  [An unconvincing system.]  \S5: ``Balance analysis case study''.  
\Cf\ \xref{Vasanthi, Arumugam, \etal}{2015a}{V.\ Vasanthi, S.\ Arumugam, Atulya K.\ Nagar, \& Sovan Mitra}, 
\xref{Perina, Buckley, and Nagar}{2017a}{C.\ Perina, N.\ Buckley, \& A.K.\ Nagar}
\xref{Mansano, Allem, \etal}{20xxa}{Rafael Esteves Mansano, Luiz Emilio Allem, Renata Raposo Del-Vecchio, \& Carlos Hoppen}.
\dated{10 Sept 2009, 12 Feb 2022}
\codes{\CSG:~\CKG:~\CAppl:~\CBal,~\CSw,~\CExp}%
\itemspace

\author{Frank Harary \& Bernt Lindstr\"om} 
\authorx{Frank Harary \& Bernt Lindstrom} 
\authorspace
\bsgitem{1981a}{} On balance in signed matroids.  \emph{J.\ Combin.\ Inform.\ System.\ Sci.}\ 6 (1981), 123--128.  
\MR{0630233} (83i:05024).  \Zbl{474.05021}.					
\annot{} Thm.\ 1:  The number of balanced signings of matroid $M$ is $\leq 2^{\rk(M)}$, with equality iff $M$ is binary.  Thm.\ 3:  Minimal deletion and negation sets coincide for all signings of $M$ iff $M$ is binary.  Thm.\ 5:  For connected binary $M$, a signing is balanced iff every circuit containing a fixed point is balanced.
\codes{\CS:~\CMatrd:~\CBal,~\CFr}%
\itemspace

\author{Frank Harary, Bernt Lindstr\"om, \& Hans-Olov Zetterstr\"om} 
\authorx{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom} 
\authorspace
\bsgitem{1982a}{} On balance in group graphs.  \emph{Networks} 12 (1982), 317--321.  
\MR{0671831} (84a:\-05055).  \Zbl{496.05052}. 						
\annot{} Implicitly characterizes balance and balancing sets in a gain graph $\Phi$ by switching (proof of Thm.\ 1).  [For balance, see also \xref{Acharya and Acharya}{1986a}{B.\ Devadas Acharya \& Mukti Acharya}, \xref{Zaslavsky}{1977a}{Thomas Zaslavsky} and (\xrefd{1989a}{Thomas Zaslavsky}, Lemma 5.3.  For abelian gains, see also \xref{Gargano and Quintas}{1985a}{Michael Gargano \& Louis V.\ Quintas}.  In retrospect we can see that the characterization of balanced gains is as the $1$-coboundaries with values in a group, which for abelian groups is essentially classical.]  Thm.\ 1:  The number of balanced gain functions.  Thm.\ 2:  Any minimal deletion set is an alteration set.  Thm.\ 3:  $l(\Phi) \leq m(1 - \#\fG^{-1})$.  Thm.\ 4:  $l(\Sigma) \leq \frac 12(m - \frac{n-1}{2})$, with strict inequality if not all degrees are even.  [Compare with \xref{Akiyama, Avis, Chv\'atal, and Era}{1981a}{J.\ Akiyama, D.\ Avis, V.\ Chvatal, \& H.\ Era}, Thm.\ 1.]
\par
\codes{\CGG,~\CSG:~\Csw(\CBal),~\CEnum(\CBal),~\CFr}%
\itemspace

\author{Frank Harary, J.\ Richard Lundgren, \& John S.\ Maybee} 
\authorspace
\bsgitem{1985a}{}  On signed digraphs with all cycles negative.  \emph{Discrete Appl.\ Math.}\ 12 (1985), 155--164.  
\MR{0808456} (87g:05108).  \Zbl{586.05019}.						
\annot{} Which digraphs $D$ can be signed so that every cycle is negative?  Three types of example.  Type 1: The vertices can be numbered $1, 2, \ldots, n$ so that the downward arcs are just $(2,1), (3,2), \ldots, (n,n-1)$.  (Strong ``upper'' digraphs; Thm.\ 2.)  Type 2:  No cycle is covered by the remaining cycles (``free cyclic'' digraphs).  This type includes arc-minimal strong digraphs.  Type 3:  A symmetric digraph, iff the underlying graph $\Gamma$ is bipartite and no two points on a common circle and in the same color class are joined by a path outside the cycle (Thm.\ 10; proved by signing $\Gamma$ via \xref{Zaslavsky}{1981b}{Thomas Zaslavsky}).  [Further work in \xref{Chaty}{1988a}{Guy Chaty}.]
\codes{\CSD:~\CBal,~\CSG}%
\itemspace

\author{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} 
\authorspace
\bsgitem{1965a}{} \emph{Structural Models:  An Introduction to the Theory of Directed Graphs}.  Wiley, New York, 1965.  
\MR{0184874} (32 \#2345).  \Zbl{139.41503} (139, p.\ 415c).				
 \annot{} In Ch.\ 10, ``Acyclic digraphs'':  ``Gradable digraphs'', pp.\ 275--280.  That means a digraph whose vertices can be labelled by integers so that $f(w) = f(v) + 1$ for every arc $(v, w)$.  [Equivalently, the Hasse diagram of a graded poset.]  [Characterized by \xref{Topp and Ulatowski}{1987a}{J.\ Topp \& W.\ Ulatowski}.]
\par
 \codes{\CGD:~\Cbal,~\CExr}%
 \annotctd{} Ch.\ 13: ``Balance in structures''.  ``Criteria for balance'', pp.\ 340--346 (\cf\ \xref{Harary}{1953a}{Frank Harary}); local balance (\xref{Harary}{1955a}{Frank Harary})).  ``Measures of structural balance'', pp.\ 346--352:  ``degree of balance'' (proportion of balanced circles; \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}); ``line-index for balance'' [frustration index] (\xref{Abelson and Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg}, \xref{Harary}{1959b}{Frank Harary}).
 \annotctd{} ``Limited balance'', pp.\ 352--355.  \xref{Harary}{1955a}{Frank Harary}); also:  Adjacency matrix $A(D, \sigma)$ of a signed digraph: entries are $0, \pm1$.  The ``valency matrix'' is the \xref{Abelson--Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} adjacency matrix $R$.  Thm.\ 13.8:  Entries of $(R-pI)^l$ show the existence of (undirected) walks of length $l$ of each sign between pairs of vertices.  [Strengthened in \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, Thm.\ 2.1.]
 \annotctd{} ``Cycle-balance and path-balance'', pp.\ 355--358: here directions of arcs are taken into account.  E.g., Thm.\ 13.11:  Every cycle is positive iff each strong component is balanced as an undirected graph.
\par
\codes{\CSG:~\CBal,~\CFr,~\CAdj:~\CExp,~\CExr}\codesctd{\CSD:~\CBal,~\CExr}%
\itemspace

\bsgitem{1968a}{}  \emph{Introduction a la th\'eorie des graphes orient\'es.  Mod\`eles structuraux}.  Dunod, Paris, 1968.  
\Zbl{176.22501} (176, p.\ 225a).  
\annot{} French edition of \xrefd{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright}.
\codes{\CGD:~\Cbal,~\CExr}%
\par
\codes{\CSG:~\CBal,~\CFr,~\CAdj:~\CExp,~\CExr}\codesctd{\CSD:~\CBal,~\CExr}%
\itemspace

\author{Frank Harary \& Edgar M.\ Palmer} 
\authorspace
\bsgitem{1967a}{} On the number of balanced signed graphs.  \emph{Bull.\ Math.\ Biophysics} 29 (1967), 759--765.  
\Zbl{161.20904} (161, p.\ 209d).  
\codes{\CSG:~\CBal:~\CEnum}%
\itemspace

\bsgitem{1973a}{}  \emph{Graphical Enumeration}.  Academic Press, New York, 1973.
\MR{0357214} (50 \#9682).  \Zbl{266.05108}.
\annot{} Four exercises and a remark concern signed graphs, balanced signed graphs, and signed trees.
\codes{\CSG:~\CEnum,~\CBal}%
\itemspace

\bsgitem{1977a}{}  (As ``F.\ Kharari and \`E.\ Palmer'')  \emph{Perechislenie grafov}.  ``Mir'', Moscow, 1977.  
\MR{0447038} (56 \#5353).  
\annot{} Russian translation of \xrefd{1973a}{Frank Harary \& Edgar M.\ Palmer}.
\codes{\CSG:~\CEnum,~\CBal}%
\itemspace

\author{Frank Harary, Edgar M.\ Palmer, Robert W.\ Robinson, \& Allen J.\ Schwenk} 
\authorspace
\bsgitem{1977a}{} Enumeration of graphs with signed points and lines.  \emph{J.\ Graph Theory} 1 (1977), 295--308.  
\MR{0465932} (57 \#5818).  \Zbl{379.05035}.					
\annot{} See \xref{Bender and Canfield}{1983a}{Edward A.\ Bender \& E.\ Rodney Canfield}.
\codes{\CSG,~\CVS:~\CEnum}%
\itemspace

\author{Frank Harary \& Michael Plantholt} 
\authorspace
\bsgitem{1983a}{}  The derived signed graph of a digraph.  \emph{Expositiones Math.}\ 1 (1983), no.\ 4, 343--347.
\MR{0782975} (86h:05056).  \Zbl{525.05030}.					
\annot{$L_{HP}$} A digraph $D$ gives a signed line graph $\Lambda_{HP}(D)$ with $V_{HP} := E(D)$ and edges $+ef$ if $e,f$ have the same head, $-ef$ if $e,f$ have the same tail.  [The negative part of $\Lambda(+D)$ in \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky} with extraverted edges made positive and introverted edges negative.]
\dated{4 Sept 2010, 17 Jan 2012}
\codes{\CSG:~\CLG,~\CBal}%
\itemspace

\author{Frank Harary \& Geert Prins} 
\authorspace
\bsgitem{1959a}{}  The number of homeomorphically irreducible trees, and other species.  \emph{Acta Math.}\ 101 (1959), 141--162.  
\MR{0101846} (21 \#653).  \Zbl{084.19304} (84, p.\ 193d).				
\par
\codes{\CSG:~\CEnum}%
\itemspace

\author{Frank Harary \& Robert W.\ Robinson} 
\authorspace
\bsgitem{1977a}{} Exposition of the enumeration of point-line-signed graphs enjoying various dualities.  
In:  R.C.\ Read and C.C.\ Cadogan, eds., \emph{Proceedings of the Second Carribean Conference in Combinatorics and Computing} (Cave Hill, Barbados, 1977), pp.\ 19--33.  
Dept.\ of Math., University of the West Indies, Cave Hill, Barbados, 1977.  
\annot{} Counts signed trees.
\dated{28 May 2017}
\codes{\CSG,~\CVS:~\CEnum}%
\itemspace

\author{Frank Harary \& Bruce Sagan} 
\authorspace
\bsgitem{1984a}{} Signed posets.  
In: \emph{Calcutta Mathematical Society Diamond-cum-Platinum Jubilee Commemoration Volume (1908--1983)}, Part I, pp.\ 3--10.  Calcutta Math.\ Soc., Calcutta, 1984.  
\MR{0845035} (87k:06003).  \Zbl{588.05048}.						
\annot{} A signed poset is a (finite) partially ordered set $P$ whose M\"obius function takes on only values in $\{0, \pm 1\}$.  $S(P)$ is the signed graph with $V = P$ and $E_{\epsilon} = \{xy : x \leq y \text{ and } \mu(x,y) = \epsilon 1$\} for $\epsilon = +, -$.  Some examples are chains, tree posets, and any product of signed posets.  Thm.\ 1 characterizes $P$ such that $\#S(P) \cong H(P)$, the Hasse diagram of $P$.  Thm.\ 3 characterizes posets for which $S(P)$ is balanced.  Thm.\ 4 gives a sufficient condition for clusterability of $S(P)$.  There are many unanswered questions, most basically  \Question\ 1.  Which signed graphs have the form $S(P)$?  [See \xref{Zelinka}{1988a}{Bohdan Zelinka} for a partial answer.]
\codes{\CSG,~\CS}%
\itemspace

\author{Frank Harary \& Marcello Truzzi} 
\authorspace
\bsgitem{1979a}{}  The graph of the zodiac:  On the persistence of the quasi-scientific paradigm of astrology.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 4 (1979), 147--160.
\MR{0564189} (82e:00004) (\qv).  
\codes{\CSG:~\CBal}%
\itemspace

\author{Katsumi Harashima}						%HARASHIMA
\authorspace
\authref  See \hyperlink{Hideo Kosako}{H.~Kosako}.
\itemspace

\author{E.\ Harburg}							%HARBURG
\authorspace
\authref  See \hyperlink{K.O.\ Price}{K.O.~Price}.
\itemspace

\author{Mela Hardin}						%HARDIN
\authorspace
\authref  See \hyperlink{Matthias Beck}{M.~Beck}.
\itemspace

\author{Nadia Hardy}						%HARDY
\authorspace
\authref  See \hyperlink{Samuel Fiorini}{S.~Fiorini}.
\itemspace

\author{Tero Harju}							%HARJU
\authorspace
\authref  See also \hyperlink{Andrzej Ehrenfeucht}{A.~Ehrenfeucht} and \hyperlink{Jurriaan Hage}{J.~Hage}.
\authrefspace

\bsgitem{2004a}{}  Tutorial on DNA computing and graph transformation.  
In: H.\ Ehrig \etal, eds., \emph{ICGT 2004}, pp.\ 434--436.  \LNCS 3256.  Springer, Berlin, 2004.
\annot{} [Vertex-]signed overlap graphs mentioned on p.\ 436.
\dated{6 Feb 2011} 
\codes{\CVS:~\CAlgor,~\CAppl}%
\itemspace

\bsgitem{2005a}{}  Combinatorial models of gene assembly.  
In:  S.B.\ Cooper, B.\ L\"owe, and L.\ Torenvliet, eds., \emph{New Computational Paradigms} (First Conf.\ Computability in Europe, CiE 2005, Amsterdam, 2005), pp.\ 188--195.  \LNCS 3526.  Springer, Berlin, 2005.
\Zbl{1113.68400}.  
\annot{} A vertex-signed graph (called a ``signed graph'') encodes the overlap of signed permutations (pp.\ 190ff.).  
\dated{6 Feb 2011} 
\par
\codes{\CVS:~\CAlgor,~\CAppl}%
\itemspace

\author{Tero Harju, Chang Li, \& Ion Petre}
\authorspace
\bsgitem{2007a}{}  Examples on the parallel complexity of signed graphs.  TUCS Tech.\ Rep.\ No.\ 811, Turku Centre for Computer Science, 20 pp.  Turku, Finland, 2007.
\par
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2008a}{}  Graph theoretic approach to parallel gene assembly.  \emph{Discrete Appl.\ Math.}\ 156 (2008), no.\ 18, 3416--3429. 
\MR{2467313} (2010c:92075).  \Zbl{1200.05238}.  
\annot{} See \xref{Harju, Li, Petre, and Rozenberg}{2005a}{Tero Harju, Chang Li, Ion Petre, \& Gregorz Rozenberg}.  The ``parallel complexity'' of a vertex-signed graph is the minimum number of operations required to reduce it to $\eset$.  The value for an all-positive or all-negative tree is low ($\leq$ 3 and 2, resp.).  \Conjecture.  That of an all-negative graph is $\leq$ 3.
\par
\codes{\CVS,~\CAppl}%
\itemspace

\bsgitem{2008b}{}  Parallel complexity of signed graphs for gene assembly in ciliates.  \emph{Soft Computing} 12 (2008), 731--737.
\Zbl{1137.92305}.  
\annot{} See \xrefd{2008a}{Tero Harju, Chang Li, \& Ion Petre}.  The parallel complexity of various examples, e.g., complete tripartite graphs with constant sign (complexity $\leq 3$), and an all-positive circle with two negative leaves hanging off each circle vertex (complexity $\leq$ 4 or 5).
\codes{\CVS,~\CAppl}%
\itemspace

\author{Tero Harju, Chang Li, Ion Petre, \& Gregorz Rozenberg}
\authorspace
\bsgitem{2005a}{}  Parallelism in gene assembly.  
In:  C.\ Ferretti, G.\ Mauri, and C.\ Zandron, eds., \emph{DNA Computing} (Proc.\ 10th Int.\ Workshop on DNA Computing, DNA10, Milan, 2004), pp.\ 138--148.  \LNCS 3384.
Springer, Berlin, 2005.
\MR{2179032} (no rev).  \Zbl{1116.68454}.  
\annot{} The signs are on vertices.  An operation is ``local complementation'' of a vertex $v$: in the neighborhood $N(v)$, negate the vertices and complement the edges.  Molecular operations formalized for vertex-signed graphs are: (1) deletion of an isolated negative vertex, (2) local complementation of a positive vertex, then deletion of the vertex, (3) a complementation in the neighborhood of two adjacent negative vertices $v,w$: complement in $N(v) \cup N(w)$, then complement in $N(v) \cap N(w)$.  (The paper has a misprint.)  The objective is to reduce the graph to $\eset$ by these operations, if possible.  One consideration is when operations can be performed ``in parallel'', i.e., independently of order of operations.
\codes{\CVS,~\CAppl}%
\itemspace

\bsgitem{2007a}{}  Complexity measures for gene assembly.  
In:  K.\ Tuyls \etal, eds., \emph{Knowledge Discovery and Emergent Complexity in Bioinformatics} (First Int.\ Workshop, KDECB 2006, Ghent, 2006), pp.\ 42--60.  Lect.\ Notes in Bioinformatics.  \LNCS 4366.  Springer, Berlin, 2007.
\annot{} \S7, ``Fourth complexity measure: Parallelism'':  A definition of parallelism in terms of  applying rules (operations) to vertex-signed graphs.
\dated{6 Feb 2011} 
\codes{\CVS:~\CAlgor}%
\itemspace

\bsgitem{2006a}{}  Parallelism in gene assembly.  \emph{Nat.\ Computing} 5 (2006), no.\ 2, 203--223.
\MR{2259034} (2007h:68043).  \Zbl{1114.68043}.  
\annot{} See \xrefd{2005a}{Tero Harju, Chang Li, Ion Petre, \& Gregorz Rozenberg}.
\codes{\CVS,~\CAppl}%
\itemspace

\author{Tero Harju, Ion Petre, \& Gregorz Rozenberg}
\authorspace
\bsgitem{2004a}{}  Tutorial on DNA computing and graph transformation.  
In:  H.\ Ehrig \etal, eds., \emph{ICGT 2004}, pp.\ 434--436.  \LNCS 3256.  Springer, Berlin, 2004.
\annot{} See \xref{Harju, Li, Petre, and Rozenberg}{2005a}{Tero Harju, Chang Li, Ion Petre, \& Gregorz Rozenberg} \etal
\codes{\CVS,~\CAppl:~\CExp}%
\itemspace

\author{Pierre de la Harpe}						%HARPE
\authorspace
\bsgitem{1994a}{}  Spin models for link polynomials, strongly regular graphs and Jaeger's Higman-Sims model.  \emph{Pacific J.\ Math.}\ 162 (1994),  no.\ 1, 57--96.
\MR{1247144} (94m:\-57014).  \Zbl{795.57002}.  
\codes{\CSGc:~\CKnot,~\CInvar}%
\itemspace

\authorx{David Harries}		 				%HARRIES
\author{David Harries \& Hans Liebeck}
\authorspace
\bsgitem{1978a}{}  Isomorphisms in switching classes of graphs.  \emph{J.\ Austral.\ Math.\ Soc.\ (A)} 26 (1978), 475--486.  
\MR{0520101} (80a:05109).  \Zbl{411.05044}.					
\annot{} Given $\Sigma = (K_n, \sigma)$ and an automorphism group $\fA$ of the switching class $[\Sigma]$, is $\fA$ ``exposable'' on $[\Sigma]$ (does it fix a representative of $[\Sigma]$)?  General techniques and a solution for the dihedral group.  Done in terms of Seidel switching of unsigned simple graphs.  (A further development from \xref{Mallows and Sloane}{1975a}{C.L.\ Mallows \& N.J.A.\ Sloane}.  [Related work in \xref{M.\ Liebeck}{1982a}{Martin W.\ Liebeck} and \xref{Cameron}{1977a}{Peter J.\ Cameron}.])
\codes{\Ckg:~\Csw,~\CTG:~\CAut}%
\itemspace

\author{Matthew Hartley}						%HARTLEY
\authorspace
\authref  See \hyperlink{Georg R.\ Walther}{G.R.~Walther}.
\itemspace

\author{Alexander K.\ Hartmann}					%HARTMANN
\authorspace
\authref  See also \hyperlink{C.\ Amoruso}{C.~Amoruso}, \hyperlink{Guy Hed}{G.~Hed}, \hyperlink{O.\ Melchert}{O.~Melchert}, and \hyperlink{Martin Pelikan}{M.~Pelikan}.
\authrefspace

\bsgitem{1998a}{}  Are ground states of 3d $\pm J$ spin glasses ultrametric?  \emph{Europhys.\ Lett.}\ 44 (1998), no.\ 2, 249--254.
\annot{} The distances of ground states in a signed cubic lattice with side $L$, measured by overlap, tested on many samples with $L\leq14$ for the appearance of approaching ultrametricity in the infinite limit of $L$.  There is such an appearance.  $L$ is too small for quantitative statements.  [Ultrametricity is a strong property that has been conjectured by Parisi \etal\  It is disavowed in \xref{Hed, Hartmann, Stauffer, and Domany}{2001a}{Guy Hed, Alexander K.\ Hartmann, Dietrich Stauffer, \& Eytan Domany}.]
Dictionary: \cf\ \xrefd{2000a}{Alexander K.\ Hartmann}.
\dated{11 Jan 2015} 
\codes{\CSG,~\CPhys:~\CFr:~\CState}%
\itemspace

\bsgitem{1999a}{}  Ground-state landscape of 2d $\pm J$ Ising spin glasses.  \emph{European Phys.\ J.\ B} 8 (1999), 619--626.
\annot{} Examples of signed square lattice graphs.  Evidence is against ultrametricity of ground states, contrary to prior findings in higher dimensions.  
Dictionary: \cf\ \xrefd{2000a}{Alexander K.\ Hartmann}.
\dated{10 Jan 2015} 
\par
{\codes{\CSG:~\CFr:~\CState:~\CAlgor,~\CPhys}\codesctd{\CAlgor:~\CExp}}%
\itemspace

\bsgitem{1999b}{}  Ground-state behavior of the three-dimensional $\pm J$ random-bond Ising model.  \emph{Phys.\ Rev.\ B} 59 (1999), no.\ 5, 3617--3623.
\annot{} Examples of signed cubic lattice graphs for varying concentrations $p := \#E^-/\#E$ of negative edges: ground states, frustration index (``ground state energy''), average overlap, etc.
Dictionary: \cf\ \xrefd{2000a}{Alexander K.\ Hartmann}.
\dated{10 Jan 2015} 
\codes{\CSG:~\CFr,~\CState:~\CAlgor,~\CPhys}%
\itemspace

\bsgitem{1999c}{}  Scaling of stiffness energy for three-dimensional $\pm J$ Ising spin glasses.  \emph{Phys.\ Rev.\ E} 59 (1999), no.\ 1, 84--87.
\annot{} ``Stiffness'' = ground-state domain wall energy.  The graph is a cubic lattice.  Many ground states are generated and compared.
	
\par
\codes{\CSG,~\CPhys:~\CFr:~\CState,~\CAlgor}%
\itemspace

\bsgitem{2000a}{\dag}  Ground-state clusters of two-, three-, and four-dimensional $\pm J$ Ising spin glasses.  \emph{Phys.\ Rev.\ E} 63 (2000), art.\ 016106, 7 pp.
\annot{} The ground states in examples of signed square, cubic, and tesseractic (4-hypercubic) lattices are found to fall into relatively few clusters.  An algorithmic method called ``ballistic search'' permits larger conclusions from smaller numbers of states.
	
Dictionary:  ``ground state'' = switching with fewest negative edges, ``ground state energy'' = $l(\Sigma)$, ``ground state graph'' has ground states $\zeta$ for vertices and an edge between ground states that differ by switching a vertex (necessarily having $d^+=d^-$), ``cluster'' = component of ground-state graph, ``overlap'' of states $q(\zeta,\zeta') := n\inv [\#(\zeta\zeta')\inv(+) - \#(\zeta\zeta')\inv(-)] = (1/n)\cdot$[number of vertices of agreement $-$ number of vertices of disagreement].
\dated{10 Jan 2015} 
\codes{\CSG:~\CFr:~\CState:~\CAlgor,~\CPhys}%
\itemspace

\bsgitem{2008a}{}  Droplets in the two-dimensional $\pm J$ Ising spin glass.  \emph{Phys.\ Rev.\ B} 77 (2008), art.\ 144418, 5 pp.
\codes{\CSG:~\CFr:~\CState:~\CAlgor,~\CPhys}%
\itemspace

\bsgitem{2011a}{}  Ground states of two-dimensional Ising spin glasses: Fast algorithms, recent developments and a ferromagnet-spin glass mixture.  \emph{J.\ Stat.\ Phys.}\ 144 (2011), 519--540.
\MR{2826631} (2012g:82059).  \Zbl{1227.82086}.  
\annot{} Review of the ground state problem, methods, and conclusions.
\par
\codes{\CSG,~\CPhys:~\CFr:~\CState,~\CAlgor:~\CExp,~\CRef}%
\itemspace

\author{Alexander K.\ Hartmann \& Federico Ricci-Tersenghi}
\authorspace
\bsgitem{2002a}{}  Direct sampling of complex landscapes at low temperatures: The three-dimen\-sion\-al $\pm J$ Ising spin glass.  \emph{Phys.\ Rev.\ E} 66 (2002), art.\ 224419, 8 pp.
\annot{} The state landscape appears to be more complex at small positive temperatures than at zero temperature.
\codes{\CSG:~\CFr:~\CState:~\CAlgor,~\CPhys}%
\itemspace

\author{Alexander K.\ Hartmann \& Heiko Rieger}
\authorspace
\bsgitem{2002a}{}  \emph{Optimization Algorithms in Physics}.  WILEY-VCH, Berlin, 2002.
\MR{1881155} (2004b:00006).  \Zbl{1003.82002}.  
\codes{\CPhys:~\Csg:~\CFr:~\CAlgor}%
\itemspace

\author{Alexander K.\ Hartmann \& Heiko Rieger, eds.}
\authorspace
\bsgitem{2004a}{}  \emph{New Optimization Algorithms in Physics}.  WILEY-VCH, Weinheim, 2004.
\MR{} 2089307 (2005d:90004).  \Zbl{1054.90002}.  
\codes{\CPhys:~\Csg:~\CFr:~\CAlgor}%
\itemspace

\author{Alexander K.\ Hartmann \& Martin Weigt}
\authorspace
\bsgitem{2005a}{}  \emph{Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics}.  Wiley-VCH, Weinheim, Germany, 2005.
\MR{2293999} (2009b:82028).  \Zbl{1094.82002}.  
\annot{} ``Example:  Ising spin glasses'': Frustration index of signed graphs on p.\ 6.  \S11.7, ``Matchings and spin glasses'':  Outlines the matching theory method (\cf\ \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai} and \xref{Barahona}{1982a}{Francisco Barahona}) for planar graphs, for calculating $l(\Sigma)$ and locating ground states (switchings with fewest negative edges).  Also, locating interesting excited states (states with more than the fewest unsatisfied edges), specifically, domain walls and droplets.  A ``domain'' is generated by negating signs of a set of edges; the vertices whose spins remain the same form one domain and the complement is the other.  The increased energy (the ``domain wall energy'') has thermodynamic implications.  [How to choose the negation set, and what can be the shapes of domain walls, are not obvious.]  A ``droplet'' in a state $s$, vis-\'a-vis a ground state $s_0$, is a component of the subgraph induced by $(ss_0)\inv(-1)$.  The sizes of droplets appear to have consequences for thermodynamics.
\dated{24 Aug 2012} 
\par
\codes{\CSG:~\CWG,~\CFr,~\CState:~\CPhys,~\CAlgor:~\CExp,~\CRef}%
\itemspace

\author{A.K.\ Hartmann \& A.P.\ Young}
\authorspace
\bsgitem{2001a}{}  Lower critical dimension of Ising spin glasses.  \emph{Phys.\ Rev.\ B} 64 (2001), art.\ 180404, 4 pp.
\annot{} For unweighted ($\pm J$) and randomly weighted (Gaussian) signed graphs, ground states are computed and compared.  The lower critical dimension is different in the two types.
\dated{22 Jan 2015} 
\par
\codes{\CPhys,~\CSG:~\CFr,~\CState}%
\itemspace

\authorx{Nora Hartsfield}		 				%HARTSFIELD
\author{Nora Hartsfield \& Gerhard Ringel}
\authorspace
\bsgitem{1989a}{} Minimal quadrangulations of nonorientable surfaces.  \emph{J.\ Combin.\ Theory Ser.\ A} 50 (1989), 186--195.  
\MR{0989193} (90j:57003).  \Zbl{665.51007}.
\annot{} ``Cascades'': see \xref{Youngs}{1968a}{J.W.T.\ Youngs}.  
\codes{\Csg:~\COri:~\CAppl}%
\itemspace

\authorx{Forough Hassanibesheli}					%HASSANIBESHELI
\author{Forough Hassanibesheli, Leila Hedayatifar, Przemys{\l}aw Gawro\'nski, Maria Stojkow, Dorota \.Zuchowska-Skiba, \& Krzysztof Ku{\l}akowski}
\authorspace
\bsgitem{2017a}{}  Gain and loss of esteem, direct reciprocity and Heider balance.  \emph{Physica A} 468 (2017), 334--339.
\MR{3580677} (no rev).  \Zbl{1400.91423}.  \arXiv{1609.03358}.			
\codes{\CSD:~\CDyn,~\CBal,~\CFr}%
\itemspace

\authorx{Keivan Hassani Monfared}					%HASSANI MONFARED
\author{K.\ Hassani Monfared, G.\ MacGillivray, D.D.\ Olesky, \& P.\ van den Driessche}
\authorspace
\bsgitem{2019a}{}  Inertias of Laplacian matrices of weighted signed graphs.  \emph{Special Matrices} 7 (2019), 327--342. 
\MR{4045536}.  \Zbl{1431.05101}.  \arXiv{1912.03844}.				
\annot{} 
Dictionary: ``flexibility'' $\tau(\Gamma,\sigma) = r(E^+)+r(e^-)-r(E)$, $r$ = rank in $\bfF(\Gamma)$.  [Hence related to connectivity of $\bfF(\Gamma)$ via sign partition of $E$.]
	
\dated{3 Sept 2019} 
\codes{\CSG,~\CWG:~\CLap:~\CEig,~\Cmatrd}%
\itemspace

\author{Keivan Hassani Monfared \& Sudipta Mallik}
\authorspace
\bsgitem{2019a}{}  An analog of matrix tree theorem for signless Laplacians.  \emph{Linear Algebra Appl.}\  560 (2019), 43--55.
\MR{3866544}.  \Zbl{1402.05139}.  \arXiv{1805.04759}.  
\annot{} Matrix tree theorem is special case $-\Gamma$ of \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Thm.\ 8A.4.  New: \# negative circles in $-\Gamma$ is $\leq \det\Lap(-\Gamma)/4$, with $=$ iff $-\Gamma$ is balanced or unicyclic.  [\Problem.  Generalize to signed graphs.]
\dated{13 Oct 2022} 
\codes{\Csg:~\CPar:~\CLap}%
\itemspace

\author{Kurt H\"assig} 						%HASSIG
\authorx{Kurt Hassig}
\authorspace
\bsgitem{1975a}{}  Theorie verallgemeinerter Fl\"usse und Potentiale.  
In: \emph{Siebente Oberwolfach-Tagung uber Operations Research} (1974), pp.\ 85--98.  Operations Research Verfahren, Band XXI.  A.~Hain, Meisenheim am Glan, 1975.  
\MR{0450137} (56 \#8434).  \Zbl{358.90070}.
\codes{\CGN:~\CInc}%
\itemspace

\bsgitem{1979a}{}  \emph{Graphentheoretische Methoden des Operations Research}.  Leitfaden der angew.\ Math.\ und Mechanik, 42.  B.G.~Teubner, Stuttgart, 1979.  
\MR{0528758} (80f:\-90002).  \Zbl{397.90061}.  
\annot{} Ch.\ 5: ``Verallgemeinerte Fluss- und Potentialdifferenzen-probleme.''  The lift matroid arises from a side condition, i.e., extra row, added to the incidence matrix of the graph.  [The side condition is expressed graphically by additive real gains.]
\codes{\CGN:~\CInc,~\CMatrd,~\CBal:~\CExp,~\CRef}%
\itemspace

\author{Refael Hassin} 						%HASSIN
\authorspace
\bsgitem{1981a}{} Generalizations of Hoffman's existence theorem for circulations.  \emph{Networks}  11 (1981), 243--254.  
\MR{0636230} (83c:90055).  \Zbl{459.90026}.
\codes{\CGN}%
\itemspace

\author{O.\ Hatami}							%HATAMI O
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{P.\ Hatami}							%HATAMI P
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.

\author{Naomichi Hatano}						%HATANO
\authorspace
\authref  See also \hyperlink{Ernesto Estrada}{E.~Estrada}.
\authrefspace

\author{Naomichi Hatano \& James E.\ Gubernatis}
\authorspace
\bsgitem{2000a}{}  A bivariate multicanonical Monte Carlo of the 3D $\pm J$ spin glass.  In:  David P.\ Landau \etal, eds., \emph{Computer Simulation Studies in Condensed-Matter Physics XII} (Proc.\ Twelfth Workshop, Athens, Ga., 1999), pp.\ 149--161.  Springer Proc.\ Phys., Vol.\ 85.  Springer, Berlin, 2000.
\annot{} Similar to \xref{Hatano and Gubernatis}{2002a}{Naomichi Hatano \& James E.\ Gubernatis}.
\dated{28 Mar 2013}
\par
\codes{\CPhys,~\Csg:~\CState(\Cfr)}%
�J Ising spin glass is described. The autocorrelation time is approximately proportional to the system size, which is a great improvement over previous spin-glass simulations. The Binder plot indicates the critical temperature T_c ? 1.3. The order-parameter distribution function P(q) exhibits a feature of the droplet picture of the low-temperature phase.
\itemspace

\bsgitem{2002a}{}  Evidence for the double degeneracy of the ground-state in the three-dimensional $\pm J$ spin glass.  \emph{Phys.\ Rev.\ B} 66 (2002), art.\ 054437, 14 pp.
\arXiv{cond-mat/0008115}.  
\annot{} A Monte-Carlo investigation of infinite signed cubic lattice graphs at zero temperature, by means of large finite cubic lattices.  Are there only two ground states, one the negative of the other, or are there many, unrelated ground states?  The paper supports the former.  
\Cf\ \xref{Hatano and Gubernatis}{2002b}{Naomichi Hatano \& James E.\ Gubernatis}.  
[Having only one ground state (up to sign reversal) means there is only one switching that minimizes $\#E^-$.  (\Conjecture.  That is not true of any signed graph, finite or infinite, except for very special, very regular graphs and signatures.)  However, zero temperature may distort the normal behavior of a signed graph.]
\dated{28 Mar 2013} 
\codes{\CPhys,~\Csg:~\CState(\Cfr)}%
\itemspace

\bsgitem{2002b}{}  Double degeneracy in the ground state of the 3D $\pm J$ spin glass.  \emph{Comput.\ Phys.\ Commun.}\ 147 (2002), no.\ 1-2, 414--418.
\Zbl{994.82557}.  
\annot{} Reply to criticism of \xref{Hatano and Gubernatis}{2002a}{Naomichi Hatano \& James E.\ Gubernatis}.
\dated{28 Mar 2013} 
\codes{\CPhys,~\Csg:~\CState(\Cfr)}%
\itemspace

\author{D.M.\ Hatch}							%HATCH
\authorspace
\authref  See \hyperlink{S.T.\ Chui}{S.T.~Chui}.
\itemspace

\authorx{Emilie Haynsworth}						%HAYNSWORTH
\author{Emilie Haynsworth \& A.J.\ Hoffman}
\authorspace
\bsgitem{1969a}{}  Two remarks on copositive matrices.  \emph{Linear Algebra Appl.}\ 2 (1969), 387--392. 
\MR{0248157} (40 \#1411).  \Zbl{185.08004}.  
\annot{}  Matrix $A(K_n,\sigma)+I$: properties as quadratic form.  Thm.:  It is copositive iff $(K_n,\sigma)$ is balanced.  \Cf\ \xref{Hoffman and Pereira}{1973a}{Alan J.\ Hoffman \& Francisco Pereira}.  
\dated{28 May 2017}
\codes{\Csg:~\Ckg:~\Cadj}%
\itemspace

\authorx{Bian He}							%HE BA
\author{Bian He, Ya-Lei Jin, \& Xiao-Dong Zhang}
\authorspace
\bsgitem{2013a}{}  Sharp bounds for the signless Laplacian spectral radius in terms of clique number.  \emph{Linear Algebra Appl.}\ 438 (2013), no.\ 10, 3851--3861.
\MR{3034503}.  \Zbl{1282.05119}.  \arXiv{1209.3214}.				
\annot{} \S4:  The incidence energy (derived from $Q := \Lap(-\Gamma)$) has a bound like that in Thm.\ 4.5.
\dated{21 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Chang-Xiang He}						%HE CX
\authorspace
\authref  See also \hyperlink{Peikang Zhang}{P.-K.~Zhang}.
\itemspace

\author{Chang-Xiang He \& Min Zhou}
\authorspace
\bsgitem{2014a}{}  A sharp upper bound on the least signless Laplacian eigenvalue using domination number.  \emph{Graphs Combin.}\ 30 (2014), 1183--1192.
\MR{3248498}.  \Zbl{1298.05203}.  
\par
\codes{\Cpar:~\CLap:~\CEig:~\Cdom}%
\itemspace

\author{Jin-Ling He}							%HE JL
\authorspace
\authref  See \hyperlink{Jia-Yu Shao}{J.-Y.~Shao}.
\itemspace

\authorx{Shengjie He}						%HE SJ
\author{Shengjie He \& Rong-Xia Hao}
\authorspace
\bsgitem{20xxa}{}  The relation between the independence number and rank of a signed graph.  Submitted.  
\arXiv{1907.07837}.  
\codes{\CSG:~\CAdj}%
\itemspace

\author{Shengjie He, Rong-Xia Hao, \& Fengming Dong}
\authorspace
\bsgitem{2020a}{}  The rank of a complex unit gain graph in terms of the matching number.  \emph{Linear Algebra Appl.}\ 589 (2020), 158--185.
\MR{4046874}.  \Zbl{1437.05141}.  \arXiv{1909.07555}.  
\annot{} Bounds on $\rk A(\Phi)$ in terms of matching number $\mu$ and cyclomatic number $\xi$.  Thm.\ 1.10: $2\mu-2\xi \leq r \leq 2\mu+\xi$.  Thms.\ 1.11--12:  Equality characterized.  Thm.\ 1.13:  Better bounds in terms of edge bipartiticity and acyclic induced subgraphs.
\dated{2 Jan 2020} 
\codes{\CGG:~\CAdj}%
\itemspace

\author{Shengjie He, Rong-Xia Hao, \& Hong-Jian Lai}
\authorspace
\bsgitem{2019a}{}  Bounds for the matching number and cyclomatic number of a signed graph in terms of rank.  \emph{Linear Algebra Appl.}\ 572 (2019), 273--291.
\MR{3926237}.  \Zbl{1411.05107}.  
\annot{} Bounds on rank: $2\mu-2\xi \leq \rk A(\Sigma) \leq 2\mu+2\xi$.  Upper and lower bounds are attained by loose cacti with certain circle signs.  [\Cf\ \xref{He, Hao, \& Dong}{2020a}{Shengjie He, Rong-Xia Hao, \& Fengming Dong}.]
\dated{17 Jan 2019} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{Shengjie He, Rong-Xia Hao, \& Aimei Yu}
\authorspace
\bsgitem{2022a}{}  Bounds for the rank of a complex unit gain graph in terms of the independence number.  \emph{Linear Multilinear Algebra} 70 (2022), no.\ 7, 1382--1402. 
\MR{4413130}.  \Zbl{1487.05162}.  \arXiv{1909.08533}.  
\annot{} Rank of $A(\Phi)$.
\codes{\CGG:~\CAdj}%
\itemspace

\bsgitem{20xxa}{}  On the inertia index of a mixed graph with the matching number.  Submitted.  
\arXiv{1909.07146}.  
\codes{\Cgg:~\CAdj}%
\itemspace

\authorx{Shushan He}						%HE SS
\author{Shushan He \& Shuchao Li}
\authorspace
\bsgitem{2012a}{}  On the signless Laplacian index of unicyclic graphs with fixed diameter.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 1, 252--261.
\MR{2859926} (2012j:05257).  \Zbl{1229.05201}.  
\codes{\Csg:~\Cpar:~\CEig}%
\itemspace

\author{Xiaocong He}						%HE XC
\authorspace
\authref  See \hyperlink{Shahid Zaman}{S.~Zaman}.
\itemspace

\author{Xiping He}							%HE XP
\authorspace
\authref  See \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\author{Patrick Headley}						%HEADLEY
\authorspace
\bsgitem{1997a}{}  On a family of hyperplane arrangements related to the affine Weyl groups.  \emph{J.\ Algebraic Combin.}\ 6 (1997), 331--338.  
\MR{1471893} (98e:52010).  \Zbl{911.52009}.					
\annot{} The characteristic polynomials of the Shi hyperplane arrangements $\cS(W)$ of type $W$ for each Weyl group $W$, evaluated computationally.  $\cS(W)$ is obtained by splitting the reflection hyperplanes of $W$ in two in a certain way; thus $\cS(A_{n-1})$ splits the arrangement representing $\Lat \bfF(K_n)$---more precisely, it represents $\Latb \{0,1\}\vec K_n$; that of type $B_n$ splits the arrangement representing $\Lat \bfF(\pm K_n\full)$, and so on.  [\Cf\ \xref{Athanasiadis}{1996a}{Christos A.\ Athanasiadis}.]
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar}%
\itemspace

\authorx{Brian Healy}						%HEALY
\author{Brian Healy \& Arthur Stein}
\authorspace
\bsgitem{1973a}{}  The balance of power in international history: Theory and reality.  \emph{J.\ Conflict Resolution} 17 (1973), no.\ 1, 33--61.
\annot{} Describes balance (incorrectly) and clusterability of a signed graph; examines the relevance of, \emph{i.a.}, signed-graphic balance.
\dated{9 Jun 2012} 
\codes{\CPsS;~\CSG:~\CBal,~\CClu:~\CExp}%
\itemspace

\author{Robert W.\ Heath Jr.}						%HEATH
\authorspace
\authref  See \hyperlink{Thomas Strohmer}{T.~Strohmer}.
\itemspace

\author{Saroja R.\ Hebbar}						%HEBBAR
\authorspace
\authref  See \hyperlink{Prabhakar Ramrao Hampiholi}{P.R.~Hampiholi}.
\itemspace

\authorx{Guy Hed}							%HED
\author{Guy Hed, Alexander K.\ Hartmann, Dietrich Stauffer, \& Eytan Domany}
\authorspace
\bsgitem{2001a}{}  Spin domains generate hierarchical ground state structure in $J=\pm1$ spin glasses.  \emph{Phys.\ Rev.\ Lett.}\ 86 (2001), no.\ 14, 3148--3151.
\annot{} Proposes an intermediate structure of ground states (switchings with smallest $\#E^-$) of a signed graph (``Ising spin glass'' with edge weights $\pm1$), not ultrametric (\cf\ \xref{Hartmann}{1998a}{Alexander K.\ Hartmann}) but ``hierarchical''.
\dated{11 Jan 2015} 
\codes{\CSG:~\CState(\Cfr),~\CPhys}%
\itemspace

\author{Leila Hedayatifar}						%HEDAYATIFAR
\authorspace
\authref  See \hyperlink{Forough Hassanibesheli}{F.~Hassanibesheli}.
\itemspace

\author{Rajneesh Hegde}						%HEGDE R
\authorspace
\authref  See \hyperlink{Anke van Zuylen}{A.~van~Zuylen}.
\itemspace

\author{Pinar Heggernes}						%HEGGERNES
\authorspace
\authref  See \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender}.
\itemspace

\author{Fritz Heider} 							%HEIDER
\authorspace
\bsgitem{1946a}{} Attitudes and cognitive organization.  \emph{J.\ Psychology} 21 (1946), 107--112.  
\annot{} No mathematics, but a formative article.  Precursor and inspiration of \xref{Harary}{1953a}{Frank Harary} and \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary} (\qv).  Signed digraphs (at most 4 vertices) and signed vertices are unsystematically present but there is no graph theory.  Complicated by multiple types of vertices and edges [dispensed with by Cartwright--Harary].  
\dated{15 Oct 2018}
\codes{\CPsS,~\Csd,~\Cvs:~\CBal}%
\itemspace

\bsgitem{1979a}{}  On balance and attribution.  
In:  Paul W.\ Holland and Samuel Leinhardt, eds., \emph{Perspectives on Social Network Research} (Proc.\ Sympos., Dartmouth Coll., Hanover, N.H., 1975), Ch.\ 2, pp.\ 11--23.  Academic Press, New York, 1979.
\annot{} How Heider came to his balance theory \xrefd{1946a}{Fritz Heider} and what it means.  Remarks on the \xref{Cartwright--Harary}{1956a}{Dorwin Cartwright \& Frank Harary} theory.
\dated{1 Oct 2018}
\codes{\CPsS}\codesctd{\CSG:~\CBal}%
\itemspace

\author{E.\ Heilbronner}						%HEILBRONNER
\authorspace
\bsgitem{1964a}{}  H\"uckel molecular orbitals of M\"obius-type conformations of annulenes.  \emph{Tetrahedron Lett.}\ 5 (1964), no.\ 29, 1923--1928.
\annot{} Introduces a negative sign into the bonds of a cyclic molecule [thus leading to M\"obius molecules and chemical signed graph theory; \cf\ \xref{Zimmerman}{1966a}{Howard E.\ Zimmerman}, \xref{Gutman}{1978a}{Ivan Gutman} \etal, \xref{Trinajsti\'c}{1983a}{Nenad Trinajstic}, \xrefd{1992a}{Nenad Trinajstic}, \xref{Rzepa}{2005a}{Henry S.\ Rzepa}, \xref{Herges}{2006a}{Rainer Herges}, \etal].  Eigenvalues of adjacency matrices of negative circles.
\dated{4 Nov 2010, 23 Nov 2012}
\par
\codes{\CChem:~\CSG:~\Cbal,~\CEig}%
\itemspace

\author{Matthias Hein}						%HEIN
\authorspace
\authref  See \hyperlink{Pedro Mercado}{P.~Mercado}.
\itemspace

\author{Peter Christian Heinig}						%HEINIG
\authorspace
\bsgitem{2011a}{}  Chio condensation and random sign matrices.  Manuscript, 2011.  
\arXiv{1103.2717}.  
\codes{\CSG:~\CRand}%
\itemspace

\author{Richard V.\ Helgason} 						%HELGASON
\authorspace
\authref  See \hyperlink{Jeff L.\ Kennington}{J.L.~Kennington}.
\itemspace

\author{Pavol Hell}							%HELL P
\authorspace
\authref  See \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}.
\itemspace

\author{I.\ Heller} 							%HELLER
\authorspace
\bsgitem{1957a}{} On linear systems with integral valued solutions.  \emph{Pacific J.\ Math.}\ 7 (1957), 1351--1364.  
\MR{0094381} (20 \#899).  \Zbl{079.01903}.
\codes{\Csg:~\CIncid}%
\itemspace

\author{I.\ Heller \& C.B.\ Tompkins} 
\authorspace
\bsgitem{1956a}{} An extension of a theorem of Dantzig's.  
In:  H.W.\ Kuhn and A.W.\ Tucker, eds., \emph{Linear Inequalities and Related Systems}, pp.\ 247--252.  
Annals of Math.\ Studies, No.\ 38.  Princeton Univ.\ Press, Princeton, N.J., 1956.  
\MR{0081871} (18, 459).  \Zbl{072.37804} (72, p.\ 378d).
\annot{} Thm.:  The incidence matrix of a signed graph where all edges are links is totally unimodular iff the signed graph is balanced.  (Not stated in terms of signed graphs.)  \Cf\ \xref{Hoffman and Gale}{1956a}{[A.J.\ Hoffman \& D.\ Gale]}, \xref{Hoffman}{1960a}{Alan J.\ Hoffman}, and \xref{Reichmeider}{1984a}{Philip F.\ Reichmeider}.
\codes{\Csg:~\CInc,~\CBal}%
\itemspace

\author{Marc Hellmuth}						%HELLMUTH
\authorspace
\authref  See \hyperlink{Turker Biyikoglu}{T.~Biyiko{\u g}lu}.
\itemspace

\author{P.S.\ Hemavathi}						%HEMAVATHI
\authorspace
\authref  See \xref{Lokesha--Hemavathi--Vijay}{2018a}{V.\ Lokesha, P.S.\ Hemavathi, \& S.\ Vijay}, \xref{Reddy--Hemavathi--Rajendra}{2014a}{P.\ Siva Kota Reddy, P.S.\ Hemavathi, \& R.\ Rajendra}, \xref{Reddy--Misra--Hemavathi}{2014a}{P.\ Siva Kota Reddy, U.K.\ Misra, \& P. S.\ Hemavathi}.
\itemspace

\author{J.L.\ van Hemmen}						%HEMMEN
\authorspace
\bsgitem{1983a}{}  Equilibrium theory of spin glasses:  mean-field theory and beyond.  
In:  J.L.\ van Hemmen and I.\ Morgenstern, eds., \emph{Heidelberg Colloquium on Spin Glasses} (Proc., Heidelberg, 1983), pp.\ 203--233.  Lect.\ Notes in Physics, Vol.\ 192.  Springer-Verlag, Berlin, 1983.
\MR{0733800} (85d:82086).  
\annot{} \S2.3, ``Frustration'':  Physics of Ising models with edges (``bonds'') that are positive, negative, or of undetermined sign.
\dated{16 Jun 2012} 
\par
\codes{\CPhys:~\Csg}%
\itemspace

\authorx{Robert L.\ Hemminger}			 		%HEMMINGER
\author{Robert L.\ Hemminger \& Joseph B.\ Klerlein}
\authorspace
\bsgitem{1979a}{}  Line pseudodigraphs.  \emph{J.\ Graph Theory} 1 (1977), 365--377.  
\MR{0465926} (57 \#5812).  \Zbl{379.05032}.
\annot{} An attempt, intrinsically unsuccessful, to represent the (signed) line graph of a digraph (see \xref{Zaslavsky}{20xxa}{Thomas Zaslavsky}) by a digraph.  [Continued by \xref{Klerlein}{1975a}{Joseph B.\ Klerlein}.]
\codes{\Csg:~\CLG,~\Cori}%
\itemspace

\author{Robert L.\ Hemminger \& Bohdan Zelinka} 
\authorspace
\bsgitem{1973a}{} Line isomorphisms on dipseudographs.  \emph{J.\ Combin.\ Theory Ser.\ B} 14 (1973), 105--121.  
\MR{0314679} (47 \#3230).  \Zbl{263.05107}.
\codes{\Csg:~\CLG,~\Cori}%
\itemspace

\author{Anthony Henderson}						%HENDERSON
\authorspace
\bsgitem{2006a}{}  Plethysm for wreath products and homology of sub-posets of Dowling lattices.  \emph{Electronic J.\ Combin.}\ 13 (2006), no.\ 1, Research art.\ R87, 25 pp.
\MR{2255429} (2007f:05187).  \Zbl{1113.05101}.						
\annot{} The subposets are $Q_n^{1 \mod d}(\fG)$ where $d>1$, whose elements are the flats $A \in \Lat \bfF(\fG K_n\full)$ such that $d$ divides the order of the unbalanced part and the number of vertices every balanced component is $\equiv 1 \mod d$.
\par
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\author{Michael Henley}						%HENLEY
\authorspace
\authref  See \hyperlink{Federico Ardila}{F.~Ardila}.
\itemspace

\author{Cheolwon Heo}						%HEO
\authorspace
\authref  See also \hyperlink{Bertrand Guenin}{B.~Guenin}.
\authrefspace

\bsgitem{2016a}{}  \emph{Recognizing Even-Cycle and Even-Cut Matroids}.  Master's thesis, Univ.\ of Waterloo, 2016.  \url{https://uwspace.uwaterloo.ca/handle/10012/10407}
\annot{} Dictionary:  ``even-cycle matroid'' = signed graphic lift matroid; ``even-cut'' = dual matroid.
\codes{\CSG:~\CMatrd,~\CDual:~\CAlgor}%
\itemspace

\bsgitem{2021a}{}  \emph{Representations of Even-cycle and Even-cut Matroids}.  Doctoral thesis, Univ.\ of Waterloo, 2021.  \url{https://uwspace.uwaterloo.ca/handle/10012/17297}
\par
\codes{\CSG:~\CMatrd,~\CDual:~\CAlgor}%
\itemspace

\author{Cheolwon Heo \& Bertrand Guenin}
\authorspace
\bsgitem{2020a}{}  Recognizing even-cycle and even-cut matroids.  In:  \emph{Integer Programming and Combinatorial Optimization}, pp.\ 182--195.  \LNCS 12125.  Springer, Cham, 2020.
\MR{4139569}.  
\annot{} Dictionary:  ``Even-cycle matroid'' = lift matroid of signed graph.
\par
\codes{\CSG:~\CMatrd:~\CAlgor}%
\itemspace

\author{Rainer Herges}						%HERGES
\authorspace
\bsgitem{2006a}{}  Topology in chemistry: Designing M\"obius molecules.  \emph{Chem.\ Rev.}\ 106 (2006), 4820--4842.
\annot{} A M\"obius molecule has a half-twist in a ring structure [hence can be modeled by an unbalanced signed graph; \cf\ \xref{Heilbronner}{1964a}{E.\ Heilbronner}, \xref{Gutman}{1978a}{Ivan Gutman} \etal, \xref{Trinajsti\'c}{1983a}{Nenad Trinajstic}, \xrefd{1992a}{Nenad Trinajstic}].  A survey of specific types of M\"obius molecules.  \S 3.1.4, ``Other M\"obius systems":  \xref{Rzepa}{2005a}{Henry S.\ Rzepa} \etal, Craig, and also \xref{Fowler}{2002a}{Patrick W.\ Fowler} have proposed that certain annulenes are intrinsically twisted (i.e., unbalanced) due to the d-orbital or p-orbital structure.
\dated{23 Nov 2012} 
\par
\codes{\CChem:~\Csg:~\Cbal:~\CExp,~\CRef}%
\itemspace

\authorx{Patricia Hersh}						%HERSH
\author{Patricia Hersh \& Ed Swartz}
\authorspace
\bsgitem{2008a}{}  Coloring complexes and arrangements.  \emph{J.\ Algebraic Combin.}\ 27 (2008), 205--214.
\MR{2375492} (2008m:05109).  \Zbl{1154.05315}.					
\annot{} Remark 19: Chromatic polynomials of signed graphs vis-\'a-vis subarrangements of the root system arrangement $\cB_n$ in Thm.\ 18, which gives properties of an $h$-vector.
\dated{1 Mar 2011} 
\codes{\CSG:~\CInv}%
\itemspace

\authorx{Daniel Hershkowitz}						%HERSHKOWITZ
\author{Daniel Hershkowitz \& Hans Schneider}
\authorspace
\bsgitem{1993a}{}  Ranks of zero patterns and sign patterns.  \emph{Linear Multilinear Algebra} 34 (1993), no.\ 1, 3--19. 
\MR{1334927} (96g:15004).  \Zbl{793.05027}.					
\annot{} Bipartite $\Sigma$ such that every matrix with sign pattern $\Sigma$ has the same rank, over each field $\neq \bbF_2$.
\dated{6 Mar 2011} 
\codes{\CSG:~\CQM}%
\itemspace

\author{J.A.\ Hertz}							%HERTZ
\authorspace
\authref  See \hyperlink{K.H.\ Fischer}{K.H.~Fischer}.
\itemspace

\authorx{Ryan Hessert}						%HESSERT
\author{Ryan Hessert \& Sudipta Mallik}
\authorspace
\bsgitem{2021a}{}  Moore-Penrose inverses of the signless Laplacian and edge-Laplacian of graphs.  \emph{Discrete Math.}\  344 (2021), no.\ 8, art.\ 112451, 15 pp.
\MR{4254527}.  \Zbl{1466.05120}.  \arXiv{2005.03702}.  
\annot{} 
[\Problem.  Combine both cases by generalizing to signed graphs.]
\dated{28 Dec 2022} 
\codes{\Csg:~\CPar:~\CIncid,~\CLap}%
\itemspace

\bsgitem{20xxa}{}  The inverse of the incidence matrix of a unicyclic graph.  Submitted.  \emph{}\ 
\arXiv{2201.02580}.  
\annot{} The Moore--Penrose inverse, if the matrix has not full rank.  [\Problem.  Extend to a (balanced) signed unicyclic graph.]
	
\dated{11 Feb 2022} 
\codes{\CPar:~\CIncid,~\CLap}%
\itemspace

\author{G\'abor Hetyei}						%HETYEI
\authorx{Gabor Hetyei}
\authorspace
\authref  See \hyperlink{Yuanan Diao}{Y.~Diao}.
\itemspace

\authorx{Chris Heunen}						%HEUNEN
\author{Chris Heunen \& Vaia Patta}
\authorspace
\bsgitem{2018a}{}  The category of matroids.  \emph{Appl.\ Categorical Structures} 26 (2018), no.\ 2, 205--237.
\MR{3770908}.  \Zbl{1403.18002}.  \arXiv{1512.01390}.				
\annot{} \S7.3, ``Graphs'', mentions possible bicircular, frame, and lift functors.
\dated{2 Oct 2018} 
\codes{\Cgg:\CMatrd,~\CBic}%
\itemspace

\author{Hector Hevia} 						%HEVIA
\authorspace
\authref  See \hyperlink{Gary Chartrand}{G.~Chartrand}.
\itemspace

\author{Farideh Heydari}						%HEYDARI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Soudabeh Dalvandi}{S.~Dalvandi}, \hyperlink{N.\ Kafai}{N.~Kafai}, and \hyperlink{Mona Souri}{M.~Souri}.
\itemspace

\author{Takayuki Hibi}						%HIBI
\authorspace
\authref  See also \hyperlink{Tetsushi Matsui}{T.~Matsui} and \hyperlink{Hidefumi Ohsugi}{H.~Ohsugi}.
\authrefspace

\author{Takayuki Hibi, Nan Li, \& Yan X Zhang}
\authorspace
\bsgitem{2013a}{}  Separating hyperplanes of edge polytopes.  \emph{J.\ Comhin.\ Theory Ser.\ A} 120 (2013), 218--231.
\MR{2971708}.  \Zbl{1255.05123}.  \arXiv{1112.5047}.				
\par
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\author{Takayuki Hibi, Aki Mori, Hidefumi Ohsugi, \& Akihiro Shikama}
\authorspace
\bsgitem{2016a}{}  The number of edges of the edge polytope of a finite simple graph.  \emph{Ars Math.\ Contemp.}\ 10 (2016), no.\ 2, 323--332.
\MR{3529294}.  \Zbl{1351.52007}.  \arXiv{1308.3530}.				
\annot{} 
[This is the antibalanced case.  \Problem.  Generalize to signed graphs, including balanced graphs.]
\codes{\Csg:~\CPar:~\CGeom}%
\itemspace

\author{Takayuki Hibi, Kenta Nishiyama, Hidefumi Ohsugi, \& Akihiro Shikama}
\authorspace
\bsgitem{2014a}{}  Many toric ideals generated by quadratic binomials possess no quadratic Gr\"ob\-ner bases.  \emph{J.\ Algebra} 408 (2014), 138--146.
\MR{3197176}.  \Zbl{1304.13040}.  
\annot{} Antibalanced graph criteria.  [\Problem.  Generalize to signed graphs.]
\dated{5 Oct 2014} 
\codes{\Csg:~\CPar:~\CAlgeb}%
\itemspace

\author{Akihiro Higashitani}						%HIGASHITANI
\authorspace
\authref  See \hyperlink{Tetsushi Matsui}{T.~Matsui}.
\itemspace

\author{Desmond J.\ Higham}						%HIGHAM
\authorspace
\authref  See \hyperlink{Ernesto Estrada}{E.~Estrada}.
\itemspace

\authorx{Yusuke Higuchi}						%HIGUCHI
\author{Yusuke Higuchi \& Iwao Sato}
\authorspace
\bsgitem{2015a}{}  A balanced signed digraph.  \emph{Graphs Combin.}\ 31 (2015), no.\ 6, 2215--2230.
\MR{3417229}.  \Zbl{1330.05080}.  
\annot{} ``Balance'' of $(D,\sigma)$ = cycle balance (\cf\ \xref{B.D.\ Acharya}{1980a}{B.\ Devadas Acharya}).  Characterized by graph zeta function.  Signed digraph covering.
\dated{6 Jul 2022} 
\codes{\CSD:~\CBal,~\CCov}%
\itemspace

\author{Timo Hiller}							%HILLER
\authorspace
\bsgitem{2017a}{}  Friends and enemies: A model of signed network formation.  \emph{Theor.\ Economics} 12 (2017), 1057--1087.
\MR{3713674}.  \Zbl{1396.91044}.  
\annot{} Treats balance via game theory.
\dated{6 Jul 2022} 
\codes{\CSG:~\CDyn,~\CBal}%
\itemspace

\author{Franziska Hinkelmann}						%HINKELMANN
\authorspace
\authref  See \hyperlink{Alan Veliz-Cuba}{A.~Veliz-Cuba}.
\itemspace

\author{K.\ Hinson}							%HINSON
\authorspace
\authref  See \hyperlink{Yuanan Diao}{Y.~Diao}.
\itemspace

\author{Sandra Hirche}						%HIRCHE
\authorspace
\authref  See \hyperlink{Dong Xue}{D.~Xue}.
\itemspace

\author{Andr\'e Hirschowitz}						%HIRSCHOWITZ A
\authorspace
\authref  See \hyperlink{Michel Hirschowitz}{M.~Hirschowitz}.
\itemspace

\authorx{Michel Hirschowitz}						%HIRSCHOWITZ M
\author{Michel Hirschowitz, Andr\'e Hirschowitz, \& Tom Hirschowitz}
\authorspace
\bsgitem{2007a}{}  A theory for game theories.  In: V.\ Arvind and S.\ Prasad, eds., \emph{FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science} (27th Int.\ Conf., New Delhi, 2007), pp.\ 192--203.  \LNCS 4855.  Springer-Verlag, Berlin, 2007.
\MR{2480201} (2010h:91057).  \Zbl{1136.68035}.				
\codes{\CSD:~\CAppl}%
\itemspace

\author{Tom Hirschowitz}						%HIRSCHOWITZ T
\authorspace
\authref  See \hyperlink{Michel Hirschowitz}{M.~Hirschowitz}.
\itemspace

\author{Petr Hlin\v{e}n\'y}						%HLINENY
\authorx{Petr Hlineny}
\authorspace
\authref  See also \hyperlink{Robert Ganian, Petr Hlineny, \& Jan Obdrzalek}{R.~Ganian}.
\authrefspace

\author{Petr Hline\v{n}\'y, Sang-Il Oum, Detlef Seese, \& Georg Gottlob}
\authorx{Petr Hlineny, Sang-Il Oum, Detlef Seese, \& Georg Gottlob}
\authorspace
\bsgitem{2008a}{}  Width parameters beyond tree-width and their applications.  \emph{Computer J.}\  51 (2008), no.\ 3, 326--362.
\annot{} \S4.2.1:  Switching changes rank-width of a graph by at most 1, contrasting with other widths in \xref{Bodlaender and Hage}{2012a}{Hans L.\ Bodlaender \& Jurriaan Hage}.
\dated{15 Jun 2022} 
\codes{\CTG:~\CSw}%
\itemspace

\authorx{Tuyen-Thanh-Thi Ho}						%HO TTT
\author{Tuyen-Thanh-Thi Ho, Hung Thanh Vu, \& Bac Hoai Le}
\authorspace
\bsgitem{2013a}{}  A decision-making based feature for link prediction in signed social networks.  In:  \emph{The 2013 RIVF International Conference on Computing and Communication Technologies -- Research, Innovation, and Vision for Future (RIVF)} (Hanoi, 2013), pp.\ 169--174.  IEEE, 2013.
\annot{} Given $\vec\Gamma$ and signs on $E(\vec\Gamma)\setm (u,v)$, $\sigma(u,v)$ is predicted by signed in-degrees of $v$ and signed outdegrees of $u$.  Justified for social networks by appeal to psychological traits.
\dated{22 Jan 2015} 
\par
\codes{\CSD:~\CPred:~\CAlgor:~\CPsS}%
\itemspace

\author{Dorit S.\ Hochbaum}						%HOCHBAUM
\authorspace
\bsgitem{1998a}{}  Instant recognition of half integrality and $2$-approximations.
In: Klaus Jansen and Jos\'e Rolim, eds., \emph{Approximation Algorithms for Combinatorial Optimization} (Aalborg, 1998), pp.\ 99--110.  \LNCS 1444.  Springer, Berlin, 1998.
\MR{1677400}.  \Zbl{911.90261}.
\annot{} Integer programs with constraints of a generalized real gain-graphic form, $\alpha x - \beta y - \gamma \leq z$, the gain being $\beta/\alpha$.  Slightly extends \xref{Hochbaum, Megiddo, Naor, and Tamir}{1993a}{Dorit S.\ Hochbaum, Nimrod Megiddo, Joseph (Seffi) Naor, \& Arie Tamir}.
\codes{\Cgn:~\CInc(\CDual):~\CAlgor}%
\itemspace

\bsgitem{1998b}{}  The $t$-vertex cover problem: extending the half integrality framework with budget constraints. 
In: Klaus Jansen and Jos\'e Rolim, eds., \emph{Approximation Algorithms for Combinatorial Optimization} (Aalborg, 1998), pp.\ 111--122.  \LNCS 1444.  Springer, Berlin, 1998.
\MR{1677404} (2000b:\-90032).  \Zbl{908.90213}.
\annot{} Integer programs as in \xrefd{1998a}{Dorit S.\ Hochbaum} with ``budget constraints''.
\par
\codes{\Cgn:~\CInc(\CDual):~\CAlgor}%
\itemspace

\bsgitem{2000a}{}  Instant recognition of polynomial time solvability, half integrality and 2-approxi\-ma\-tions.  
In: Klaus Jansen and Samir Khuller, eds., \emph{Approximation Algorithms for Combinatorial Optimization} (Saarbr\"ucken, 2000), pp.\ 2--14.  \LNCS 1913.  Springer, Berlin, 2000. 
\MR{1850069} (no rev).  \Zbl{976.90123}.
\annot{} Integer programs as in \xrefd{1998a}{Dorit S.\ Hochbaum}.  There is a polynomial-time solution via a minimum cut, or else a half-integral partial solution.
\par
\codes{\Cgn:~\CInc(\CDual):~\CAlgor}%
\itemspace

\bsgitem{2002a}{}  Solving integer programs over monotone inequalities in three variables: a framework for half integrality and good approximations.  
O.R.\ for a United Europe (Budapest, 2000).  \emph{European J.\ Operational Res.}\ 140 (2002), no.\ 2, 291--321.
\MR{1899053} (2003e:90052).  \Zbl{1001.90050}.
\annot{} Constraints of a generalized positive-real gain-graphic form, $\alpha x - \beta y - \gamma \leq z$, the gain being $\beta/\alpha$, contrasting $\alpha,\beta\geq0$ to the intrinsically hard case where a negative coefficient is allowed but a half-integral approximate solution is easy. 
\codes{\Cgn:~\CInc(\CDual):~\CAlgor}%
\itemspace

\author{Dorit S.\ Hochbaum, Nimrod Megiddo, Joseph (Seffi) Naor, \& Arie Tamir}
\authorspace
\bsgitem{1993a}{}  Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality.  \emph{Math.\ Programming Ser.\ B} 62 (1993), 69--83.  
\MR{1247607} (94k:90050).  \Zbl{802.90080}.					
\annot{} Approximate solution of integer linear programs with real, dually gain-\-graph\-ic coefficient matrix.  [See \xref{Sewell}{1996a}{E.C.\ Sewell}.]
\codes{\CGN:~\CInc(\CDual):~\CAlgor}%
\itemspace

\author{Dorit S.\ Hochbaum \& Joseph (Seffi) Naor}
\authorspace
\bsgitem{1994a}{}  Simple and fast algorithms for linear and integer programs with two variables per inequality.  \emph{SIAM J.\ Computing} 23 (1994), 1179--1192.  
\MR{1303329} (95h:90066).  \Zbl{831.90089}.					
\annot{} Linear and integer programs with real, dually gain-graphic coefficient matrix: feasibility for linear programs, solution of integer programs when the gains are positive (``monotone inequalities''), and identification of ``fat'' polytopes (that contain a sphere larger than a unit hypercube).
\par
\codes{\CGN:~\CInc(\CDual):~\CAlgor,~\CRef}%
\itemspace

\author{Winfried Hochst\"attler}						%HOCHSTATTLER
\authorx{Winfried Hochstattler}
\authorspace
\authref  See also \hyperlink{Luis Goddyn}{L.~Goddyn}.
\authrefspace

\author{Winfried Hochst\"attler, Robert Nickel, \& Britta Peis}
\authorx{Winfried Hochstattler, Robert Nickel, \& Britta Peis}
\authorspace
\bsgitem{2006a}{}  Two disjoint negative cycles in a signed graph.  CTW2006 -- Cologne-Twente Workshop on Graphs and Combinatorial Optimization.  \emph{Electronic Notes Discrete Math.}\ 50 (2006), 107--111.
\MR{2307287} (no rev).  \Zbl{1134.05319}.						
\annot{} Incidence matrix used to find the circles in slow polynomial time.  Use of graphic structure is explored.
\codes{\CSG:~\CStr:~\CCircles:~\CAlgor,~\CInc}%
\itemspace

\author{Herv\'e Hocquard}						%HOCQUARD
\authorx{Herve Hocquard}
\authorspace
\authref  See \hyperlink{Florent Foucaud}{F.~Foucaud}.
\itemspace

\author{Cornelis Hoede} 						%HOEDE
\authorspace
\bsgitem{1981a}{}  The integration of cognitive consistency theories.
Memorandum nr.\ 353, Dept.\ of Appl.\ Math., Twente University of Tech., Enschede, The Netherlands, Oct., 1981.
\codes{\CPsS:~\CGen}\codesctd{\CSG,~\CVS:~\CBal}%
\itemspace

\bsgitem{1982a}{}  Anwendungen von Graphentheoretischen Methoden und Konzepten in den Socialwissenschaften.  
Memorandum nr.\ 390, Dept.\ of Appl.\ Math., Twente University of Tech., Enschede, the Netherlands, May, 1982.
\annot{} Teil 4:  ``Kognitive Konsistenz.'' 
\codes{\CPsS:~\CGen:~\CExp}%
\itemspace

\bsgitem{1992a}{\dag}  A characterization of consistent marked graphs.  \emph{J.\ Graph Theory} 16 (1992), 17--23.  
\MR{1147800} (93b:05141).  \Zbl{748.05081}.
\annot{} Characterizes when one can sign the vertices of a graph so every circle has positive sign product, solving the problem of \xref{Beineke and Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}.  
Given $\Gamma$, $\mu: V \to \{+,-\}$, and a spanning tree $T$:  $(\Gamma,\mu)$ is consistent iff the fundamental circles with respect to $T$ are positive and the endpoints of the intersection of two fundamental circles have the same sign.  A polynomial-time algorithm ensues.
[The definitive word until \xref{Joglekar, Shah, and Diwan}{2010a}{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan}.  Does not include signed edges.]
\dated{rev.\ 11 Sept 2010, 2 May 2012}
\codes{\CVS:~\CBal:~\CStr}%
\itemspace

\author{Jan B.\ Hoek}						%HOEK
\authorspace
\authref  See \hyperlink{Boris N.\ Kholodenko}{B.N.~Kholodenko}.
\itemspace

\authorx{P.\ Hoever}							%HOEVER
\author{P.\ Hoever, W.F.\ Wolff, \& J.\ Zittartz}
\authorspace
\bsgitem{1981a}{}  Random layered frustration models.  \emph{Z.\ Phys.\ B} 41 (1981), 43--53.
\MR{0600279} (81m:82027).  
\annot{} Physics of Ising models on a planar square lattice.  Exact solutions for partition function, free energy, ground state energy.  The transition temperature depends only on the average edge sign, $(\#E^+-\#E^-)/\#E$.  Switching is implicit (``substituting spins'').  Model (a):  all horizontal edges are $+$ (attainable by switching); if horizontally periodic these are ``random layered frustration'' models.  Model (b):  Assumed switched to minimize $\#E^-$.
Dictionary:  ``plaquette'' = quadrilateral, ``frustration index'' = sign of plaquette.
\annotctd{} They conjecture thermodynamic consequences if the ground states ($s:V\to\signnos$ with $l(\Sigma)$ frustrated edges) are connected in the state graph $\signnos^V$.  [\Question.  For which $\Sigma$ are the ground states connected?]
\dated{16 Jun, 28 Aug 2012} 
\codes{\CPhys:~\CSG:~\Csw}%
\itemspace

\author{Peter D.\ Hoff}						%HOFF
\authorspace
\bsgitem{2005a}{}  Bilinear mixed-effects models for dyadic data.  \emph{J.\ Amer.\ Statistical Assoc.}\ 100, No.\ 469 (2005), 286--295.
\MR{2156838} (no rev).  \Zbl{1117.62353}.  
\codes{\CSG,~\CPsS:~\CBal}%
\itemspace

\author{Alan J.\ Hoffman} 						%HOFFMAN AJ*
\authorspace
\authref  See also \hyperlink{D.R.\ Fulkerson}{D.R.~Fulkerson}, \hyperlink{David Gale}{D.~Gale}, and \hyperlink{Emilie Haynsworth}{E.~Haynsworth}.
\authrefspace

\bsgitem{1960a}{}  Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In:  Richard Bellman and Marshall Hall Jr., eds., \emph{Combinatorial Analysis}, pp.\ 113--127.  Proc.\ Sympos.\ Appl.\ Math., Vol.\ 10.  American Mathematical Soc., Providence, R.I., 1960.
\MR{0114759} (22 \#5578).  \Zbl{096.00606} (96, p.\ 6f).				
\codes{\Csg:~\Cinc,~\Cbal:~\CExp}%
\itemspace

\bsgitem{1970a}{}  $-1 - \sqrt{2}$?  
In: Richard Guy \etal, eds., \emph{Combinatorial Structures and Their Applications} (Proc.\ Calgary Int.\ Conf., 1969), pp.\ 173--176.  
Gordon and Breach, New York, 1970.  
\Zbl{262.05133}.  
\codes{\CLG}%
\itemspace

\bsgitem{1972a}{}  Eigenvalues and partitionings of the edges of a graph.  \emph{Linear Algebra Appl.}\ 5 (1972), 137--146.  
\MR{0300937} (46 \#97).  \Zbl{247.05125}.					
\codes{\CPar:~\CEig,~\CFr}%
\itemspace

\bsgitem{1974a}{} On eigenvalues of symmetric ($+1,-1$) matrices.  \emph{Israel J.\ Math.}\ 17 (1974), 69--75.  
\MR{0349709} (50 \#2202).  \Zbl{281.15003}.					
\annot{} Eigenvalues of signed complete graphs.
\codes{\Csg:~\Ckg:~\CEig}%
\itemspace

\bsgitem{1975a}{}  Spectral functions of graphs.  
In: \emph{Proceedings of the International Congress of Mathematicians} (Vancouver, 1974), Vol.\ 2, pp.\ 461--463.  Canad.\ Math.\ Congress, Montreal, 1975.  
\MR{0434886} (55 \#7850).  \Zbl{344.05164}.					
\codes{\CTG,~\CEig}%
\itemspace

\bsgitem{1976a}{} On spectrally bounded signed graphs.  (Abstract.)  
In: \emph{Transactions of the Twenty-First Conference of Army Mathematicians} (White Sands, N.M., 1975), pp.\ 1--5.  ARO Rep.\ 76-1.  
U.S.\ Army Research Office, Research Triangle Park, N.C., 1976.  
\MR{0547323} (58 \#27648).  
\annot{} Abstract of \xrefd{1977b}{Alan J.\ Hoffman}.  Also, bounding the least eigenvalue in terms of principal submatrices.
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{1977a}{}  On graphs whose least eigenvalue exceeds $-1 - \sqrt{2}$.  \emph{Linear Algebra Appl.}\ 16 (1977), 153--165.  
\MR{0469826} (57 \#9607).  \Zbl{354.05048}.					
\annot{} Introduces generalized line graphs.  [They are the reduced line graphs of signed graphs of the form $-\Gamma$ with any number of negative digons attached to each vertex; see \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, Ex.\ 7.6; \xrefd{20xxa}{Thomas Zaslavsky}].
\codes{\CLG}%
\itemspace

\bsgitem{1977b}{}  On signed graphs and gramians.  \emph{Geometriae Dedicata} 6 (1977), 455--470.  
\MR{0463211} (57 \#3167).  \Zbl{407.05064}.					
\annot{} $\Sigma$ is a signed simple graph.  Let $\lambda$ be the least eigenvalue of $A(\Sigma)$.  Can (*) $A(\Sigma) - \lambda I - KK\trans$ be zero for some $K$ with all entries $0, \pm1$?  When $\lambda=-2$, $K$ exists [equivalently, $\Sigma$ is a reduced line graph of a signed graph; \cf\ \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}], with finitely many exceptions; the proof uses root systems; \cf\ \xref{Cameron, Goethals, Seidel, and Shult}{1976a}{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}.  In general, no $K$ may give zero, but the minimum, over all $K$, of the largest element of (*) is bounded by a function of $\lambda$.
\par
\codes{\CSG:~\CLG:~\CAdj,~\CEig}%
\itemspace

\author{[A.J.\ Hoffman \& D.\ Gale]} 
\authorspace
\bsgitem{1956a}{} Appendix [to the paper of \xref{Heller and Tompkins}{1956a}{I.\ Heller \& C.B.\ Tompkins}].  
In: H.W.~Kuhn and A.W.~Tucker, eds., \emph{Linear Inequalities and Related Systems}, pp.\ 252--254.  
Annals of Math.\ Studies., No.\ 38.  Princeton Univ.\ Press, Princeton, N.J., 1956.
\par
\codes{\Csg:~\CInc:~\Cbal}%
\itemspace

\author{Alan J.\ Hoffman \& Peter Joffe} 
\authorspace
\bsgitem{1978a}{} Nearest $\cS$-matrices of given rank and the Ramsey problem for eigenvalues of bipartite $\cS$-graphs.  
In: \emph{Probl\`emes Combinatoires et Th\'eorie des Graphes} (Colloq.\ Int., Orsay, 1976), pp.\ 237--240.  
Colloques Int.\ du CNRS, 260.  Editions du C.N.R.S., Paris, 1978.  
\MR{0539983} (81b:05080).  \Zbl{413.05031}.
\codes{\CSG:~\CEig}%
\itemspace

\author{Alan J.\ Hoffman \& Francisco Pereira} 
\authorspace
\bsgitem{1973a}{} On copositive matrices with $-1,0,1$ entries.  \emph{J.\ Combinatorial Theory Ser.\ A} 14 (1973), 302--309.  
\MR{0316482} (47 \#5029).  \Zbl{273.15019}.					
\annot{}  Matrices $A(\Sigma)+I$ for simple $|\Sigma|$: properties as quadratic forms.  Generalizes \xref{Haynsworth and Hoffman}{1969a}{Emilie Haynsworth \& A.J.\ Hoffman}.
\dated{28 May 2017}
\par
\codes{\Csg:~\Cadj}%
\itemspace

\authorx{Dean Hoffman}						%HOFFMAN D
\author{Dean Hoffman \& Heather Jordon}
\authorspace
\bsgitem{2006a}{}  Signed graph factors and degree sequences.  \emph{J.\ Graph Theory} 52 (2006), no.\ 1, 27--36.
\MR{2214439} (2006k:05174).  \Zbl{1117.05089}.						
\annot{} The net degree of a vertex in $\Sigma$ is $d^+(v) - d^-(v)$.  [This is best viewed as degree in an all-negative bidirected graph; \cf\ p.\ 35.]  
Thms.\ 2.3 (for $\Sigma$) and 4.1 (for a bidirected graph $\Beta$, called a ``mixed signed graph'') are an interesting $f$-factor theorem in terms of net degrees.  
Thm.\ 4.1:  Given $f: V \to \bbZ$, an ``$f$-factor'' is a subgraph whose net in-degree vector $= f$.  For disjoint $S, T \subseteq V$ and a component $Q$ of $\Beta \setminus (S \cup T)$, $J(Q,S,T)$ is computed in terms of $f$ and in-degrees and out-degrees of edges among $Q, S, T$.  $q(S,T)$ is the number of $J$-odd components $Q$.  An $f$-factor exists iff $q$ satisfies an inequality.  
Thm.\ 3.2:  Fixing the maximum edge multiplicity, an Erd\H{o}s--Gallai-type characterization of net degree sequences---simplifying the theorem of \xref{Michael}{2002a}{T.S.\ Michael}.  Thm.\ 4.2:  Net in-degree sequences of bidirected simple graphs.
[More in \xref{Jordon, McBride, and Tipnis}{2009a}{Heather Jordon, Richard McBride, \& Shailesh Tipnis}.] 
\dated{14 Oct 2009}
\par
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\author{Thomas R.\ Hoffman}						%HOFFMAN TR
\authorspace
\authref  See also \hyperlink{David M.\ Duncan}{D.M.~Duncan}.
\authrefspace

\author{Thomas R.\ Hoffman \& James P.\ Solazzo}
\authorspace
\bsgitem{2012a}{}  Complex equiangular tight frames and erasures.  \emph{Linear Algebra Appl.}\ 437 (2012), no.\ 2, 549--558. 
\MR{2921716}.  \Zbl{1246.42028}.  \arXiv{1107.2267}.				
\codes{\Cgg:~\Ckg:~\CAdj}%
\itemspace

\bsgitem{2018a}{}  Complex two-graphs.  \emph{Houston J.\ Math.}\ 44 (2018), no.\ 1, 283--300.
\MR{3796449}.  \Zbl{1391.05165}.  \arXiv{1408.0334}.				
\par
\codes{\Cgg:~\CKG:~\CAdj,~\CTG}%
\itemspace

\author{Karl Heinz Hoffmann}						%HOFFMANN KH
\authorspace
\bsgitem{2002a}{}  The statistical physics of energy landscapes:  From spin glasses to optimization.
In:  K.H.\ Hoffmann and M.\ Schreiber, eds., \emph{Computational Statistical Physics}, Ch. 4, pp.\ 57--76.  Springer, Berlin, 2002.
\Zbl{1160.82330}.  
\annot{} Expository, accessible.  Main example is the Ising model: fixed weighted $\Sigma$ and variable $\zeta: V \to \{\pm1\}$ with Hamming distance and energy function $e: \{\pm1\}^V \to \bbR$ which is mountainous, i.e, many local minima (valleys) with low or high intermediate values $e(\zeta)$.  Ground states (minimizing $e(\zeta)$) are therefore hard to find computationally.  [\Cf, e.g., \hyperlink{E.E.\ Vogel}{Vogel \etal}]  Random ``thermal variation'' of $\zeta$ leads to slow ``relaxation'' from higher to lower local minima, in theory and practice.
\dated{7 Aug 2018} 
\codes{\Csg,~\Cwg:~\CVS:~\CStr,~\CPhys,~\CAppl:~\CExp}%
\itemspace

\authorx{Franz H\"ofting}						%HOFTING
\authorx{Franz Hofting}
\author{Franz H\"ofting \& Egon Wanke}
\authorx{Franz Hofting \& Egon Wanke}
\authorspace
\bsgitem{1993a}{}  Polynomial algorithms for minimum cost paths in periodic graphs.  
In:  Vijaya Ramachandran \etal, eds., \emph{Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms} (Austin, Tex., 1993),  pp.\ 493--499.  
Assoc.\ for Computing Machinery, New York, and Soc.\ for Industrial and Appl.\ Math., Philadelphia, 1993.  
\MR{1213262} (93m:05184).  \Zbl{801.68133}.					
\annot{} Given a finite gain digraph $\Phi$ (the ``static graph'') with gains in $\bbZ^d$ and a rational cost for each edge, find a minimum-cost walk (``path'') in its canonical covering graph $\tilde\Phi$ with given initial and final vertices.  
\par
\codes{\CGD(\CCov):~\CAlgor}%
\itemspace

\bsgitem{1994a}{}  Polynomial time analysis of toroidal periodic graphs.  
In:  Serge Abiteboul and Eli Shamir, eds., \emph{Automata, Languages and Programming} (Proc.\ 21st Int.\ Colloq., ICALP 94, Jerusalem, 1994), pp.\ 544--555.  \LNCS 820.  Springer-Verlag, Berlin, 1994.  
\MR{1334129} (96c:05164).  
\annot{} Take a gain digraph $\Phi$ (the ``static graph'') with gains in $\bbZ_\alpha = \bbZ_{\alpha_1} \times \cdots \times \bbZ_{\alpha_d}$ (where $\alpha = (\alpha_1,\cdots,\alpha_d)$) and its canonical covering digraph $\tilde\Phi$ (the ``toroidal periodic graph'').  Treated algorithmically via integer linear programming and linear Diophantine equations:  existence of directed paths (NP-complete, but polynomial-time if $\Phi$ is strongly connected) and number of strongly connected components of $\tilde \Phi$.  
\par
\codes{\CGD(\CCov):~\CAlgor,~\CGeom}%
\itemspace

\bsgitem{1995a}{}  Minimum cost paths in periodic graphs.  \emph{SIAM J.\ Computing} 24 (1995), 1051--1067.
\MR{1350758} (96d:05061).  \Zbl{839.05063}.					
\annot{} Full version of \xrefd{1993a}{Franz Hofting \& Egon Wanke}.  The min-cost problem is expressed as an integer linear program.  Various conditions under which the problem is NP-hard, even a very restricted version without costs (Thms.\ 3.3, 3.5), or polynomial-time solvable (e.g.: without costs, when $\Phi$ is an undirected gain graph: Thm.\ 3.4; with costs, when $d$ is fixed: Thm.\ 4.5).
\codes{\CGD,~\CGG(\CCov):~\CAlgor,~\CGeom,~\CRef}%
\itemspace

\bsgitem{2000a}{}  Polynomial-time analysis of toroidal periodic graphs.  \emph{J.\ Algorithms} 34 (2000), no.\ 1, 14--39.
\MR{1732196} (2001k:68111).  \Zbl{958.68129}.
\annot{} Full version of \xrefd{1994a}{Franz Hofting \& Egon Wanke}.
\codes{\CGD(\CCov):~\CAlgor,~\CGeom}%
\itemspace

\author{Leslie Hogben}						%HOGBEN
\authorspace
\authref  See also \hyperlink{Steve Butler}{S.~Butler}.
\authrefspace

\bsgitem{2005a}{}  Spectral graph theory and the inverse eigenvalue problem of a graph.  \emph{Electronic J.\ Linear Algebra}\ 14 (2005), 12--31.
\MR{2202430} (2006k:05133).  \Zbl{1162.05333}.  
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\authorx{Paul W.\ Holland}			 			%HOLLAND
\author{Paul W.\ Holland \& Samuel Leinhardt}
\authorspace
\bsgitem{1970a}{}  A method for detecting structure in sociometric data.  \emph{Amer.\ J.\ Sociology} 76 (1970), no.\ 3, 492--513.
\annot{} A formulation of structure in terms of weak partial ordering, i.e., transitivity, hence triads (triples of elements).  Refers to \xrefd{1970b}{Paul W.\ Holland \& Samuel Leinhardt} for specific structures, e.g., structural balance of \xref{Cartwright--Harary}{1956a}{Dorwin Cartwright \& Frank Harary} and clustering of \xref{Davis}{1967a}{James A.\ Davis}.  The types of triples allowed determine what specific model applies.  P.\ 495 states conditions for structural balance or clustering.
\dated{26 Dec 2012} 
\codes{\CPsS:~\Csg:~\CBal,~\CClu}%
\itemspace

\bsgitem{1970b}{}  A unified treatment of some structural models for sociometric data.  Tech.\ Rep., Carnegie-Mellon University, 1970.
\annot{} See \xrefd{1970a}{Paul W.\ Holland \& Samuel Leinhardt}.
\dated{26 Dec 2012} 
\codes{\CPsS:~\Csg:~\CBal,~\CClu}%
\itemspace

\bsgitem{1971a}{}  Transitivity in structural models of small groups.  \emph{Comparative Group Studies} 2 (1971), 107--124.  
\codes{\CPsS:~\CSG:~\CBal}
\itemspace

\author{Paul W.\ Holland \& Samuel Leinhardt, eds.}
\authorspace
\bsgitem{1979a}{}  \emph{Perspectives on Social Network Research} (Proc.\ Math.\ Soc.\ Sci.\ Board Adv.\ Res.\ Sympos.\ on Social Networks held at Dartmouth 
College, Hanover, N.H., September 18--21, 1975).  Academic Press, New York, 1979.  
\codes{\CPsS,~\CSG}%
\itemspace

\authorx{Roderick B.\ Holmes}						%HOLMES
\author{Roderick B.\ Holmes \& Vern I.\ Paulsen}
\authorspace
\bsgitem{2004a}{}  Optimal frames for erasures.  \emph{Linear Algebra Appl.}\ 377 (2004), 31--51.
\MR{2021601} (2004j:42028).  \Zbl{1042.46009}.					
\annot{} Adjacency matrices of cube-root-of-unity gain graphs.	
\dated{20 Jun 2011} 
\codes{\Cgg:~\Cadj}%
\itemspace

\author{[Hein van der Holst]}						%HOLST
\authorspace
\authref  See \hyperlink{Hein van der Holst}{H.~van~der~Holst} (under `V').
\itemspace

\author{Hai-Yan Hong}						%HONG HH
\authorspace
\authref  See \hyperlink{Yi-Zheng Fan}{Y.-Z.~Fan}.
\itemspace

\author{Sungpyo Hong}						%HONG SP
\authorspace
\authref  See \hyperlink{Jin Ho Kwak}{J.H.~Kwak}.
\itemspace

\author{Yiguang Hong}						%HONG YG
\authorspace
\authref  See \hyperlink{Deyuan Meng}{D.-Y.~Meng}.
\itemspace

\authorx{Yuan Hong}							%HONG Y
\author{Yuan Hong \& Xiao-Dong Zhang}
\authorspace
\bsgitem{2005a}{}  Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees.  \emph{Discrete Math.}\ 296 (2005), no.\ 2-3, 187--197.
\MR{2154712} (2006g:05127).  \Zbl{1068.05044}.  
\annot{} Thm.\ 2: If some neighbors of $v$ in $\Gamma$ are regrafted onto $u$, forming $\Gamma'$, and if $x_u \geq x_v$ in the Perron vector of $\Lap(-\Gamma)$, then $\lambda_{\min}(\Lap(-\Gamma)) < \lambda_{\min}(\Lap(-\Gamma')).$
\dated{24 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Shlomo Hoory}						%HOORY
\author{Shlomo Hoory, Nathan Linial, \& Avi Wigderson}
\authorspace
\bsgitem{2006a}{}  Expander graphs and their applications.  \emph{Bull.\ Amer.\ Math.\ Soc.\ (N.S.)} 43 (2006), no.\ 4, 439--561.
\MR{2247919} (2007h:68055).  \Zbl{1147.68608}.					
\annot{} \S6, ``Spectrum and expansion in lifts of graphs'':  covering graphs of permutation gain graphs, and from \xref{Bilu and linial}{2006a}{Yonatan Bilu \& Nathan Linial} of signed graphs.  \S6.1, ``Covering maps and lifts'':  Covering graphs of permutation gain graphs, presented as symmetric digraphs with invertible arc gains.  \S2.6, ``Eigenvalues - old and new'':  Prop.\ 6.3.  The covering graph's eigenvalues include those of the (underlying) base graph $\Gamma$ and its eigenvectors sum to $0$ on fibers.  Prop.\ 6.4.  The signed covering graph's eigenvalues are those of $\Gamma$ and those of $(\Gamma,\sigma)$.  \S6.4, ``Nearly-Ramanujan graphs by way of 2-lifts'':  Conjectured and proven eigenvalue ranges when the base graph is a Ramanujan graph.
Dictionary:``signing'' of $A(\Gamma)$ means $A(\Gamma,\sigma)$ for any edge signature.  ``2-lift'' = double covering graph.  
\dated{25 Aug 2011} 
\codes{\Csg:~\CCov,~\CEig:~\CExp}%
\itemspace

\author{John Hopcroft}						%HOPCROFT
\authorspace
\authref  See \hyperlink{Thorsten Joachims}{T.~Joachims}.
\itemspace

\author{Carlos Hoppen}						%HOPPEN
\authorspace
\authref  See \hyperlink{Rafael Esteves Mansano}{R.E.~Mansano}.
\itemspace

\author{Tsuyoshi Horiguchi}						%HORIGUCHI
\authorspace
\authref  See also \hyperlink{Ojiro Nagai}{O.~Nagai}.
\authrefspace

\bsgitem{1986a}{}  Fully frustrated Ising model on a square lattice.  \emph{Progress Theor.\ Phys.\ Suppl.}\ No.\ 87 (1986), 33--42.
\MR{0884854} (88g:82063).  
\annot{} On the square lattice, physical quantities for periodic signed graphs with up to four edge weights.  A fairly general model of which several previous ones are special cases.
\dated{22 Jan 2015} 
\codes{\CPhys:~\CSG,~\CWG}%
\itemspace

\author{A.\ Hosseiny}						%HOSSEINY
\authorspace
\authref  See \hyperlink{A.\ Kargaran}{A.~Kargaran}.
\itemspace

\author{Jiangyou Hou}						%HOU JY
\authorspace
\authref  See \hyperlink{Dan Hu}{D.~Hu}.
\itemspace

\author{Ting Hou}							%HOU T
\authorspace
\authref  See \hyperlink{Hua Gao}{H.~Gao} and \hyperlink{Xianzhu Liu}{X.-Z.~Liu}.
\itemspace

\author{Yaoping Hou}						%HOU YP
\authorspace
\authref  See also \hyperlink{Xiaodan Chen}{X.D.~Chen}, \hyperlink{Yu Chen}{Y.~Chen}, \hyperlink{Qiao Guo}{Q.~Guo}, \hyperlink{Li Ou}{L.~Ou}, \hyperlink{Dijian Wang}{D.-J.~Wang}, and \hyperlink{Zhuang Xiong}{Z.~Xiong}.
\authrefspace

\bsgitem{2005a}{}  Bounds for the least Laplacian eigenvalue of a signed graph.  \emph{Acta Math.\ Sinica (Engl.\ Ser.)}\ 21 (2005), no.\ 4, 955--960.
\MR{2156977} (2006d:05120).  \Zbl{1080.05060}.					
\codes{\CSG:~\CEig,~\CBal}%
\itemspace

\author{Yaoping Hou, Jiongsheng Li, \& Yongliang Pan}
\authorspace
\bsgitem{2003a}{}   On the Laplacian eigenvalues of signed graphs.  \emph{Linear Multilinear Algebra} 51 (2003), 21--30.
\MR{1950410} (2003j:05084).  \Zbl{1020.05044}.					
\annot{} Properties of (mainly) largest eigenvalue $\lambda_{\min}(\Sigma)$ of the Laplacian matrix $\Lap(\Sigma)$ of a signed simple graph.  Thms.\ 2.5--2.6 repeat standard criteria for balance [with a sign error in (3) of each].  
Main results:  
\annotctd{} Upper bounds, all in terms of underlying graph:  
Lemma 3.1:  For connected $\Gamma$, $\lambda_{\max}(\Gamma,\sigma) \leq \lambda_{\max}(-\Gamma)$, $=$ iff $\sigma$ is antibalanced (e.g., $-\Gamma$).  
Thm.\ 3.4:  $\lambda_{\max}(\Sigma) \leq 2(n-1)$, $=$ iff $\Sigma \sim -K_n$.  
Thm.\ 3.5:  $\lambda_{\max}(\Sigma) \leq$ (1) max edge degree + 2, (2) max(vertex degree + average neighbor degree), (3) a combination of these degrees; $=$ iff $\Sigma$ is antibalanced and $|\Sigma|$ is semiregular bipartite. 
\annotctd{} Lower bounds:  Cor.\ 3.8:  $\lambda_{\max}(\Sigma^+) + \lambda_{\max}(\Sigma^-) \geq \lambda_{\max}(\Sigma) \geq \lambda_{\max}(\Sigma^+),$ $\lambda_{\max}(\Sigma^-)$.  
Thm.\ 3.9:  If $\Sigma$ has a vertex of degree $n-1$, then $\lambda_{\max}(\Sigma) \geq \lambda_{\max}(+|\Sigma|)$, with equality iff $\Sigma$ is balanced.  
Thm.\ 3.10:  $\lambda_{\max}(\Sigma) \geq 1 + \max_v d_{|\Sigma|}(v)$.  
\annotctd{} Interlacing:  Lemma 3.7 (special case):  $\lambda_i(\Sigma) \geq \lambda_i(\Sigma \setminus e) \geq \lambda_{i+1}(\Sigma)$, where $\lambda_1 \geq \lambda_2 \geq \cdots$.  
\annotctd{} \Problems\ about existence of cospectral unbalanced signed graphs.
\par
\codes{\CSG:~\CEig}%
\itemspace

\author{Yaoping Hou, Zikai Tang, \& Dijian Wang}
\authorspace
\bsgitem{2019a}{}  On signed graphs with just two distinct adjacency eigenvalues.  \emph{Discrete Math.}\ 342 (2019), no.\ 12, art.\ 111615.
\MR{3990025}.  \Zbl{1422.05049}.  
\annot{} Classifies those with maximum degree $\leq 4$ (all are regular, by \xref{Ramezani}{20xxa}{Farzaneh Ramezani}).  
For 3-regular there are $+K_4$< $-K_4$, and $Q_3$ as in \xref{Ghasemian and Fath-Tabar}{2017a}{E.\ Ghasemian \& G.H.\ Fath-Tabar}.  
For 4-regular there are $+K_5$, $-K_5$, and triangle-free examples $Q_4$ with all $C_4$'s negative, $S_{14}$, and for $n\geq3$, $T_{2n}$ (with eigenvalues $\pm2$), contrary to Ghasemian and Fath-Tabar.  For eigenvalues in $[-2,2]$ they give an elementary proof of part of \xref{McKee and Smyth}{2007a}{James McKee \& Chris Smyth}.
\dated{29 May 2018, rev 3 Feb 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2019b}{}  On signed graphs with just two distinct Laplacian eigenvalues.  \emph{Appl.\ Math.\ Comput.}\ 351 (2019), 1--7.
\MR{3903669} (no rev).  \Zbl{1428.05193}.  
\annot{} Classifies those with maximum degree $\leq 4$.  The irregular ones are not obtained from \xrefd{2019a}{Yaoping Hou, Zikai Tang, \& Dijian Wang}.
\dated{3 Feb 2021} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Yao Ping Hou \& Li Juan Wei}
\authorspace
\bsgitem{1999a}{}  Whitney numbers of the second kind for Dowling lattices.  (In Chinese.)  \emph{Acta Sci.\ Natur.\ Univ.\ Norm.\ Hunan.}\ 22 (1999), No.\ 3, 6--10.
\MR{1746888} (2000k:\-05017).  \Zbl{948.05004}.					
\annot{} Combinatorial proof of an explicit formula for $W_k$ [possibly the standard one?].  Studies ``associated numbers'' $W_k^r$.  Proved: $W_{n-k} \leq W_k$ for $k \leq 3$ [this must be an error for $W_k \leq W_{n-k}$ and must have some restriction on $n$; well known for $k=1$].
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{C.H.\ Houghton}						%HOUGHTON
\authorspace
\bsgitem{1977a}{}  Completely 0-simple semigroups and their associated graphs and groups.  \emph{Semigroup Forum} 14 (1977), no.\ 1, 41--67.
\MR{0453897} (56 \#12150).  \Zbl{358.20071}.					
\annot{} \Cf\ \xref{Rees}{1940a}{David Rees}, \xref{Graham}{1968a}{R.L.\ Graham}.
\codes{\Cgg:~\Csw:~\CAlgeb}%
\itemspace

\author{R.M.F.\ Houtappel}						%HOUTAPPEL
\authorspace
\bsgitem{1950a}{}  Statistics of two-dimensional hexagonal ferromagnetics with ``Ising''- interaction between nearest neighbours only.  \emph{Physica} 16 (1950), 391--392.
\Zbl{038.41903} (38, p.\ 419c).  
\annot{} Announcement of \xrefd{1950b}{R.M.F.\ Houtappel}.
\dated{19 Jun 2012}
\codes{\CPhys,~\CWG,~\Csg}%
\itemspace

\bsgitem{1950b}{}  Order-disorder in hexagonal lattices.  \emph{Physica} 16 (1950), 425--455.
\MR{0039632} (12, 576j).  \Zbl{038.13903} (38, p.\ 139c).				
\annot{} Ising spins, i.e.\ $\zeta: V \to \signnos$, in the triangular and honeycomb (hexagonal) lattice graphs on a torus.  Different edge weights (``bond strengths'') and signs are allowed in the three directions.  The all-negative triangular signature (i.e., ``antiferromagnetic'' with equal weights) is an exceptional case.  Switching the triangular lattice (p.\ 449, bottom) permits assuming that two chosen directions are positive.  Exceptional weights are the antibalanced triangular lattice with equal smaller weights, e.g., all weights equal (p.\ 449, bottom).  The honeycomb cannot be exceptional [because it is balanced] (p.\ 451).
[See also \xref{Newell}{1950a}{G.F.\ Newell}, \xref{I.\ Sy\^ozi}{1950a}{Itiro Syozi}, \xref{Wannier}{1950a}{G.H.\ Wannier}.]
\dated{20 Jun 2012} 
\par
\codes{\CPhys,~\CWG,~\Csg:~\Csw}%
\itemspace

\author{Cho-Jui Hsieh}						%HSIEH CJ
\authorspace
\authref  See also \hyperlink{Kai-Yang Chiang}{K.-Y.~Chiang}.
\authrefspace

\author{Cho-Jui Hsieh, Kai-Yang Chiang, \& Inderjit S.\ Dhillon}
\authorspace
\bsgitem{2012a}{}  Low rank modeling of signed networks.  In:  \emph{Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining} (KDD '12, Beijing, 2012), pp.\ 507--515.  ACM, New York, 2012.
\par
\codes{\CSG:~\CBal,~\CClu:~\CAlgor}%
\itemspace

\author{L.\ Hsu}							%HSU L
\authorspace
\authref  See \hyperlink{E.\ Kaszkurwicz}{E.~Kaszkurwicz}.
\itemspace

\author{Bin Hu}							%HU B
\authorspace
\authref  See also \hyperlink{Chuanchao Huang}{C.-C.~Huang}.
\authrefspace

\author{Bin Hu, Zhi-Hong Guan, Xiao-Wei Jiang, Ming Chi, Rui-Quan Liao, \& Chao-Yang Chen}
\authorspace
\bsgitem{2016a}{}  Event-driven multi-consensus of multi-agent networks with repulsive links.  \emph{Inform.\ Sci.}\ 173 (2016), 110--123.
\Zbl{1429.68310}.  
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\authorx{Chunyang Hu}						%HU CY
\author{Chunyang Hu \& Shiping Liu}
\authorspace
\bsgitem{20xxa}{}  Vertex isoperimetry on signed graphs and spectra of non-bipartite Cayley graphs.  \emph{}\ 
\arXiv{2306.05306}.  
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\authorx{Dan Hu}							%HU D
\author{Dan Hu, Hajo Broersma, Jiangyou Hou, \& Shenggui Zhang}
\authorspace
\bsgitem{2021a}{}  On the spectra of general random mixed graphs.  \emph{Electronic J.\ Combin.}\ 28 (2021),  no.\ 1, art.\ P1.3.
\MR{4245236}.  \Zbl{1468.05156}.  
\annot{} Random $\{1,\pm\mathrm{i}\}$-gain graphs.  Probabilistic upper bound.  Approximation of normalized Laplacian spectrum.
\dated{8 Feb 2021} 
\par
\codes{\Cgg:\CRand:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Dan Hu, Xueliang Li, Xiaogang Liu, \& Shenggui Zhang}
\authorspace
\bsgitem{2017a}{}  The spectral distribution of random mixed graphs.  \emph{Linear Algebra Appl.}\ 519 (2017), 343--365.
\MR{3606276}.  \Zbl{1357.05082}.  
\annot{} A random $\{1,\pm\mathrm{i}\}$-gain graph is $\Phi_n$ with probabilities on each edge of $p^2$ for gain $1$, $p(1-p)$ for each gain $\pm\mathrm{i}$, and $(1-p)^2$ for no edge, where $p = p(n) \in (0,1)$.  Thm.\ 2:  The limiting spectral distribution of $\Phi_n\cdot(1/\sqrt{np(2-p-p^3)})$ as $n \to \infty$ is the standard semicircle distribution.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CRand:~\CAdj:~\CEig}%
\itemspace

\authorx{Fu-Tao Hu}							%HU FT
\author{Fu-Tao Hu \& Mei-Yu Sun}
\authorspace
\bsgitem{20xxa}{}  Induced subgraph and eigenvalues of some signed graphs.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ (to appear).
\annot{} Certain Cartesian products $\Sigma_1 \times \Sigma_2$.	
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\authorx{Guang Hu}							%HU G
\author{Guang Hu \& Wen-Yuan Qiu}
\authorspace
\bsgitem{2009a}{}  Extended Goldberg polyhedral links with odd tangles.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 61 (2009), no.\ 3, 753--766.
\Zbl{1189.92027}.  
\annot{} See \xref{Flapan}{1995a}{Erica Flapan}.
\dated{4 Nov 2010} 
\codes{\Csg:~\CTop,~\CChem}%
\itemspace

\author{Jiangping Hu}						%HU JP
\authorspace
\authref  See \hyperlink{Yanzhi Wu}{Y.-Z.~Wu} and \hyperlink{Yulong Zhou}{Y.-L.~Zhou}.
\itemspace

\authorx{Lili Hu}							%HU LL
\author{Lili Hu \& Xiangwen Li}
\authorspace
\bsgitem{2018a}{}  Every signed planar graph without cycles of length from 4 to 8 is 3-colorable.  \emph{Discrete Math.}\ 341 (2018), no.\ 2, 513--519.
\MR{3724119}.  \Zbl{1376.05067}.  
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{2018b}{}  Nowhere-zero flows on signed wheels and signed fans.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 41 (2018), no.\ 4, 1697--1709.
\MR{3854484}.  \Zbl{1404.05073}.  
\par
\codes{\CSG:~\CFlows,~\COri}%
\itemspace

\author{Wentao Hu}							%HU WT
\authorspace
\authref  See \hyperlink{Qiumin Guo}{Q.M.~Guo}.
\itemspace

\author{Bobo Hua}							%HUA BB
\authorspace
\authref  See \hyperlink{Fatihcan M.\ Atay}{F.M.~Atay}.
\itemspace

\author{Hongbo Hua}						%HUA HB
\authorspace
\bsgitem{2007a}{}  Bipartite unicyclic graphs with large energy.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 58 (2007), no.\ 1, 57--73. 
\MR{2335478} (2008d:05101).  \Zbl{1224.05301}.  
\annot{} Fix $n \geq 13$.  For connected unicyclic $\Gamma$ such that $-\Gamma$ is balanced, excluding circles and balloons (``tadpoles'', ``lollipops''), the maximum energy occurs for a hexagon attached by an edge to the third vertex of a path.
[\Problem.  Replace ``bipartite'' by ``signed'', i.e., allow unbalanced signed graphs.]  
\dated{24 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Chuanchao Huang}						%HUANG CC
\author{Chuanchao Huang, Bin Hu, Ruixian Yang, \& Guangmei Wu}
\authorspace
\bsgitem{2018a}{}  SNMFP: A two-stage approach to community detection in signed networks.  \emph{Physica A} 510 (2018), 754--764.
\codes{\CSG:~\CClu:~\CAdj,~\CDyn:~\CAlgor}%
\itemspace

\author{Hao Huang}							%HUANG H
\authorspace
\bsgitem{2019a}{}  Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture. \emph{ Ann.\ Math.}\ (2) 190 (2019), no.\ 3, 949--955.
\MR{4024566}.  \Zbl{1427.05116}.  \arXiv{1907.00847}.				
\annot{}  Lemma 2.2 has $A(Q_n,\sigma)$ in which every quadrilateral is negative.  Lemma 2.3 is the [previously known] bound $\lambda_{\max}(\Sigma) \leq \Delta(|\Sigma|)$.
\dated{31 Jul 2019} 
\codes{\Csg:~\CAdj,~\CEig}%
\itemspace

\author{He Huang}							%HUANG H
\authorspace
\authref  See \hyperlink{Hanyuan Deng}{H.Y.~Deng}.
\itemspace

\author{Jin Huang}							%HUANG J
\authorspace
\authref  See \hyperlink{Chen Chen}{C.~Chen}.
\itemspace

\authorx{Jing Huang}							%HUANG J
\author{Jing Huang, Shuchao Li, \& Hua Wang}
\authorspace
\bsgitem{2018a}{}  Relation between the skew-rank of an oriented graph and the independence number of its underlying graph.  \emph{J.\ Combin.\ Optim.}\ 36 (2018), 65--80.
\MR{3811192}.  \Zbl{1398.05093}.  \arXiv{1704.06867}.				
\annot{} Skew adjacency matrix $= \mathrm{i}\inv A(\Phi)$ where gain graph $\Phi$ has $\phi(\overarrow{v_iv_j}) = \mathrm{i}$.
\dated{15 Jul 2019} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Qiongxiang Huang}						%HUANG QX
\authorspace
\authref  See \hyperlink{Lu Lu}{L.~Lu} and \hyperlink{Jianfeng Wang}{J.F.~Wang}.
\itemspace

\authorx{Rong Huang}						%HUANG R
\author{Rong Huang, Jianzhou Liu, \& Li Zhu}
\authorspace
\bsgitem{2011a}{}  A structural characterization of real $k$-potent matrices.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 4, 433--439.
\MR{2802524} (2012d:15023).  \Zbl{1237.15029}.					
\codes{\CQM:~\CSD}%
\itemspace

\author{Sumin Huang}						%HUANG SM
\authorspace
\authref  See \hyperlink{Xiangyu Ren}{X.-Y.~Ren}.
\itemspace

\author{Tao Huang}							%HUANG T
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Ting-Zhu Huang}						%HUANG TZ
\authorspace
\authref  See \hyperlink{Gui-Xian Tian}{G.X.~Tian}, \hyperlink{Jingming Zhang}{J.M.~Zhang}, and \hyperlink{Ling Zhang}{L.~Zhang}.
\itemspace

\author{Yihua Huang}						%HUANG YH
\authorspace
\authref  See \hyperlink{Yu-Bin Gao}{Y.-B.~Gao}.
\itemspace

\author{Yufei Huang}							%HUANG YF
\authorspace
\authref  See also \hyperlink{Chaohua Liang}{C.H.~Liang}.
\authrefspace

\author{Yufei Huang, Bolian Liu, \& Siyuan Chen}
\authorspace
\bsgitem{2012a}{}  The generalized $\tau$-bases of primitive non-powerful signed digraphs with $d$ loops.  \emph{Graphs Combin.}\ 28 (2012), 227--242.
\MR{2891644}.  \Zbl{1256.05093}.  
\codes{\CSD:~\Cqm}%
\itemspace

\author{Yufei Huang, Bolian Liu \& Yingluan Liu}
\authorspace
\bsgitem{2011a}{}  The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences.  \emph{Discrete Math.}\ 311 (2011), no.\ 6, 504--511.
\MR{2799902} (2012a:05189).  \Zbl{1222.05130}.					
\annot{} The largest spectral radius and the extremal graphs.
\dated{19 Nov 2011} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Zexi Huang}						%HUANG ZX
\author{Zexi Huang, Arlei Silva, \& Ambuj Singh}
\authorspace
\bsgitem{2022a}{}  POLE: Polarized embedding for signed networks.  In:  \emph{Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining} (WSDM '22, 2022, Tempe, Ariz.), 11 pp.  ACM, New York, 2022.
\arXiv{2110.09899}.  
\codes{\CSG:~\Cfr:~\CAlgor,~\CPsS}%
\itemspace

\author{B.A.\ Huberman}						%HUBERMAN
\authorspace
\authref  See \hyperlink{Eduardo Fradkin}{E.~Fradkin}.
\itemspace

\author{Falk H\"uffner}						%HUFFNER
\authorx{Falk Huffner}
\authorspace
\authref  See also \hyperlink{Sebastian Bocker}{S.~B\"ocker}.
\authrefspace

\author{Falk H\"uffner, Nadja Betzler, \& Rolf Niedermeier}
\authorx{Falk Huffner, Nadja Betzler, \& Rolf Niedermeier}
\authorspace
\bsgitem{2007a}{}  Optimal edge deletions for signed graph balancing.  
In:  Camil Demetrescu, ed., \emph{Experimental Algorithms} (6th Int.\ Workshop, WEA 2007, Rome, 2007), pp.\ 297--310.  \LNCS 4525.  Springer-Verlag, Berlin, 2007.
\Zbl{1203.68125}.  
\annot{} An improved algorithm for frustration index of $-\Sigma$.	
Dictionary:  ``balanced'' = antibalanced, i.e., balance of $-\Sigma$; ``2-coloring that minimizes inconsistencies with given edge labels'' = switching function that minimizes $|E^-(-\Sigma)|$.
\dated{10 Sept 2011}
\codes{\CSG:~\CFr:~\CAlgor}%
\itemspace

\bsgitem{2010a}{}  Separator-based data reduction for signed graph balancing.  \emph{J.\ Combin.\ Optim.}\ 20 (2010), no.\ 4, 335--360.
\MR{2734305} (2011j:05325).  \Zbl{1206.90201}.					
\annot{} Dictionary:  ``balanced'' = antibalanced, i.e., balance of $-\Sigma$.
\par
\codes{\CSG:~\CFr:~\CAlgor}
\itemspace

\author{Falk H\"uffner, Christian Komusiewicz, \& Andr\'e Nichterlein}
\authorx{Falk Huffner, Christian Komusiewicz, \& Andre Nichterlein}
\authorspace
\bsgitem{2015a}{}  Editing graphs into few cliques: Complexity, approximation, and kernelization schemes.  In:  Frank Dehne \etal, eds., \emph{Algorithms and Data Structures} (Proc.\ 14th Int.\ Symp., WADS 2015, Victoria, B.C., 2015), pp.\ 410--421.  \LNCS 9214.  Springer, Cham, 2015.
\MR{3677580}.  \Zbl{06502370}.  
\codes{\Csg:~\Ckg:~\CClu,~\CFr:~\CAlgor}%
\itemspace

\author{Florian Hug}							%HUG
\authorspace
\authref  See \hyperlink{Irina E.\ Bocharova}{I.E.~Bocharova}.
\itemspace

\authorx{JiSun Huh}							%HUH JS
\author{JiSun Huh, Sangwook Kim, \& Boram Park}
\authorspace
\bsgitem{2021a}{}  On toric ideals arising from signed graphs.  \emph{J.\ Algebraic Combin.}\ 53 (2021), 1265--1298.
\MR{4263651}.  \Zbl{1478.13040}.  \arXiv{1810.02082}.  
\annot{} Unifies toric ideals of graphs and digraphs (\cf\ \xref{Villarreal}{1995a}{Rafael H.\ Villarreal}, \xref{Ohsugi and Hibi}{1999a}{Hidefumi Ohsugi \& Takayuki Hibi}, \xref{Reyes, Tatakis, and Thoma}{2012a}{Enrique Reyes, Christos Tatakis, \& Apostolos Thoma}, \etal) by generalization to bidirected graphs $\Beta$ (Beta).  
A positive closed walk $W = e_1\cdots e_l$ is broken at incoherent vertices into segments $W_1,\ldots,W_k$ ($k\geq1$) and gives binomial $B_W := \prod E(W_\text{odd}) - \prod E(W_\text{even})$, where $W_\text{odd} := W_1 \cup W_3 \cup \cdots$ and $W_\text{even} := W_2 \cup W_4 \cup \cdots$.  The toric ideal $I_\Beta \subseteq \bbZ[E]$ is generated by those binomials.  
\annotctd{} \S3.1, ``Primitive binomials''. Thm.\ 3.1 (restated):  $B_W$ is primitive iff $W$ is an edge-simple walk that consists of a cactus whose outer circles are negative and inner circles are positive.
\annotctd{} [See \xref{Reyes, Tatakis, and Thoma}{2012a}{Enrique Reyes, Christos Tatakis, \& Apostolos Thoma} for two proposed signed-graph problems.]
\annotctd{} Dictionary:  ``signed graph'' = bidirected graph $\Beta$; ``balanced vertex'' in a walk = coherent vertex; ``even-, odd-signed'' walk = positive, negative walk; $A(\Beta)$ = incidence matrix $\Eta$; $r(\Beta) = \nul \Eta$.
\par
\codes{\Csg:~\COri:~\CInc,~\CAlgeb}%
\itemspace


\author{Axel Hultman}						%HULTMAN
\authorspace
\bsgitem{2002a}{}  Polygraph arrangements.  \emph{European J.\ Combin.}\ 23 (2002), 937--948.
\MR{1938350} (2003i:05137).  \Zbl{1018.52009}.  
\annot{} \S 5, ``A Dowling generalization''.  	
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2007a}{}  The topology of spaces of phylogenetic trees with symmetry.  \emph{Discrete Math.}\ 307 (2007), no.\ 14, 1825--1832.
\MR{2316821} (2008a:05055).  \Zbl{1109.92031}.					
\annot{} Introduces Dowling trees: ``Natural Dowling analogues of the complex of phylogenetic trees''.  
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2007b}{}  Link complexes of subspace arrangements.  
\emph{European J.\ Combin.}\ 28 (2007), no.\ 3, 781--790.
\MR{2300759} (2007m:52029).  \Zbl{1113.52038}.  \arXiv{math/0507314}.			
\annot{} Interprets chromatic polynomials of signed graphs in terms of Hilbert polynomials.  
\codes{\CSG:~\CInvar}%
\itemspace

\author{John Hultz} 							%HULTZ
\authorspace
\authref  See also \hyperlink{Fred Glover}{F.~Glover}.
\authrefspace

\author{John Hultz \& D.\ Klingman} 
\authorspace
\bsgitem{1979a}{}  Solving singularly constrained generalized network problems.  \emph{Appl.\ Math.\ Optim.}\ 4 (1978), 103--119.  
\MR{0475831} (57 \#15414).  \Zbl{373.90075}.
\par
\codes{\CGN:~\CMatrd:~\Cbases}%
\itemspace

\authorx{Norman P.\ Hummon}						%HUMMON
\author{Norman P.\ Hummon \& Patrick Doreian}
\authorspace
\bsgitem{2003a}{\dag}  Some dynamics of social balance processes: bringing Heider back into balance theory.  \emph{Social Networks} 25 (2003), 17--49.
\newcommand\vS{\vec\Sigma}
\annot{} Presents a model for evolution of balance and clusterability (as in \xref{Davis}{1967a}{James A.\ Davis}) of a signed digraph and explores it via computer simulations.  
\annotctd{} Definitions:  Given a signed digraph $\vS$ and a partition $\pi$ of $V$, define the `clusterability' $c(\vS,\pi) :=$ (\# negative edges within blocks of $\pi$) $+$ (\# positive edges between blocks).  Define $\pi(\vS) :=$ any $\pi$ that minimizes $c(\vS,\pi)$.  Define $\vS(v_i) := \{\vec{v_iv_j} \in \vec E(\vS)\}$ with signs.  ($\vS$ models relations in a social group $V$.  $\vS_i$ is the graph of relations perceived by $v_i$.)
\annotctd{} Initial conditions:  Fixed $\#V$, fixed ``contentiousness'' $p :=$ the probability that an initial edge is negative, a fixed ``communication'' rule, random $\vS^0$ and, for each $v_i \in V$, $\vS_i^0 := \vS^0$.
At time $t+1$, $\vS_i^t(v_i)$ changes to $\vS_i^{t+1}(v_i)$ to minimize $d(d(\vS_i^{t+1},\pi(\vS^t))$.  Then $\vS_j^{t+1}(i)$ changes to $\vS_i^{t+1}(v_i)$ for some $v_j$ (depending on $\vS_i$ and the communication rule).
\annotctd{} Computer simulations examined the types of changes and emerging clusterability of $\vS^t$ or $\vS_i^t$ as $t$ increases, under four different communication rules, random initial conditions with various $p$, and $\#V=3,5,7,10$.  The outcomes are highly suggestive (see \S4; $p$ seems influential).  
[\Problem.  Predict the outcomes in terms of initial conditions through a mathematical analysis.]
\dated{26 Apr 2009}
\par
\codes{\CSD,~\Csg:~\CBal,~\CClu:~\CAlgor}\codesctd{\CPsS}%
\itemspace

\author{Norman P.\ Hummon \& T.J.\ Fararo}
\authorspace
\bsgitem{1995a}{}  Assessing hierarchy and balance in dynamic network models.  \emph{J.\ Math.\ Sociology} 20 (1995), 145--159.
\Zbl{858.92032}.  
\itemspace

\author{David J.\ Hunter}						%HUNTER
\authorspace
\bsgitem{2012a}{}  \emph{Essentials of Discrete Mathematics}.  Second ed.  Jones \& Bartlett Learning, Sudbury, Mass., 2012.  Third ed., 2017.
\Zbl{1236.00002}.  
\annot{} \S6.2.4, ``Signed graphs and balance''.  Elementary.
\dated{12 Jan 2018} 
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\author{John E.\ Hunter} 						%HUNTER
\authorspace
\bsgitem{1978a}{}  Dynamic sociometry.  \emph{J.\ Math.\ Sociology} 6 (1978), 87--138.  
\MR{0504069} (58 \#20631).  
\codes{\CSG:~\CBal,~\CClu}%
\itemspace

\authorx{Bofeng Huo}						%HUO BF
\author{Bofeng Huo, Shengjin Ji, Xueliang Li, \& Yongtang Shi}
\authorspace
\bsgitem{2011a}{}  Solution to a conjecture on the maximal energy of bipartite bicyclic graphs.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 4, 804--810.
\MR{2807234} (2012e:05232).  \Zbl{1220.05073}.					
\annot{} [\Question.  Do the results generalize to $A(\Sigma)$ for antibalanced signed bicyclic graphs?]
\dated{8 Sept 2016} 
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Bofeng Huo, Xueliang Li, \& Yongtang Shi}
\authorspace
\bsgitem{2011a}{}  Complete solution to a problem on the maximal energy of unicyclic bipartite graphs.  \emph{Linear Algebra Appl.}\ 434 (2011), no.\ 5, 1370--1377.
\MR{2763594} (2011m:05176).  \Zbl{1205.05146}.					
\annot{} [\Question.  Do the results generalize to $A(\Sigma)$ for antibalanced signed unicyclic graphs?]
\dated{21 Mar 2011} 
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\authorx{Li Fang Huo}						%HUO LF
\author{Li Fang Huo \& Yu Bin Gao}
\authorspace
\bsgitem{2010a}{}  Local bases of two class of primitive nonpowerful signed digraphs with girth 2.  \emph{Math.\ Pract.\ Theory} 40 (2010), no.\ 10, 235--239.
\MR{2730313} (no rev).  
\par
\codes{\CSD:~\CAdj,~\Cqm}%
\itemspace

\author{C.A.J.\ Hurkens} 						%HURKENS
\authorspace
\bsgitem{1989a}{}  On the existence of an integral potential in a weighted bidirected graph.  \emph{Linear Algebra Appl.}\ 114/115 (1989), 541--553.  
\MR{0986893} (90c:05142).  \Zbl{726.05050}.						
\annot{} Given: a bidirected graph $\Beta$ (with no loose or half edges or positive loops) and an integer weight $b_e$ on each edge.  Wanted: an integral vertex weighting $x$ such that $\incid(B)\transpose x \leq b$, where $\incid(B)$ is the incidence matrix.  Such $x$ exists iff (i) every coherent circle or handcuff walk has nonnegative total weight and (ii) each doubly odd Korach walk (a generalization of a coherent handcuff that has a cutpoint dividing it into two parts, each with odd total weight) has positive total weight.  This improves a theorem of \xref{Schrijver}{1991a}{Alexander Schrijver} and is best possible.
Dictionary: ``path'' (``cycle'') = coherent (closed) walk.
\codes{\Csg:~\COri:~\CInc}%
\itemspace

\author{Jake Huryn}							%HURYN
\authorspace
\authref  See \hyperlink{Oscar Coppola}{O.~Coppola}.
\itemspace

\author{David A.\ Huse}						%HUSE
\authorspace
\authref  See \hyperlink{Creighton K.\ Thomas}{C.K.~Thomas}.
\itemspace

\author{Joan Hutchinson}						%HUTCHINSON
\authorspace
\authref  See \hyperlink{Dan Archdeacon}{D.~Archdeacon}.
\itemspace

\author{Daniel Huttenlocher}						%HUTTENLOCHER
\authorspace
\authref  See \hyperlink{Jure Leskovec}{J.~Leskovec}.
\itemspace

\author{Tony Huynh}							%HUYNH
\authorspace
\authref  See also \hyperlink{Michele Conforti}{M.~Conforti} and \hyperlink{Jim Geelen}{J.~Geelen}.
\authrefspace

\bsgitem{2009a}{}  (As Tony Chi Thong Huynh)  \emph{The Linkage Problem for Group-labelled Graphs}.  Doctoral thesis, University of Waterloo, 2009. 
\annot{} Gains are in $\GF(q)^\times$ or sometimes in a finite abelian group.  
Dictionary:  ``group-labelled graph'' = gain graph, ''Dowling matroid'' = frame matroid (not Dowling geometry), ``shifting'' = switching.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Tony Huynh, Felix Joos, \& Paul Wollan}
\authorspace
\bsgitem{2019a}{}  A unified Erd\H{o}s--P\'osa theorem for constrained cycles.  \emph{Combintorica} 39 (2019), no.\ 2, 91--133.
\MR{3936194}.  \Zbl{1438.05198}.  \arXiv{1605.07082}.				
\annot{} Double gain graph: gains in $\fG_1 \times \fG_2$.  Doubly nonneutral circle: both gains $\neq1$.
\codes{\CGG:~\CCircles:~\CStr}%
\itemspace

\author{Tony Huynh, Andrew D.\ King, Sang-Il Oum, \& Maryam Verdian-Rizi}
\authorspace
\bsgitem{2017a}{}  Strongly even cycle decomposable graphs.  \emph{J.\ Graph Theory} 84 (2017), no.\ 2, 158--175.
\MR{3601124}.  \arXiv{1209.0160}.  \Zbl{1354.05073}.  				
\annot{} $\Gamma$ is ``strongly even cycle decomposable'' iff every signed graph $(\Gamma,\sigma)$ decomposes into positive circles (confusingly called ``even cycles'').
Dictionary:  \cf\ \xref{Huynh, Oum, and Verdian-Rizi}{2017a}{Tony Huynh, Sang-il Oum, \& Maryam Verdian-Rizi}.
\dated{26 Dec 2012} 
\codes{\CSG:~\CCircles}%
\itemspace

\author{Tony Huynh, Sang-il Oum, \& Maryam Verdian-Rizi}
\authorspace
\bsgitem{2017a}{}  Even cycle decompositions of graphs with no odd-$K_4$-minor.  \emph{European J.\ Combin.}\ 65 (2017), 1--14.
\MR{3679832}.  \Zbl{1369.05172}.  \arXiv{1211.1868}.				
\annot{} Despite the name, decomposability of $\Sigma$ inton(edge-disjoint) positive circles.
Dictionary:  ``even'' = positive, ``odd'' = negative (hence, unnecessary confusion), ``even-length'' = even, ``re-signing'' = switching, ``parity'' of vertex = parity of $d^-(v)$.
\dated{26 Dec 2012} 
\par
\codes{\CSG:~\CCircles}%
\itemspace


\label{LETTER I} \hypertarget{LETTER I}{}	% LETTER  I


\author{Dawn Iacobucci}						%IACOBUCCI
\authorspace
\bsgitem{1994a}{}  Graphs and matrices.  In:  Stanley Wasserman \& Katherine Faust, eds., \emph{Social Network Analysis:  Methods and Applications}, Ch.\ 4, pp.\ 92--166.  Structural Anal.\ Soc.\ Sci., 8.  Cambridge Univ.\ Press, Cambridge, 1994.
\annot{} See \xref{Wasserman and Faust}{1994a}{Stanley Wasserman \& Katherine Faust, eds.}.
\dated{21 Jan 2019} 
\par
\codes{\CPsS,~\CSG,~\CSD:~\CBal,~\CExp,~\CRef}%
\itemspace

\author{Giovanni Iacono}						%IACONO
\authorspace
\authref  See also \hyperlink{Giuseppe Facchetti}{G.~Facchetti} and \hyperlink{Nicola Soranzo}{N.~Soranzo}.
\authrefspace

\author{Giovanni Iacono \& Claudio Altafini}
\authorspace
\bsgitem{2010a}{}  Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks.  \emph{BMC Systems Biol.}\ 4 (2010), art.\ 83, 14 pp.\ + suppl.
	
\codes{\CSD,~\CSG:~\CFr,~\CSw,~\CState,~\CAlgor,~\CBiol}%
\itemspace

\author{G.\ Iacono, F.\ Ramezani, N.\ Soranzo, \& C.\ Altafini}
\authorspace
\bsgitem{2010a}{}  Determining the distance to monotonicity of a biological network: a graph-theoretical approach.  \emph{IET Systems Biol.}\ 4 (2010), no.\ 3, 223--235.
	
\par
\codes{\CSD,~\CSG:~\CFr,~\CSw,~\CAlgor,~\CBiol}%
\itemspace

\author{Toshihide Ibaraki} 						%IBARAKI
\authorspace
\authref  See also \hyperlink{Yves Crama}{Y.~Crama} and \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}.
\authrefspace

\author{T.\ Ibaraki \& U.N.\ Peled} 
\authorspace
\bsgitem{1981a}{}  Sufficient conditions for graphs to have threshold number 2.  
In: Pierre Hansen, ed., \emph{Studies on Graphs and Discrete Programming} (Proc.\ Workshop, Brussels, 1979), pp.\ 241--268.
North-Holland Math.\ Studies, 59.  Ann.\ Discrete Math., 11.  North-Holland, Amsterdam, 1981.  
\MR{0653829} (84f:05056).  \Zbl{479.05058}.
\codes{\Cpar:~\Cori}%
\itemspace

\author{Takashi Iino}							%IINO
\authorspace
\authref  See \hyperlink{Takeo Yoshikawa}{T.~Yoshikawa}.
\itemspace

\author{Takeo Ikai}							%IKAI
\authorspace
\authref   See \hyperlink{Hideo Kosako}{H.~Kosako}.
\itemspace

\authorx{Yoshiko T.\ Ikebe}						%IKEBE
\author{Yoshiko T.\ Ikebe \& Akihisa Tamura}
\authorspace
\bsgitem{2003a}{}  Polyhedral proof of a characterization of perfect bidirected graphs.  \emph{IEICE Trans.\ Fundam.\ Electronic Commun.\ Computer Sci.}\ E86-A (2003), no.\ 5, 1000--1007.
\codes{\Csg:~\COri:~\CGeom}%
\itemspace

\bsgitem{20xxa}{}  Perfect bidirected graphs.  Manuscript.  
\annot{} A transitively closed bidirection of a simple graph is perfect iff its underlying graph is perfect.  (See \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg} for definitions.)  [Also proved by \xref{Sewell}{1996a}{E.C.\ Sewell}.]
\codes{\Csg:~\COri:~\CInc,~\CGeom}%
\itemspace

\authorx{Rintaro Ikeshita}						%IKESHITA
\author{Rintaro Ikeshita \& Shin-Ichi Tanigawa}
\authorspace
\bsgitem{2018a}{\dag}  Count matroids of group-labeled graphs.  \emph{Combinatorica} 38 (2018), no.\ 5, 1101--1127.
\MR{3884781}.  \Zbl{1424.05039}.  \arXiv{1507.01259}.				
\codes{\CGG:~\CMatrd}%
\itemspace

\authorx{Victor Il'ev}							%ILEV
\author{Victor Il'ev, Svetlana Il'eva, \& Alexander Kononov}
\authorspace
\bsgitem{2016a}{}  Short survey on graph correlation clustering with minimization criteria.  In:  Yury Kochetov \etal, eds., \emph{Discrete Optimization and Operations Research} (9th Int.\ Conf., DOOR 2016, Vladivostok, 2016), pp.\ 25--36.  \LNCS 9869.  Springer, [Cham], 2016.
\MR{3577727}.  \Zbl{1380.68314}.  
\codes{\Csg:~\CClu}%
\itemspace

\author{Svetlana Il'eva}						%ILEVA
\authorspace
\authref  See \hyperlink{Victor Il'ev}{V.~Il'ev}.
\itemspace

\author{Aleksandar Ili\'c}						%ILIC A
\authorx{Aleksandar Ilic}
\authorspace
\authref  See \hyperlink{Lihua Feng}{L.H.~Feng}, \hyperlink{Guihai Yu}{G.H.~Yu}, and \hyperlink{Bo Zhou}{B.~Zhou}.
\itemspace


\author{Denis Petrovich Ilyutko}						%ILYUTKO
\authorspace
\authref  See \hyperlink{Vassily Olegovich Manturov}{V.O.~Manturov}.
\itemspace

\author{Nicole Immorlica}						%IMMORLICA
\authorspace
\authref  See \hyperlink{Erik D.\ Demaine}{E.~Demaine}.
\itemspace

\author{Takehiro Inohara}						%INOHARA
\authorspace
\bsgitem{1999a}{}  On conditions for a meeting not to reach a recurrent argument.  \emph{Appl.\ Math.\ Comput.}\ 101 (1999), 281--298.
\MR{1677966} (99k:90010).  \Zbl{942.91019}.					
\par
\codes{\CSD,~\CPsS}
\itemspace

\bsgitem{2000a}{}  Meetings in deadlock and decision makers with interperception.  \emph{Appl.\ Math.\ Comput.}\ 109 (2000), 121--133.
\MR{1738208} (2000m:91035).  \Zbl{1042.91010}.					
\par
\codes{\CSD,~\CPsS}
\itemspace

\bsgitem{2002a}{}  Characterization of clusterability of signed graph in terms of Newcomb's balance of sentiments.  \emph{Appl.\ Math.\ Comput.}\  133 (2002), no.\ 1, 93--104.
\MR{1923185} (2003i:05064).  \Zbl{1023.05072}.					
\annot{} Assumption: all $\sigma(i,i)=+$.  Thm.\ 3: A signed complete digraph is clusterable iff $\sigma(i,j)=-$ or $\sigma(j,k)=\sigma(i,k)$ for every triple $\{i,j,k\}$ of vertices (not necessarily distinct).  [The notation is unnecessarily complicated.]
\codes{\CSD:~\CClu,~\CPsS}%
\itemspace

\bsgitem{2003a}{}  Clusterability of groups and information exchange in group decision making with approval voting system.  \emph{Appl.\ Math.\ Comput.}\ 136 (2003), no.\ 1, 1--15.
\MR{1935595} (2004b:91059).  \Zbl{1042.91086}.						
\codes{\CSD:~\CKG:~\CBal,~\CClu,~\CPsS}%
\itemspace

\bsgitem{2004a}{}  Quasi-clusterability of signed graphs with negative self evaluation.  \emph{Appl.\ Math.\ Comput.}\ 158 (2004), no.\ 1, 201--215.
\MR{2091243} (2005f:05072).  \Zbl{1055.05074}.					
\par
\codes{\CSD:~\CClu,~\CPsS}%
\itemspace

\bsgitem{2004b}{}  Signed graphs with negative self evaluation and clusterability of graphs.  \emph{Appl.\ Math.\ Comput.}\ 158 (2004), no.\ 2, 477--487.
\MR{2094633} (2005f:05073).  \Zbl{1054.05048}.					
\codes{\CSD:~\CClu,~\CPsS}%
\itemspace

\bsgitem{2007a}{}  Relational dominant strategy equilibrium as a generalization of dominant strategy equilibrium in terms of a social psychological aspect of decision making.  \emph{European J.\ Oper.\ Res.}\ 182 (2007), 856--866.
\Zbl{1121.90355}.  
\codes{\CSD,~\CPsS}
\itemspace

\author{Takehiro Inohara, Shingo Takahashi, \& Bunpei Nakano}
\authorspace
\bsgitem{1998a}{}  On conditions for a meeting not to reach a deadlock.  \emph{Appl.\ Math.\ Comput.}\ 90 (1998), 1--9.
\MR{1485601}.  \Zbl{907.90014}.							
\codes{\CSD,~\CPsS}
\itemspace

\bsgitem{2000a}{}  Credibility of information in `soft' games with interperception of emotions.  \emph{Appl.\ Math.\ Comput.}\ 115 (2000), 23--41.
\MR{1779380} (2001e:91037).  \Zbl{1046.91004}.					
\codes{\CSD,~\CPsS}
\itemspace

\author{Jun-ichi Inoue}						%INOUE
\authorspace
\authref  See \hyperlink{Sei Suzuki}{S.~Suzuki}.
\itemspace

\authorx{Yuri J.\ Ionin}						%IONIN
\author{Yuri J.\ Ionin \& Mohan S.\ Shrikhande}
\authorspace
\bsgitem{2006a}{}  \emph{Combinatorics of Symmetric Designs}.  Cambridge Univ.\ Press, Cambridge, Eng., 2006.
\MR{2234039} (2008a:05001).  \Zbl{1114.05001}.					
\annot{} \S7.3, ``Switching in strongly regular graphs'':  Graph switching and two-graphs.
\codes{\CTG,~\CSw:~\CExp}%
\itemspace

\authorx{Jerad Ipsen}						%IPSEN
\author{Jerad Ipsen \& Sudipta Mallik}
\authorspace
\bsgitem{20xxa}{}  Incidence and Laplacian matrices of wheel graphs and their inverses.  Submitted.  \emph{}\ 
\arXiv{2201.02579}.  
\annot{} Moore--Penrose inverse of $\Lap(-\Gamma)$.
\dated{13 Oct 2022} 
\par
\codes{\Csg:~\CPar:~\CIncid,~\CLap}%
\itemspace

\author{Ali Iranmanesh}						%IRANMANESH
\authorspace
\authref  See \hyperlink{Jalal Askari}{J.~Askari}.
\itemspace

\author{Masao Iri}							%IRI
\authorspace
\authref  See also \hyperlink{J.\ Shiozaki}{J.~Shiozaki}.
\authrefspace

\author{Masao Iri \& Katsuaki Aoki}
\authorspace
\bsgitem{1980a}{}  A graphical approach to the problem of locating the origin of the system failure.  \emph{J.\ Operations Res.\ Soc.\ Japan} 23 (1980), 295--312.  
\MR{0606141} (82c:90041).  \Zbl{447.90036}.  
\codes{\CSD,~\CVS:~\CAppl}
\itemspace

\author{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama} 
\authorspace
\bsgitem{1976a}{}  [A graphical approach to the problem of locating the system failure.] (In Japanese.) [???] 76(135) (1976), 63--68.
\codes{\CSD,~\CVS:~\CAppl}%
\itemspace

\bsgitem{1979a}{}  An algorithm for diagnosis of system failures in the chemical process.  \emph{Computers and Chem.\ Eng.}\ 3 (1979), 489--493 (1981).  
\annot{} The process is modelled by a signed digraph with some nodes $v$ marked by $\mu(v) \in \{+,-,0\}$.  (Marks $+, -$ indicate a failure in the process.)  Object: to locate the node which is origin of the failure.  An oversimplified description of the algorithm:  $\mu$ is extended arbitrarily to $V$.  Arc $(u, v)$ is discarded if $0 \neq \mu(u)\mu(v) \neq \sigma(u,v)$.  If the resulting digraph has a unique initial strongly connected component $S$, the nodes in it are possible origins.  Otherwise, this extension provides no information.  (I have overlooked: special marks on ``controlled'' nodes; speedup by stepwise extension and testing of $\mu$.)  
[Continued in \xref{Shiozaki, Matsuyama, O'Shima, and Iri}{1985a}{J.\ Shiozaki, H.\ Matsuyama, E.\ O'Shima, \& M.\ Iri}.]
[This article and \xrefd{1976a}{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama} seem to be the origin of a whole literature.  See e.g.\ \xref{Chang and Yu}{1990a}{Chung-Chien Chang \& Cheng-Ching Yu}, \xref{Kramer and Palowitch}{1987a}{M.A.\ Kramer \& B.L.\ Palowitch, Jr.}]  
\codes{\CSD,~\CVS:~\CAppl,~\CAlgor}%
\itemspace


\authorx{Lucas Isenmann}						%ISENMANN
\author{Lucas Isenmann \& Timoth\'ee Pecatte}
\authorx{Lucas Isenmann \& Timothee Pecatte}
\authorspace
\bsgitem{2017a}{}  Mo\"bius stanchion systems.  \emph{Electronic Notes Discrete Math.}\ 62 (2017), 177--182.
\MR{3746719}.  \Zbl{1383.05069}.  
\annot{} Signed plane graph $\Sigma$ is orientation embedded via the plane vertex rotations in surface $S$.  Thm.:  All single-face embeddings are connected via two operations: negating certain edges, and simultaneously negating certain edge pairs.  The edges involved depend on the 1-face walk.
Dictionary:  ``M\"obius edge'' = negative edge; ``painting walk'' = face walk in $S$. ``M\"obius stanchion system (MSS)'' = $\Sigma$ with single-face embedding in $S$ [\cf\ \xref{\v{S}ir\'a\v{n} and \v{S}koviera}{1991a}{Jozef Siran \& Martin Skoviera}].  
\dated{2 Nov 2017} 
\codes{\Csg:~\CTop}%
\itemspace

\author{Toru Ishihara}						%ISHIHARA
\authorspace
\bsgitem{2000a}{}  Cameron's construction of two-graphs.  \emph{Discrete Math.}\ 215 (2000), 283--291.
\MR{1746466} (2000k:05090).  \Zbl{959.05099}.						
\annot{} A new proof of \xref{Cameron}{1994a}{Peter J.\ Cameron}.
\codes{\CTG}%
\itemspace

\bsgitem{2002a}{} Signed graphs associated with the lattice $A_n$.  \emph{J.\ Math.\ Univ.\ Tokushima} 36 (2002), 1--6 (2003).
\MR{1974060} (2004c:05086).  \Zbl{1032.05061}.
\annot{} A signed graph corresponding to a base of $A_n$ is a [signed] path of cliques and locally switches to a path.  (For local switching see \xref{Cameron, Seidel, and Tsaranov}{1994a}{P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov}.)
\codes{\CSG:~\CGeom}\codesctd{\CSG:~\CSw:~\CGen}%
\itemspace

\bsgitem{2004a}{} Local switching of some signed graphs.  \emph{J.\ Math.\ Univ.\ Tokushima} 38 (2004), 1--7. 
\MR{2123167} (2005m:05110).  \Zbl{1067.05032}.
\annot{} Which signed graphs locally switch to a tree?  Examples only.
\par
\codes{\CSG:~\CSw:~\CGen}%
\itemspace

\bsgitem{2005a}{}  Local switching of signed induced cycles.  \emph{J.\ Math.\ Univ.\ Tokushima} 39 (2005), 1--5.
\MR{2194305} (2006i:05077).  
\annot{} Converting an induced circle to a path by local switching.
\par
\codes{\CSG:~\CSw:~\CGen}%
\itemspace

\bsgitem{2007a}{}  Signed graphs and Hushimi trees.  \emph{J.\ Math.\ Univ.\ Tokushima} 41 (2007), 13--23.
\MR{2380208} (no rev).  \Zbl{1138.05316}.  
\annot{} Local switching between trees.
\dated{28 Dec 2011} 
\codes{CSG}%
\itemspace

\author{Sorin Istrail}							%ISTRAIL
\authorspace
\bsgitem{2000a}{}  Statistical mechanics, three-dimensionality and NP-completeness.  I.\ Universality of intractability for the partition function of the Ising model across non-planar lattices.  
In:  \emph{Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing} (STOC, Portland, Ore., 2000), pp.\ 87--96.  ACM, New York, 2000.
\MR{2114521} (no rev).  
\annot{} Extends \xref{Barahona}{1982a}{Francisco Barahona} on finite signed lattice graphs to the computational complexity of (a) ground states (i.e., frustration index) and (more difficult) (b) partition function (generating function of frustrated edges over all states), for signed infinite lattice graphs.  [An infinite lattice graph is (apparently) a graph drawn in $\bbE^d$, crossings allowed, that has translational symmetry in $d$ independent directions.]  General conclusion:  For nonplanar ones they are NP-hard.  
Thm.\ 1:  A lattice graph in $d=2, 3$ is planar iff it does not contain a certain $d=2$ lattice graph $K_0$, the ``Basic Kuratowskian''.  Lem.\ 2:  Every 3-regular graph has a subdivision contained in $K_0$.  
\S5, ``Computational complexity of the 3D Ising models'':  Lattice graphs with signs, subgraphs thereof, all-positive subgraphs, all-negative subgraphs.  Thm.\ 2:  For every subgraph of a signed non-planar infinite lattice graph, computing $l(\Sigma)$ for finite sublattice graphs $\Sigma$ is NP-hard.  Thm.\ 3:  For every subgraph of an all-negative non-planar infinite lattice graph, computing $l(\Sigma)$ for finite sublattice graphs $\Sigma$ is NP-hard.  \S5.3, ``Ising models with $\{-J,+J\}$ interactions'':  For every signed non-planar infinite lattice graph, computing $l(\Sigma)$ for finite sublattice graphs $\Sigma$ is NP-hard; the proof is postponed to ``the full version of the paper'' [which has not appeared].
\dated{21 Aug 2012} 
\codes{\CSG,~\CPhys:~\CFr}%
\itemspace

\author{Gabriel Istrate}						%ISTRATE
\authorspace
\bsgitem{2009a}{}  On the dynamics of social balance on general networks (with an application to XOR-SAT).  Machines, Computations and Universality, Part II.  \emph{Fund.\ Inform.}\ 91 (2009), no.\ 2, 341--356.  
\MR{2516378} (2010f:68140).  \Zbl{1181.91282}.				
\annot{} Imbalance measured by triangles.  Repeatedly change signs of edges of a fixed graph.  Looks for recurrent states and time to become balanced.
\dated{5 May 2010} 
\codes{\CSG:~\CFr:~\CDyn}%
\itemspace

\author{C.\ Itzykson} 							%ITZYKSON
\authorspace
\authref  See \hyperlink{R.\ Balian}{R.~Balian}.
\itemspace

\author{P.L.\ Ivanescu [P.L.\ Hammer]} 					%IVANESCU
\authorspace
\authref  See \hyperlink{Egon Balas}{E.~Balas} and \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}.
\itemspace

\author{Sousuke Iwai} 						%IWAI
\authorspace
\authref  See also \hyperlink{Osamu Katai}{O.~Katai}.
\authrefspace

\author{Sousuke Iwai \& Osamu Katai} 
\authorspace
\bsgitem{1978a}{} Graph-theoretic models of social group structures and indices of group structures.  (In Japanese.)  \emph{Systems and Control} (\emph{Shisutemu to Seigyo})  22 (1978), 713--722.  
\MR{0540551} (80d:92038).  
\codes{CPsS:~\CExp}
\itemspace

\author{Ravi Iyengar [Satteluri R.K.\ Iyengar]}				%IYENGAR
\authorx{Ravi Iyengar}
\authorspace
\authref  See \hyperlink{Avi Ma'ayan}{A.~Ma'ayan}.
\itemspace

\author{Hiroshi Iyetomi}						%IYETOMI
\authorspace
\authref  See \hyperlink{Takeo Yoshikawa}{T.~Yoshikawa}.
\itemspace


\label{LETTER J} \hypertarget{LETTER J}{}	% LETTER  J


\author{Mike Jackanich}						%JACKANICH
\authorspace
\authref  See \hyperlink{Matthias Beck}{M\~Beck}.
\itemspace

\author{Bill Jackson}							%JACKSON
\authorspace
\authref   See \hyperlink{Peter J.\ Cameron}{P.J.~Cameron} and \hyperlink{Jorgen Bang-Jensen}{J.~Bang-Jensen}.
\itemspace

\author{Michael S.\ Jacobson}						%JACOBSON
\authorspace
\authref  See \hyperlink{Arthur H.\ Busch}{A.H.~Busch} and \hyperlink{Ralph J.\ Faudree}{R.J.~Faudree}.
\itemspace

\author{Fabien Jacques}						%JACQUES
\authorspace
\authref  See also \hyperlink{Christopher Duffy}{C.~Duffy}.
\authrefspace

\author{Fabien Jacques \& Alexandre Pinlou}
\authorspace
\bsgitem{2021a}{}  The chromatic number of signed graphs with bounded maximum average degree.  In:  Jaroslav Ne\v{s}et\v{r}il \etal, eds., \emph{Extended Abstracts EuroComb 2021} (European Conf.\ Combin.\ Graph Theory Appl., virtual), pp. 657--662.  Trends Math., Vol.\ 14.  Birkh\"auser, Cham, 2021.
\annot{} Extended abstract of \xrefd{2022a}{Fabien Jacques \& Alexandre Pinlou}.
\codes{\CSG:~\CHom,~\CCol}%
\itemspace

\bsgitem{2022a}{}  The chromatic number of signed graphs with bounded maximum average degree.  \emph{Discrete Appl.\ Math.}\ 316 (2022), 43--59.
\MR{4413252}.  
\arXiv{2104.11121}.  
\codes{\CSG:~\CHom,~\CCol}%
\itemspace

\author{Fran\c{c}ois Jaeger}						%JAEGER*
\authorx{Francois Jaeger}
\authorspace
\bsgitem{1992a}{}  On the Kauffman polynomial of planar matroids.  
In: Jaroslav Ne\v{s}et\v{r}il and Miroslav Fiedler, eds., \emph{Fourth Czechoslovak Symposium on Combinatorics, Graphs and Complexity} (Prachatice, 1990), pp.\ 117--127.  
Ann.\ Discrete Math., Vol.\ 51.  North-Holland, Amsterdam, 1992.  
\MR{1206253} (94d:57016).  \Zbl{763.05021}.						
\annot{} (This is not the colored Tutte polynomial of \xref{Kauffman}{1989a}{Louis H.\ Kauffman}.)  
\annotctd{} Jaeger shows that the Kauffman polynomial, originally defined for link diagrams and here transformed to an invariant of signed plane graphs, depends only on the edge signs and the circle matroid.  It can also be reformulated to be essentially independent of signs.  \Problem.  Define a similar invariant for more general matroids.
\par
\codes{\CSGc,~\CS(\CMatrd):~\CInvar,~\CKnot}%
\itemspace

\author{G.R.\ Jafari}							%JAFARI
\authorspace
\authref  See \hyperlink{A.\ Kargaran}{A.~Kargaran} and \hyperlink{R.\ Masoumi}{R.~Masoumi}.
\itemspace

\author{N.\ Jafari Rad}						%JAFARI RAD
\authorspace
\authref  See \hyperlink{N.\ Kafai}{N.~Kafai}.
\itemspace

\author{R.\ Jagadeesh}						%JAGADEESH
\authorspace
\authref  See \hyperlink{M.R.\ Rajesh Kanna}{M.R.~Rajesh Kanna}.
\itemspace

\author{John C.\ Jahnke} 						%JAHNKE
\authorspace
\authref  See \hyperlink{Julian O.\ Morrissette}{J.O.~Morrissette}.
\itemspace

\author{Kamal Jain}							%JAIN K
\authorspace
\authref  See \hyperlink{Anke van Zuylen}{A.~van~Zuylen}.
\itemspace

\author{Rashmi Jain}							%JAIN R
\authorspace
\authref  See also \hyperlink{Mukti Acharya, Rashmi Jain, \& Sangita Kansal}{M.~Acharya}.
\authrefspace

\author{Rashmi Jain, Mukti Acharya, \& Sangita Kansal}
\authorspace
\bsgitem{2015a}{}  $\cC$-cycle compatible splitting signed graphs $\fS(S)$ and $\Gamma(S)$.  \emph{European J.\ Pure Appl.\ Math.}\ 8 (2015), no.\ 4, 469--477.
\MR{3418484}.  \Zbl{1440.05110}.  
\codes{\CSG:~\CVS}%
\itemspace

\bsgitem{2018a}{}  Vertex equitable labeling of signed bistars.  \emph{Nat.\ Acad.\ Sci.\ Lett.}\ 41 (2018), no.\ 4, 239--242.
\MR{3842237}.  
\codes{}%
\itemspace

\bsgitem{2021a}{}  Characterizations of line-cut signed graphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.}\ 44 (2021), no.\ 2, 147--150.
\MR{4233614}.  
\codes{\CSG:~\CLG(\CGen),~\CVS}%
\itemspace

\author{Rashmi Jain, Sangita Kansal, \& Mukti Acharya}
\authorspace
\bsgitem{2017a}{}  $\cC$-consistent and $\cC$-cycle compatible dot-line signed graphs.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 469--478.
\MR{3754837}.  \Zbl{1383.05143}.  
\codes{\CSG:~\CLG}%
\itemspace

\author{Mahdi Jalili}							%JALILI
\authorspace
\authref  See \hyperlink{Pouya Esmailian}{P.~Esmailian}, \hyperlink{Amin Javari}{A.~Javari}, and \hyperlink{Mohsen Shahriari}{M.~Shahriari}.
\itemspace

\authorx{Fareeha Jamal}						%JAMAL
\author{Fareeha Jamal \& Mehtab Khan}
\authorspace
\bsgitem{2020a}{}  Extremal iota energy of a subclass of bicyclic digraphs and sidigraphs.  \emph{Asian-European J.\ Math.}\ 13 (2020), no. 7, art.\ 2050127, 13 pp.
\MR{4140792}.  
\annot{} \Cf\ \xref{Hafeez and Khan}{2018a}{Sumaira Hafeez \& Mehtab Khan}.
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\authorx{Robert Janczewski}						%JANCZEWSKI
\author{Robert Janczewski, Krzysztof Turowski, \& Bart{\l}omiej Wr\'oblewski}
\authorx{Robert Janczewski, Krzysztof Turowski, \& Bartlomiej Wroblewski}
\authorspace
\bsgitem{20xxa}{}  Edge coloring of graphs of signed class 1 and 2.  Submitted.  \emph{}\ 
\arXiv{2205.15425}.  
\annot{} $\chi'(\Sigma)=\Delta(|\Sigma|)+0$ or $1$ by \xref{Behr}{2020a}{Richard Behr}.  Is $\chi'(\Gamma,\sigma)$ independent of $\sigma$?  If so, $\Gamma$ is said to be in class $1^\pm$ or $2^\pm$, respectively.  Partial characterizations.
\dated{4 Jun 2022} 
\codes{\CSG:~\CECol}%
\itemspace

\authorx{Hye Jin Jang}						%JANG HJ
\author{Hye Jin Jang, Jack Koolen, Akihiro Munemasa, \& Tetsuji Taniguchi}
\authorspace
\bsgitem{2014a}{}  On fat Hoffman graphs with smallest eigenvalue at least $-3$.  \emph{Ars Math.\ Contemp.}\  7 (2014), no.\ 1, 105--121.
\MR{3047614}.  \Zbl{1301.05219}.  \arXiv{1211.3929}.				
\codes{\CSG:~\CEig}%
\itemspace

\author{Abdul Salam Jarrah}						%JARRAH
\authorspace
\authref  See \hyperlink{Eduardo D.\ Sontag}{E.~Sontag}.
\itemspace

\authorx{John J.\ Jarvis}			 			%JARVIS
\author{John J.\ Jarvis \& Anthony M.\ Jezior}
\authorspace
\bsgitem{1972a}{} Maximal flow with gains through a special network.  \emph{Operations Res.}\ 20 (1972), 678--688.  
\MR{0317739} (47 \#6286).  \Zbl{241.90021}.  
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{A.\ Javanmard}						%JAVANMARD
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\authorx{Amin Javari}						%JAVARI
\author{Amin Javari \& Mahdi Jalili}	
\authorspace
\bsgitem{2014a}{}  Cluster-based collaborative filtering for sign prediction in social networks with positive and negative links.  \emph{ACM Trans.\ Intelligent Sys.\ Tech.}\ 5 (2014), no.\ 2, art.\ 24, 19 pp.
\codes{\CSG:~\CClu,~\CPred:~\CAlgor}%
\itemspace

\author{M.\ Javarsineh}						%JAVARSINEH
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{C.\ Jayaprakash}						%JAYAPRAKASH
\authorspace
\authref  See \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\itemspace

\author{C.\ Jayasekaran [Chelliah Jayasekaran]}				%JAYASEKARAN
\authorx{C.\ Jayasekaran}
\authorspace
\authref  See also \hyperlink{G.\ Sumathy}	{G.~Sumathy} and \hyperlink{V.\ Vilfred}{V.~Vilfred}.
\authrefspace

\bsgitem{2007a}{}  \emph{A Study on Self Vertex Switchings of Graphs}, Ph.D.\ dissertation, Manonmanium Sundaranar University, Tirunelveli, India, 2007.
\annot{} A self vertex switching is a Seidel (graph) switching $\Gamma^v \cong \Gamma$ for $v \in V$, or it is $v$ [better called a ``self-switching vertex'', \cf\ MR for \xref{Vilfred and Jayasekaran}{2009a}{V.\ Vilfred \& C.\ Jayasekaran}].  [\Cf\ articles of \hyperlink{Jurriaan Hage}{J.\ Hage}.]
\dated{26 Sept 2012} 
\par
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2012a}{}  Self vertex switchings of unicyclic graphs.  \emph{Graph Theory Notes N.Y.}\ 62 (2012), 29--38.
\MR{3012272}.  
\annot{} See \xref{Jayasekaran}{2007a}{C.\ Jayasekaran}.  Characterizes unicyclic graphs with a self-switching vertex.
\dated{26 Sept 2012} 
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2012b}{}  Self vertex switchings of connected unicyclic graphs.  \emph{J.\ Discrete Math.\ Sci.\ Cryptogr.}\ 15 (2012), no.\ 6, 377--388.
\MR{3060112} (no rev).  \Zbl{1350.05077}.  
\par
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2016a}{}  Self vertex switchings of trees.  \emph{Ars Combin.}\ 127 (2016), 33--43.
\MR{3559076}.  \Zbl{1474.05047}.  
\annot{} See \xref{Jayasekaran}{2007a}{C.\ Jayasekaran}.  Characterizes trees with a self-switching vertex.
\dated{26 Sept 2012} 
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2016b}{}  Self vertex switchings of disconnected unicyclic graphs.  \emph{Ars Combin.}\ 129 (2016), 51--62.
\MR{3560853}.  \Zbl{1413.05318}.  
\codes{\Ctg:~\CSw}%
\itemspace

\author{C.\ Jayasekaran \& M.\ Ashwin Shijo}
\authorspace
\bsgitem{2021a}{}  Anti-duplication self vertex switching in some graphs.  \emph{Malaya J.\ Mat.}\ 9 (2021), no.\ 1, 338--342.
\MR{4229922}.  
\codes{\Ctg:~\CSw}%
\itemspace

\author{C.\ Jayasekaran, J.\ Christabel Sudha, \& M.\ Ashwin Shijo}
\authorspace
\bsgitem{2021a}{}  2-Vertex self switching of forest.  \emph{Nonlinear Stud.}\ 28 (2021), no.\ 3, 749--759.
\MR{4318145}.  
\codes{\Ctg:~\CSw}%
\itemspace

\author{C.\ Jayasekaran \& G.\ Sumathy}
\authorspace
\bsgitem{2016a}{}  Characterization of disconnected two-cyclic graphs with a self vertex switching.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 41 (2016), no.\ 1-3, 123--140.
\MR{3676449}.  \Zbl{1375.05258}.  
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2014a}{}  Self vertex switching of connected two-cyclic graphs.  \emph{J.\ Discrete Math.\ Sci.\ Cryptogr.}\ 17 (2014), no.\ 2, 157--179.
\MR{3241015} (no rev).  \Zbl{1348.05114}.  
\codes{\Ctg:~\CSw}%
\itemspace

\author{Clark Jeffries} 						%JEFFRIES
\authorspace
\bsgitem{1974a}{} Qualitative stability and digraphs in model ecosystems.  \emph{Ecology} 55 (1974), 1415--1419.  
\annot{} Sufficient (and necessary) conditions for sign stability in terms of negative cycles and a novel color test.  Proofs are sketched or (for necessity) absent.  [Necessity is proved in \xref{Logofet and Ul'yanov}{1982a}{D.O.\ Logofet \& N.B.\ Ul'yanov}, \xrefd{1982b}{D.O.\ Logofet \& N.B.\ Ul'yanov}.]
\codes{\CSD:~\CSta}%
\itemspace

\bsgitem{1993a}{}  Some matrix patterns arising in queuing theory.  
In: Richard A.\ Brualdi, Shmuel Friedland, and Victor Klee, eds., \emph{Combinatorial and Graph-Theoretical Problems in Linear Algebra}, pp.\ 165--174.  
IMA Vols.\ Math.\ Appl., 50.  Springer-Verlag, New York, 1993. 
\MR{1240961} (94e:15056).  \Zbl{789.60069}.
\annot{} In a weighted symmetric digraph, a cycle is ``balanced'' if the product of its weights equals the weight product of the inverse cycle (p.\ 171).  If all cycles of length $\geq 3$ are balanced, stability multipliers exist in an associated differential system (Thms.\ 9, 10).  [For weights $a_{ij}$, define gains $\phi(v_i,v_j) := a_{ij}/a_{ji}$.  Then ``balance'' is balance in the gain graph.  \Question:  What can be made of this?]
\dated{13 Apr 2009}
\codes{\Cgg:~\Cbal}%
\itemspace

\author{Clark Jeffries, Victor Klee, \& Pauline van den Driessche} 
\authorspace
\bsgitem{1977a}{} When is a matrix sign stable?  \emph{Canad.\ J.\ Math.}\ 29 (1977), 315--326.  
\MR{0447288} (56 \#5603).  \Zbl{383.15005}.					
\codes{\CSD:~\CSta}%
\itemspace

\author{Clark Jeffries \& P.\ van den Driessche}
\authorspace
\bsgitem{1988a}{}  Eigenvalues of matrices with tree graphs.  
\emph{Linear Algebra Appl.}\ 101 (1988), 109--120. 
\MR{0941299} (89i:05198).  \Zbl{686.05037}.					
\annot{} $A$ is a real matrix whose bipartite graph is a forest.  The signed digraph $\vec\Sigma(A)$ yields information about eigenvalues.  Controllability of solutions of $\dot x(t) = Ax(t)$ may be deduced from $\vec\Sigma(A)$.
\dated{24 July 2010} 
\par
\codes{\CQM:~\CSD}%
\itemspace

\bsgitem{1991a}{}  Qualitative stability and solvability of difference equations.  \emph{Linear Multilinear Algebra} 30 (1991), no.\ 4, 275--282.
\MR{1129184} (92m:39011).  
\par
\codes{\CQM:~\CSD,~\CSol,~\CSta}%
\itemspace

\authorx{Eva Jel{\'\i}nkov\'a}						%JELINKOVA
\authorx{Eva Jelinkova}
\author{Eva Jel{\'\i}nkov\'a \& Jan Kratochv{\'\i}l}
\authorx{Eva Jelinkova \& Jan Kratochvil}
\authorspace
\bsgitem{2008a}{}  On switching to $H$-free graphs.  
In:  Hartmut Ehrig \etal, eds., \emph{Graph Transformations} (4th Int.\ Conf., ICGT 2008, Leicester, U.K., 2008), pp.\ 379--395.  \LNCS 5214.  Springer-Verlag, Berlin, 2008.
\Zbl{1175.68298}.  
\annot{} Characterizing graph switching classes that contain a graph with no $H$ subgraph, for some particular graph $H$.  [An example of \xref{Kratochv{\'\i}l, Ne\v{s}et\v{r}il, \& Z\'yka}{1992a}{Jan Kratochvil, Jaroslav Nesetril, \& Ondrej Zyka}.]
\dated{21 Mar 2011} 
\codes{\CTG:~\CSw}%
\itemspace

\author{Robin Jenkins}						%JENKINS
\authorspace
\authref  See \hyperlink{Peter Abell}{P.~Abell}.
\itemspace

\author{Kevin Jennings}						%JENNINGS
\authorspace
\authref  See \hyperlink{Cian O'Brien}{C.~O'Brien}.
\itemspace

\author{Pablo Jensen}						%JENSEN P
\authorspace
\authref  See \hyperlink{Sergio Gomez}{S.~G\'omez}.
\itemspace

\authorx{Paul A.\ Jensen}			 			%JENSEN PA
\author{Paul A.\ Jensen \& J.\ Wesley Barnes}
\authorspace
\bsgitem{1980a}{}  \emph{Network Flow Programming}.  Wiley, New York, 1980.
\MR{0579183} (82f:90096).  \Zbl{502.90057}.  
Repr.: Robert E.\ Krieger, Melbourne, Fla., 1987.  
\MR{0934708} (89a:90152).  
\annot{} \S1.4:  ``The network-with-gains model.''  \S2.8: ``Networks with gains---\-ex\-am\-ple applications.''  Ch.~9:  ``Network manipulation algorithms for the generalized network.''  Ch.~10:  ``Generalized minimum cost flow problems.''
\codes{\CGN:~\CMatrd(\Cbases)}%
\annotctd{} \S5.5:  ``Negative cycles.'' 
\codes{\COG:~\CMatrd(\Cbases)}%
\itemspace

\bsgitem{1984a}{}  \emph{Potokovoe programmirovanie}.  Radio i Svyaz, Moskva, 1984.  
\Zbl{598.90035}.  
\annot{} Russian translation of \xrefd{1980a}{Paul A.\ Jensen \& J.\ Wesley Barnes}.
\codes{\CGN:~\CMatrd(\Cbases)}\codesctd{\COG:~\CMatrd(\Cbases)}%
\itemspace

\author{P.A.\ Jensen \& Gora Bhaumik} 
\authorspace
\bsgitem{1977a}{}  A flow augmentation approach to the network with gains minimum cost flow problem.  \emph{Management Sci.}\ 28 (1976/77), no.\ 6 (Feb., 1977), 631--643.  
\MR{0441300} (55 \#14163a).  \Zbl{352.90024}.
\codes{\CGN}%
\itemspace

\authorx{Tommy R.\ Jensen}						%JENSEN TR
\author{T.R.\ Jensen \& F.B.\ Shepherd}
\authorspace
\bsgitem{1995a}{}  Note on a conjecture of Toft.  \emph{Combinatorica} 15 (1995), no.\ 3, 373--377.
\MR{1357283} (96h:05073).  \Zbl{832.05038}.  
\annot{} Proves \xref{Toft's}{1975a}{Bjarne Toft} conjecture for a 4-critical graph with a degree-3 vertex, whence for line graphs.
\dated{2 Nov 2017} 
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\author{Tommy R.\ Jensen \& Bjarne Toft}
\authorspace
\bsgitem{1995a}{} \emph{Graph Coloring Problems}.  Wiley, New York, 1995.  
\MR{1304254} (95h:05067).  \Zbl{950.45277}.  						
\annot{} \S8.14: ``$t$-perfect graphs.''  Related to $-\Gamma$ with no subgraph homeomorphic to $-K_4$ (no ``odd-$K_4$'').  \Cf\ \xref{Gerards and Schrijver}{1986a}{A.M.H.\ Gerards \& A.\ Schrijver}, \xref{Gerards and Shepherd}{1998b}{A.M.H.\ Gerards \& F.B.\ Shepherd}.
\codes{\Csg:~\CPar:~\CGeom,~\CStr}%
\annotctd{} \S13.4: ``Bouchet's 6-flow conjecture'' (for signed graphs).  See \xref{Bouchet}{1983a}{Andre Bouchet}, \xref{Khelladi}{1987a}{Abdelkader Khelladi}.
\codes{\CSG:~\CFlows}%
\annotctd{} \S15.9: ``Square hypergraphs.''  Related to nonexistence of even cycles in a digraph and to sign nonsingularity.  See \xref{Seymour}{1974a}{P.D.\ Seymour} and \xref{Thomassen}{1985a}{Carsten Thomassen}, \xrefd{1986a}{Carsten Thomassen}, \xrefd{1992a}{Carsten Thomassen}.
\codes{\Csd:~\CPar:~\Cbal,~\CSol:~\CExp}%
\itemspace

\authorx{Mark Jerrum}						%JERRUM
\authorspace
\authref  See also \hyperlink{Heng Guo}{H.~Guo}.
\authrefspace

\author{Mark Jerrum \& Alistair Sinclair}
\authorspace
\bsgitem{1990a}{}  Polynomial-time approximation algorithms for the Ising model (extended abstract).  In:  Michael S.\ Paterson, ed., \emph{Automata, Languages and Programming} (Proc.\ 17th Int.\ Colloq., Warwick, 1990), pp.\ 462--475.  \LNCS 443.  Springer-Verlag, Berlin, 1990.
\MR{1076810} (91e:68004) (book).  \Zbl{764.65091}.				
\annot{} Extended abstract of \xrefd{1993a}{Mark Jerrum \& Alistair Sinclair}.
\dated{26 Jun 2011} 
\codes{\Csg:~\CFr,~\CPhys}%
\itemspace

\bsgitem{1993a}{}  Polynomial-time approximation algorithms for the Ising model.  \emph{SIAM J.\ Comput.}\ 22 (1993), no.\ 5, 1087--1116.
\MR{1237164} (94g:82007).  \Zbl{782.05076}.					
\annot{} \S6, ``Completeness results'':  The problem {\sc Ising} is to find the partition function $\sum_{\zeta: V \to \signnos} 2^{-\beta H(\Sigma^\zeta)}$ of a signed simple graph $\Sigma$, where $H(\Sigma^\zeta) = \sum_{vw \in E} \sigma^\zeta(vw)$.  Thm.\ 14 suggests nonexistence of certain approximation algorithms.
\dated{26 Jun 2011} 
\codes{\Csg:~\CFr,~\CPhys}%
\itemspace

\author{R.H.\ Jeurissen} 						%JEURISSEN
\authorspace
\bsgitem{1975a}{}  Covers, matchings and odd cycles of a graph.  \emph{Discrete Math.}\ 13 (1975), 251--260.  
\MR{0412039} (54 \#168).  \Zbl{311.05129}.
\annot{} Involves the negative-circle edge-packing number of $-\Gamma$.
\codes{\Cpar:~\CFr}%
\itemspace

\bsgitem{1981a}{}  The incidence matrix and labellings of a graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 30 (1981), 290--301. 
\MR{0624546} (83f:05048).  \Zbl{409.05042}, (\Zbl{457.05047}).			
\annot{} The rank of the incidence matrix of a signed graph, in arbitrary characteristic, generalizing the all-negative results of \xref{Doob}{1974a}{Michael Doob}.  Employs column operations on the incidence matrix.  Application to magic labellings, where at each vertex a number (in a ring) is specified; the value of an edge is added if it enters the vertex and subtracted if it departs.  
\S5, ``Generalizations'': ``Mixed'' graphs, really signed graphs.
\S6: A new proof of \xref{Doob}{1973a}{Michael Doob}'s theorem on the multiplicity of $-2$ as a line-graph eigenvalue in arbitrary characteristic.
\codes{\Csg,~\Cori:~\CInc,~\CEig(\CLG)}%
\itemspace

\bsgitem{1983a}{}  Disconnected graphs with magic labellings.  \emph{Discrete Math.}\ 43 (1983), 47--53.  
\MR{0680303} (84c:05064).  \Zbl{499.05053}.						
\annot{} The graphs, called ``mixed'', are bidirected graphs without introverted edges.
Dictionary: `` `bipartite' '' = balanced (as a signed graph; the term ``balanced'' is herein used with another meaning).
\codes{\Csg,~\Cori:~\Cinc}%
\itemspace

\bsgitem{1983b}{}  Pseudo-magic graphs.  \emph{Discrete Math.}\ 43 (1983), 207--214.  
\MR{0685628} (84g:\-05122).  \Zbl{514.05054}.					
\annot{} Mostly, the graphs are all-negative signed graphs (oriented to be extroverted).
\S5, ``Labelings of mixed graphs'', discusses bidirected graphs without introverted edges; as in the undirected problem, the (signed-graphically) balanced and unbalanced cases differ.
\codes{\Csg,~\Cori:~\CInc}%
\itemspace

\bsgitem{1988a}{}  Magic graphs, a characterization.  \emph{European J.\ Combin.}\ 9 (1988), 363--368.  
\MR{0950055} (89f:05138).  \Zbl{657.05065}.					
\annot{} Connected graphs with magic labellings are classified, separately for bipartite and nonbipartite graphs [as one might expect, due to the connection with the incidence matrix of $-\Gamma$; see \xref{Stewart}{1966a}{B.M.\ Stewart}].
\par
\codes{\Cpar:~\Cinc}%
\itemspace

\author{William S.\ Jewell} 						%JEWELL
\authorspace
\bsgitem{1962a}{}  Optimal flow through networks with gains.  \emph{Operations Res.}\ 10 (1962), 476--499.  
\MR{0144784} (26 \#2325).  \Zbl{109.38203} (109, p.\ 382c).	
\codes{\CGN}%
\itemspace

\author{P.\ Jeyalakshmi}						%JEYALAKSHMI
\authorspace
\bsgitem{2021a}{}  Domination in signed graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ 13 (2021), article 2050094, 11 pp.
\MR{4201751}.  \Zbl{1460.05079}.  
\annot{} $S \subseteq V$ dominates if every $v \notin S$ has more positive than negative neighbors in $S$.  Thm.\ 2.3:  Upper bound on $\#E^-$ if $\exists$ dominating set.  [Independent of $\#E$; should be strengthened.]  Thm.\ 3.1:  $\#S \geq n/(1+\Delta^+)$.  Thm.\ 3.4:  Upper bound on $\min\#S$.  Characterizations for $\min\#S \leq 4$.  Examples: paths and circles.
\dated{27 Dec 2020} 
\codes{\CLab:~\CSG:~\CDom}%
\itemspace

\author{P.\ Jeyalakshmi, K.\ Karuppasamy. \& S.\ Arockiaraj}
\authorspace
\bsgitem{2021a}{}  Independent domination in signed graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ 13 (2021), no.\ 1, art.\ 2050094, 11 pp.
\MR{4324432}.  \Zbl{1478.05065}.  
\par
\codes{\CLab:~\CSG:~\CDom}%
\itemspace

\author{Anthony M.\ Jezior} 						%JEZIOR
\authorspace
\authref  See \hyperlink{John J.\ Jarvis}{J.J.~Jarvis}.
\itemspace

\authorx{Samuel Jezn\'y}						%JEZNY
\authorx{Samuel Jezny}
\author{Samuel Jezn\'y \& Mari\'an Trenkler}
\authorx{Samuel Jezny \& Mari\'an Trenkler}
\authorspace
\bsgitem{1983a}{}  Characterization of magic graphs.  \emph{Czechoslovak Math.\ J.}\ 33(108) (1983), 435--438.  
\MR{0718926} (85c:05030).  \Zbl{571.05030}.						
\annot{} A weak characterization of magic graphs.  [See \xref{Jeurissen}{1988a}{R.H.\ Jeurissen} for a stronger characterization.]
\codes{\Cpar:~\CInc}%
\itemspace

\authorx{Sai Ji}							%JI S
\author{Sai Ji, Dachuan Xu, Donglei Du, \& Ling Gai}
\authorspace
\bsgitem{2020a}{}  Approximation algorithm for the balanced 2-correlation clustering problem on well-proportional graphs.  In:  Zhao Zhang \etal, eds., \emph{Algorithmic Aspects in Information and Management} (14th Int.\ Conf., AAIM 2020, Jinhua, China), pp.\ 97--107.  \LNCS 12290.  Springer, Cham, 2020.
\Zbl{1482.68176}.  
\annot{} Two clusters of equal size in $(K_n,\sigma)$ with fewest bad edges.  (Not a balanced signed graph.)
\dated{17 Sept 2022} 
\codes{\CSG:~\CKG:~\CClu:~\CAlgor}%
\itemspace

\author{Sai Ji, Dachuan Xu, Min Li, \& Yishui Wang}
\authorspace
\bsgitem{2019a}{}  Approximation algorithm for the correlation clustering problem with non-uni\-form hard constrained cluster sizes.  In:  Ding-Zhu Du, Lian Li, \etal, eds., \emph{Algorithmic Aspects in Information and Management} (13th Int.\ Conf., AAIM 2019, Beijing), pp.\ 159--168.  \LNCS 11640.  Springer, Cham, 2019.
\annot{} Part of \xrefd{2022a}{Sai Ji, Dachuan Xu, Min Li, \& Yishui Wang}.
\codes{\CSG:~\CKG:~\CClu:~\CAlgor}%
\itemspace

\bsgitem{2022a}{}  Approximation algorithms for two variants of correlation clustering problem.  \emph{J.\ Combin.\ Optim.}\ 43 (2022), 933--952.
\MR{4452857}.  \Zbl{1495.90155}.  
\annot{} Clustering in signed $K_n$ with fewest bad edge.  Model 1:  Random signs with specified probabilities.  Model 2:  Specified maximum cluster sizes.
\dated{17 Sept 2022} 
\par
\codes{\CSG:~\CKG:~\CClu:~\CRand:~\CAlgor}\codesctd{\CSG:~\CKG:~\CClu:~\CAlgor}%
\itemspace

\author{Shengjin Ji}							%JI SL
\authorspace
\authref  See \hyperlink{Bofeng Huo}{B.~Huo}.
\itemspace

\author{Zhijian Ji}							%JI ZJ
\authorspace
\authref  See \hyperlink{Hua Gao}{H.~Gao} and \hyperlink{Xianzhu Liu}{X.-Z.~Liu}.
\itemspace

\authorx{Peng Jia}							%JIA P
\author{Peng Jia, Noah E.\ Friedkin, \& Francesco Bullo}
\authorspace
\bsgitem{2016a}{}  The coevolution of appraisal and influence networks leads to structural balance.  \emph{IEEE Trans.\ Network Sci.\ Engin.}\ 3 (2016), no.\ 4, 286--298.
\MR{3589253} (no rev).  
\codes{\CSD:~\CKG:~\CBal,~\CClu:~\CDyn}%
\itemspace

\author{Guangfeng Jiang}						%JIANG GF
\authorspace
\authref  See also \hyperlink{Qiumin Guo}{Q.M.~Guo} and  \hyperlink{Weili Guo}{W.L.~Guo}.
\authrefspace

\author{Guangfeng Jiang \& Jianming Yu}
\authorspace
\bsgitem{2004a}{}  Supersolvability of complementary signed-graphic hyperplane arrangements.  \emph{Australasian J.\ Combin.}\ 30 (2004), 261--276.
\MR{2080474} (2005j:05042).  \Zbl{1054.05049}.
\annot{} Characterizes supersolvability of $\bfF(K_n,\sigma)$.  [A special case of \xref{Zaslavsky}{2001a}{Thomas Zaslavsky}.]
\codes{\CSG:~\CGeom:~\Cmatrd}%
\itemspace

\author{Guangfeng Jiang, Jianming Yu, \& Jianghua Zhang}
\authorspace
\bsgitem{2008a}{}  Poincar\'e polynomial of a class of signed complete graphic arrangements.  
In: Kazuhiro Konno  and Viet Nguyen-Khac, eds., \emph{Algebraic Geometry in East Asia---Hanoi 2005} (Proc.\ 2nd Int.\ Conf., Hanoi, 2005), pp.\ 289--297.  
Adv.\ Stud.\ Pure Math., Vol.\ 50.  Math.\ Soc.\ of Japan, Tokyo, 2008. 
\MR{2409562} (2009j:52024).  \Zbl{1144.52025}.					
\annot{} The chromatic polynomial of $K_n(-K_3)$, i.e., $+K_n$ with a triangle made all negative, factors integrally except for a cubic factor.
[See \xref{Zaslavsky}{1982c}{Thomas Zaslavsky}, \S7, for a graph-theoretic treatment of such examples.  One expects a direct proof by adding positive vertices in sequence to $-K_3$.  \Problem.  Evaluate $\chi_\Sigma(\lambda)$ where $\Sigma$ is $\Sigma_1$ with a new vertex positively adjacent to all vertices of $\Sigma_1$.]
\dated{25 Feb 2012} 
\codes{\CSG:~\CGeom,~\CInvar}%
\itemspace

\authorx{Huidi Jiang}							%JIANG HD
\author{Huidi Jiang \& Hongwei Zhang}
\authorspace
\bsgitem{2020a}{}  Output sign-consensus of heterogeneous multiagent systems over fixed and switching signed graphs.  \emph{Int.\ J.\ Robust Nonlinear Control} 30 (2020), no.\ 5, 1938--1955.
\MR{4086453} (no rev).  \Zbl{1465.93197}.  
\annot{} \Cf\ \xref{Sun, Zhang, and Lewis}{2021a}{Zhenyu Sun, Hongwei Zhang, \& Frank L.\ Lewis}.
\codes{\CSD}%
\itemspace

\author{Jing-Jing Jiang}						%JIANG JJ
\authorspace
\authref  See \hyperlink{Shang Wang Tan}{S.W.~Tan} and X.L.~Wu.
\itemspace

\author{Jonathan Q.\ Jiang}						%JIANG JQ
\authorspace
\bsgitem{2015a}{}  Stochastic block model and exploratory analysis in signed networks.  \emph{Phys.\ Rev.\ E} 91 (2015), art.\ 062805, 11 pp.
\arXiv{1501.00594}.  
\codes{\CSG}%
\itemspace

\author{Lan Jiang}							%JIANG L
\authorspace
\authref  See \hyperlink{Ly Dinh}{L.~Dinh}.
\itemspace

\author{Xiao-Wei Jiang}						%JIANG XW
\authorspace
\authref  See \hyperlink{Bin Hu}{B.~Hu}.
\itemspace

\authorx{Ye Jiang}							%JIANG Y
\author{Ye Jiang, Hongwei Zhang, He Cai, \& Jie Chen}
\authorspace
\bsgitem{2017a}{}  Sign-consensus of multi-agent systems over fast switching signed graphs.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 1993--1998.  IEEE, 2017.
\codes{\CSG}%
\itemspace

\author{Ye Jiang, Hongwei Zhang, \& Jie Chen}
\authorspace
\bsgitem{2017a}{}  Sign-consensus of linear multi-agent systems over signed directed graphs.  \emph{IEEE Trans.\ Industrial Electronics} 64 (2017), no.\ 6, 5075--5083.
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\bsgitem{2017b}{}  Sign-consensus of multi-agent systems over signed graphs with intermittent communications.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 5973--5977.  IEEE, 2017.
\par
\codes{\CSG:~\CBal:~\CDyn}%
\itemspace

\authorx{Yiting Jiang}						%JIANG YT
\author{Yiting Jiang, Daphne Der-Fen Liu, Yeong-Nan Yeh, \& Xuding Zhu}
\authorspace
\bsgitem{2019a}{}  Colouring of generalized signed triangle-free planar graphs.  \emph{Discrete Math.}\ 342 (2019), no.\ 3, 836--843.
\MR{3885203}.  \Zbl{1403.05049}.  
\annot{} See \xref{Jin, Wong, and Zhu}{20xxa}{Ligang Jin, Tsai-Lien Wong, \& Xuding Zhu}.  ``Signs'' are permutation gains.
\codes{\CGG:~\CCol}%
\itemspace

\author{Yiting Jiang \& Xuding Zhu}
\authorspace
\bsgitem{2020a}{}  4-colouring of generalized signed planar graphs.  \emph{Electronic J.\ Combin.}\ 27 (2020), no.\ 3, art.\ 3.31, 23 pp.
\MR{4245144}.  \Zbl{1446.05032}.  
\annot{} ``Good'' gains in $\fS_4$ that use $\fe$ must be in $\fV_4$.
Dictionary: ``generalized signed'' = permutation gains in $\fS_4$; 4-coloring = permutation gain coloring by $[4]$.
\dated{21 May 2022} 
\codes{\CGG:~\CCol}%
\itemspace

\authorx{Zilin Jiang}							%JIANG ZL
\author{Zilin Jiang \& Alexandr Polyanskii}
\authorspace
\bsgitem{20xxa}{}  Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below.  Submitted.  \emph{}\ 
\arXiv{2111.10366}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, \& Yufei Zhao}
\authorspace
\bsgitem{20xxa}{}  Spherical two-distance sets and eigenvalues of signed graphs.  Submitted.  \emph{}\ 
\arXiv{2006.06633}.  
\annot{} Fix $-1 \leq \beta < 0 \leq \alpha < 1$.  A two-distance set is $X \subseteq S^d$ such that all $\cos\angle(x,y) \in \{\alpha,\beta\}$.  Thm.\ 1.11:  $\max \#X \leq d + m$ where $m$ asymptotically$_{d \to \infty} =$ max multiplicity of eigenvalue $(1-\alpha)/(\alpha-\beta)$ in a certain family of signed graphs.
\annotctd{} Dictionary:  ``Chromatic number $\chi^\pm(\Sigma)$'' = cluster number, i.e., min \#\ of clusters.
\annotctd{} [\Questions\ for clusterable $\Sigma$:  How can it be switched to remain clusterable?  How does the cluster number change?]
\dated{7 Nov 2020} 
\codes{\CSG:~\CClu,~\CAdj:~\CEig}%
\itemspace

\authorx{Bao Jiao}							%JIAO B
\author{Bao Jiao, Yang Chun, \& Tianyong Qiang (as Tianyongqiang)}
\authorspace
\bsgitem{2010a}{}  Signless Laplacians of finite graphs.  In:   \emph{2010 International Conference on Apperceiving Computing and Intelligence Analysis} (ICACIA, Chengdu, 2010), pp.\ 440--443.  IEEE, 2010.
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Licheng Jiao}						%JIAO LC
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai} and \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\author{Qiang Jiao}							%JIAO Q
\authorspace
\authref  See \hyperlink{Xue Lin}{X.~Lin}.
\itemspace

\author{Yang Jiao}							%JIAO Y
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.-S.~Wu}.
\itemspace

\authorx{Jes\'us Arturo Jim\'enez Gonz\'alez}				%JIMENEZ
\authorx{Jesus Arturo Jimenez Gonzalez}
\author{Jes\'us Arturo Jim\'enez Gonz\'alez \& Andrzej Mr\'oz}
\authorspace
\bsgitem{20xxa}{}  Bidirected graphs, integral quadratic forms and some Diophantine equations.  \emph{}\ 
\arXiv{2304.12555}.  
\codes{\Csg:~\COri:~\CIncid,~\CAlgeb}%
\itemspace

\author{Ra\'ul D.\ Jim\'enez}						%JIMENEZ
\authorx{Raul D.\ Jimenez}
\authorspace
\authref  See \hyperlink{Oscar Rojo}{O.~Rojo}.
\itemspace

\author{Chao Jin}							%JIN C
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\author{Ligang Jin}							%JIN LG
\authorspace
\authref  See also \hyperlink{Yingli Kang}{Y.-L.~Kang}.
\authrefspace

\author{Ligang Jin, Yingli Kang, \& Eckhard Steffen}
\authorspace
\bsgitem{2016a}{} Choosability in signed planar graphs.  \emph{European J.\ Combin.}\ 52A (2016), 234--243.
\MR{3425977}.  \Zbl{1327.05082}.  \arXiv{1502.04561}.				
\codes{\CSG:~\CCol}%
\itemspace

\author{Ligang Jin, Tsai-Lien Wong, \& Xuding Zhu}
\authorspace
\bsgitem{2021a}{}  Colouring of $S$-labelled planar graphs.  \emph{European J.\ Combin.}\ 92 (2021), art.\ 103198, 6 pp.
\MR{4142159}.  \Zbl{1458.05072}.  
\annot{} 
Dictionary:  ``$S$-labelled'' = permutation gains in $S$.
\codes{\CGG:~\CCol}%
\itemspace

\bsgitem{20xxa}{}  Colouring of generalized signed planar graphs.  Submitted.\emph{}\ 
\arXiv{1811.08584}.							
\annot{} 
Dictionary:  ``weak $L$-coloring'' of $\Gamma$ = $L$-list coloring of $+\Gamma$; ``generalized signed graph'' = $\fS_n$-gain graph.
\Cf\ \xref{Jiang, Liu, Yeh, and Zhu}{2019a}{Yiting Jiang, Daphne Der-Fen Liu, Yeong-Nan Yeh, \& Xuding Zhu}, \xref{X.D.\ Zhu}{20xxb}{Xuding Zhu}.
\codes{\CGG:~\CCol}%
\itemspace

\authorx{Xian'an Jin}							%JIN XA
\author{Xian'an Jin \& Fuji Zhang}
\authorspace
\bsgitem{2005a}{}  The Kauffman brackets for equivalence classes of links.  \emph{Adv.\ Appl.\ Math.}\ 34 (2005), no.\ 1, 47--64. 
\MR{2102274} (2005j:57009).  \Zbl{1060.05041}.					
\annot{} They compute the Read--Whitehead chain polynomial of a sign-colored graph in which, for each divalent vertex, the two incident edges have the same color.  This is applied to get the Kauffman bracket of small link diagrams.
[\Cf\ \xref{W.L.\ Yang and Zhang}{2007a}{Weiling Yang \& Fuji Zhang}.]
\codes{\CSGc:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{2007a}{}  The replacements of signed graphs and Kauffman brackets of link families.  \emph{Adv.\ Appl.\ Math.}\ 39 (2007), no.\ 2, 155--172.
\MR{2333646} (2009b:57005).  \Zbl{1129.57004}.  \arXiv{math/0511326}.		
\codes{\CSGc:~\CInvar,~\CKnot}%
\itemspace

\author{Ya-Lei Jin}							%JIN YL
\authorspace
\authref  See \hyperlink{Bian He}{B.A.~He}.
\itemspace

\authorx{Thorsten Joachims}						%JOACHIMS
\author{Thorsten Joachims \& John Hopcroft}
\authorspace
\bsgitem{2005a}{}  Error bounds for correlation clustering.  In:  Luc De Raedt and Stefan Wrobel, eds., \emph{ICML 2005: Proceedings, Twenty-Second International Conference on Machine Learning} (Bonn, 2005), pp.\ 385--392.  ACM, New York, 2005.
MR none.  
\codes{\Csg:~\CClu:~\CAlgor}%
\itemspace

\author{Herbert Jodlbauer}						%JODLBAUER
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.H.~Yu}.
\itemspace

\author{Peter Joffe} 							%JOFFE
\authorspace
\authref  See \hyperlink{Alan J.\ Hoffman}{A.J.~Hoffman}. 
\itemspace

\authorx{Manas Joglekar}						%JOGLEKAR
\author{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan}
\authorspace
\bsgitem{2010a}{}    Balanced group labeled graphs.  
In:  \emph{International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTBC-2010)} (Cochin, 2010) [Summaries], pp.\ 120--121.  
Dept.\ of Math., Cochin Univ.\ of Science and Technology, 2010.
\annot{} Extended abstract of \xrefd{2012a}{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan}.
\dated{13 Jan 2012}
\codes{\CSG,~\CVS:~\CBal}%
\itemspace

\bsgitem{2012a}{\dag\dag}  Balanced group labeled graphs.  Recent Trends in Graph Theory and Combinatorics (Cochin, 2010).  \emph{Discrete Math.}\ 312 (2012), no.\ 9, 1542--1549.
\MR{2899887}.  \Zbl{1239.05162}.  
\annot{} $\Gamma$ has weights $w: V \cup E \to \fA$ where $\fA$ is an abelian group.  ($\fA=\bbZ_2$ is signs.)  ``Balance'' = harmony: the sum around every circle $= 0$.  Thm.\ 1:  There are $\#\fA^{\#V+t-c(\Gamma)}$ harmonious labellings, where $t:=$ number of edge 3-components of $\Gamma$.  Lemma 2.  If $(\Gamma,w)$ is balanced and $u,v$ are edge 3-connected in $\Gamma$, then $2w(P)=w(u)+w(v)$ for every $uv$-path.  
\dated{30 Aug 2010}
\codes{\CGGw:~\CBal}%
\annotctd{} Thm.\ 3 is a construction for all edge 2-connected $\Gamma$ such that $\exists$\ harmonious sign labelling, not all $+$.  
[This is the definitive characterization of consistent vertex signatures as in \xref{Beineke and Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}, improving on \xref{Hoede}{1992a}{Cornelis Hoede}.]
\dated{30 Aug 2010}
\codes{\CSG,~\CVS:~\CBal}%
\itemspace

\author{Rolf Johannesson}						%JOHANNESSON
\authorspace
\authref  See \hyperlink{Irina E.\ Bocharova}{I.E.~Bocharova}.
\itemspace

\author{Karl H.\ Johansson}						%JOHANSSON K
\authorspace
\authref  See \hyperlink{Guodong Shi}{G.-D.~Shi}.
\itemspace

\author{Mikael Johansson}						%JOHANSSON M
\authorspace
\authref  See \hyperlink{Guodong Shi}{G.-D.~Shi}.
\itemspace

\author{David John}							%JOHN
\authorspace
\bsgitem{1998a}{}  Minimal edge cuts to induce balanced signed graphs.  
Proc.\ Twenty-ninth Southeastern Int.\ Conf.\ Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1998).  \emph{Congressus Numer.}\ 132 (1998), 5--8.
\MR{1676490} (99j:\-05178).  \Zbl{991.53332}.	
\annot{} Polynomial-time algorithms to decide balance of a signed graph [this has long been known; see e.g.\ \xref{Hansen}{1978a}{Pierre Hansen}] and allegedly to find the minimum number of negative edges whose deletion makes the graph balanced [the frustration index].  Contract the positive edges, leaving a graph consisting of the negative edges.  To detect balance, look for bipartiteness of the contraction.  [Inferior to the standard algorithm.]  For negative frustration index, find a maximum cut of the contraction.  [Something is wrong, since Max Cut is NP-complete and frustration index contains Max Cut.  I believe the algorithm finds a nonmaximum cut.]
\codes{\CSG:~\CBal,~\CFr:~\CAlgor}%
\itemspace

\author{Eugene C.\ Johnsen}						%JOHNSEN
\authorspace
\bsgitem{1986a}{}  Structure and process:  Agreement models for friendship formation.  \emph{Social Networks} 8 (1986), no.\ 3, 257--306.
\MR{0860770} (87m:92093).  
\annot{} Signed complete digraphs $(\overset{\leftrightarrow}{K}_n,\sigma)$.  A list of permitted isomorphism types of triads (order-3 induced subgraphs) (a ``microstructure'') implies a list of possible $\sigma$'s (a ``macrostructure'').  Ex.\ 2.1:  Permitting symmetrically signed triads with 1 or 3 positive arc pairs gives balanced signed complete graphs as in \xref{Cartwright and Harary's model}{1956a}{Dorwin Cartwright \& Frank Harary}.  Ex.\ 2.2:  Symmetric signs with 0, 1, or 3 positive arc pairs give clusterable signed complete graphs as in \xref{Davis's model}{1967a}{James A.\ Davis}.  Other examples are positive digraph models from the literature, with negative arcs inserted to complete the digraph.  Throughout, empirical data are used to prune potential examples.
\annotctd{} \S 3, ``Submodels and their substantive interpretation:  $\sigma = (\sigma_t)_t$ evolves in [my simplification] discrete time $t \in \bbZ$ according to some combination of four ``processes''.  The corresponding equilibrium signatures are the macrostructure and give the permitted triads.  The processes:  \\
(1) \ $\sigma_t(ac)=\sigma_t(bc) \implies \sigma_{t+1}(ab)=\sigma_{t+1}(ba)=+.$  \\
\hspace*{1.6em} $\sigma_t(ab)=-$ or $\sigma_t(ba)=- \implies \sigma_{t+1}(ac)\neq\sigma_{t+1}(bc).$\\[4pt]
(2) \ $\sigma_t(ac)=\sigma_t(bc) \implies \sigma_{t+1}(ab)=+$ or $\sigma_{t+1}(ba)=+.$  \\
\hspace*{1.6em} $\sigma_t(ab)=\sigma_t(ba)=- \implies \sigma_{t+1}(ac)\neq\sigma_{t+1}(bc).$\\[4pt]
(3) \ $\sigma_t(ab)=+$ or $\sigma_t(ba)=+ \implies \sigma_{t+1}(ac)=\sigma_{t+1}(bc).$  \\
\hspace*{1.6em} $\sigma_t(ac)\neq\sigma_t(bc) \implies \sigma_{t+1}(ab)=\sigma_{t+1}(ba)=-.$\\[4pt]
(4) \ $\sigma_t(ab)=\sigma_t(ba)=+ \implies \sigma_{t+1}(ac)=\sigma_{t+1}(bc).$  \\
\hspace*{1.6em} $\sigma_t(ac)\neq\sigma_t(bc) \implies \sigma_{t+1}(ab)=-$ or $\sigma_{t+1}(ba)=-.$
\annotctd{} \S 4, ``Agreement--friendship processes related to affect'':  The equilibrium triads for each process and some combinations (``microprocesses'') and the corresponding macrostructure.  E.g., (1) with (3) gives the balance model.  Some combinations do not yield macrostructures.  Five combinations are ``core''.
Later \S\S:  Further analysis of the core combinations (``core microprocesses'').
\dated{27 Dec 2012} 
\codes{\CPsS,~\CSD:~\CKG:~\CStr}%
\itemspace

\bsgitem{1989a}{}  The micro-macro connection:  Exact structure and process.  
In:  Fred Roberts, ed., \emph{Applications of Combinatorics and Graph Theory to the Biological and Social Sciences}, pp.\ 169--201.  IMA Vols.\ Math.\ Appl., Vol.\ 17.  Springer-Verlag, New York, 1989.  
\MR{1009376} (90g:92089) (\qv).  \Zbl{725.92026} (\qv).				
\annot{} An elaborate classificatory analysis of ``triads'' (signed complete directed graphs of 3 vertices) vis-\'a-vis ``macrostructures'' (signed complete directed graphs) with reference to structural interactions and implications of triadic numerical restrictions on ``dyads'' (s.c.d.g.\ of 2 vertices).  Connections to certain models of affect in social psychology.  [``Impenetrability!  That's what \emph{I} say!''  ``Would you tell me, please,'' said Alice, ``what that means?'']
\codes{~\CKG,~\CSD,~\CSG:~\CBal,~\CPsS:~\CExp}%
\itemspace

\author{Eugene C.\ Johnsen \& H.\ Gilman McCann} 
\authorspace
\bsgitem{1982a}{}  Acyclic triplets and social structure in complete signed digraphs.  \emph{Social Networks} 3 (1982), 251--272.  
\annot{} Balance and clustering analyzed via triples rather than edges.  [Possible because the digraph is complete.  A later analysis via triples is in \xref{Doreian and Krackhardt}{2001a}{Patrick Doreian \& David Krackhardt}.]
\codes{\CSD:~\CBal,~\CClu}%
\itemspace

\author{Charles R.\ Johnson}						%JOHNSON CR
\authorspace
\authref  See also \hyperlink{Peter J.\ Cameron}{P.J.~Cameron} and \hyperlink{Carolyn Eschenbach}{C.A.~Eschenbach}.
\authrefspace

\bsgitem{1983a}{}  Sign patterns of inverse nonnegative matrices.  \emph{Linear Algebra Appl.}\ 55 (1983), 69--80.
\MR{0719863} (86i:15001).  \Zbl{519.15008}.					
\codes{\CSG:~\CSol}%
\itemspace

\author{Charles R.\ Johnson, Frank Thomson Leighton, \& Herbert A.\ Robinson}
\authorspace
\bsgitem{1979a}{}  Sign patterns of inverse-positive matrices.  \emph{Linear Algebra Appl.}\ 24 (1979), 75--83.
\codes{\CSG:~\CSol}%
\itemspace

\author{Charles R.\ Johnson \& John Maybee}
\authorspace
\bsgitem{1991a}{}  Qualitative analysis of Schur complements.  
In: \emph{Applied Geometry and Discrete Mathematics:  The Victor Klee Festschrift}, pp.\ 359--365.  
DiMACS Ser.\ Discrete Math.\ Theor.\ Computer Sci., Vol.\ 4.  Amer.\ Math.\ Soc., Providence, 1991.  
\MR{1116363} (92h:15004).  \Zbl{742.15009}.
\annot{} In square matrix $A$ let $A[S]$ be the principal submatrix with rows and columns indexed by $S$.  Thm.\ 1:  Assume $A[S]$ is sign-nonsingular in standard form and $i, j \notin S$.  Then the $(i, j)$ entry of the Schur complement of $A[S]$ has sign determined by the sign pattern of $A$ iff, in the signed digraph of $A$, every path $i \rightarrow j$ via $S$ has the same sign.
\codes{\CQM:~\Csd}%
\itemspace

\author{Charles R.\ Johnson, William D.\ McCuaig, \& David P.\ Stanford}
\authorspace
\bsgitem{1995a}{}  Sign patterns that allow minimal semipositivity.  \emph{Linear Algebra Appl.}\ 223--224 (1995), 363--373.
\MR{1340701} (96g:15021).  \Zbl{829.15017}.					
\codes{\CSG:~\CQM:~\CSol}%
\itemspace

\author{Charles R.\ Johnson, Michael Neumann, \& Michael J.\ Tsatsomeros}
\authorspace
\bsgitem{1996a}{}  Conditions for the positivity of determinants.  \emph{Linear Multilinear Algebra} 40 (1996), 241--148.
\MR{1382081} (97a:15014).  \Zbl{866.15001}.					
\codes{\CSD:~\CQM}%
\itemspace

\author{Charles R.\ Johnson, D.D.\ Olesky, Michael Tsatsomeros, \& P.\ van den Driessche}
\authorspace
\bsgitem{1993a}{}  Spectra with positive elementary symmetric functions.  \emph{Linear Algebra Appl.}\ 180 (1993), 247--261.  
\MR{1206420} (94a:15028).  \Zbl{778.15006}.					
\annot{} Suppose the signed digraph $D$ of an $n \times n$ matrix has longest cycle length $k$ and all cycles of $-D$ are negative.  Theorem: If $k = n - 1$, the eigenvalues lie in a domain subtending angle $< 2\pi/k$.  This is known for $k = 2$ but false for $k = n - 3$.
\codes{\CQM,~\CSD}%
\itemspace

\author{Charles R.\ Johnson, Frank Uhlig, \& Dan Warner}
\authorspace
\bsgitem{1982a}{}  Sign patterns, nonsingularity, and the solvability of $Ax = b$.  \emph{Linear Algebra Appl.}\ 47 (1982), 1--9.
\MR{0672727} (84h:15005).  \Zbl{488.15002}.					
\codes{\CSG:~\CSol}%
\itemspace

\author{David S.\ Johnson}						%JOHNSON DS
\authorspace
\bsgitem{1983a}{}  The NP-completeness column: An ongoing guide.  \emph{J.\ Algorithms} 4 (1983), 87--100.
\annot{} \S4, Problem 3, ``Ground state of a spin glass'': Is the ``ground state spin energy'' of a weighted signed graph $\leq K$?  NP-complete for weights $\pm1$ on a two-layer cubic lattice; \cf\ \xref{Barahona}{1982a}{Francisco Barahona}.  Related problems.
\dated{18 Jun 2012} 
\codes{\CSG:~\Cwg,~\CFr,~\CAlgor}%
\itemspace

\author{Ellis L.\ Johnson} 						%JOHNSON EL
\authorspace
\authref  See also \hyperlink{Jack Edmonds}{J.~Edmonds}, \hyperlink{Huiling Gan}{H.L.~Gan}, and \hyperlink{Gilles Gastou}{G.~Gastou}.
\authrefspace

\bsgitem{1965a}{}  Programming in networks and graphs.  Report ORC 65-1, Operations Research Center, University of California, Berkeley, Calif., Jan.\ 1965.  
\annot{} \S7:  ``Flows with gains.'' \S8:  ``Linear programming in an undirected graph.'' \S9:  ``Integer programming in an undirected graph.''
\par
\codes{\CGN:~\CInc,~\CMatrd(\Cbases)}\codesctd{\Cecyc:~\CInc,~\CMatrd(\Cbases),~\CAlgor}%
\itemspace

\bsgitem{1966a}{}  Networks and basic solutions.  \emph{Operations Res.}\ 14 (1966), 619--623.  
\codes{\CGN}
\itemspace

\author{Ellis L.\ Johnson \& Sebastiano Mosterts}
\authorspace
\bsgitem{1987a}{}  On four problems in graph theory.  \emph{SIAM J.\ Algebraic Discrete Methods} 8 (1987), 163--185.  
\MR{0881177} (88d:05097).  \Zbl{614.05036}.					
\annot{} Two of the problems:  Given a signed graph (edges called ``even'' and ``odd'' rather than ``positive'' and ``negative'').  The co-postman problem is     to find a minimum-cost deletion set (of edges).  The ``odd circuit'' problem is to find a minimum-cost negative circle.  The Chinese postman problem is described in a way that involves cobalance and ``switching'' around a circle.  
\codes{\CSG:~\CFr(\CGen),~\CInc}%
\itemspace

\author{Ellis L.\ Johnson \& Manfred W.\ Padberg} 
\authorspace
\bsgitem{1982a}{}  Degree-two inequalities, clique facets, and biperfect graphs.  
In:  Achim Bachem, Martin Grotschel, and Bernhard Korte, eds., \emph{Bonn Workshop on Combinatorial Optimization} (Fourth, 1980), pp.\ 169--187.  
North-Holland Math.\ Studies, 66.  Ann.\ Discrete Math., 16.  North-Holland, Amsterdam, 1982.  
\MR{0686306} (84j:05085).  \Zbl{523.52009}.					
\annot{} Geometry of the bidirected stable set polytope $P(\Beta)$ (which generalizes the stable set polytope to bidirected graphs), defined as the convex hull of $0,1$ solutions of $x_i + x_j \leq 1$, $-x_i - x_j \leq -1$, $x_i \leq x_j$ for extroverted, introverted, and directed edges of $\Beta$.  (Thus, undirected graphs correspond to extroverted bidirected graphs.)  It suffices to treat transitively closed bidirections of simple graphs ([unfortunately] called ``bigraphs'').  [Such a bidirected graph must be balanced.]  A ``biclique'' $(S_+, S_-)$ is the Harary bipartition of a balanced complete subgraph ($S_+$, $S_-$ are the source and sink sets of the subgraph).  It is ``strong'' if no external vertex has an edge directed out of every vertex of $S_+$ and an edge directed into every vertex of $S_-$.  Strong bicliques generate facet inequalities of the polytope.  Call $\Beta$ perfect if these facets (and nonnegativity) determine $P(\Beta)$.  $\Gamma$ is ``biperfect'' if every transitively closed bidirection $\Beta$ of $\Gamma$ is perfect.  Conjectures:  $\Gamma$ is biperfect iff it is perfect.  $\Gamma$ is perfect iff some transitively closed bidirection is perfect.  [Both proved by \xref{Sewell}{1996a}{E.C.\ Sewell} and independently by \xref{Ikebe and Tamura}{20xxa}{Yoshiko T.\ Ikebe \& Akihisa Tamura}.  See e.g.\ \xref{Tamura}{1997a}{Akihisa Tamura}
for further work.]
\codes{\Csg:~\COri:~\CInc,~\CGeom,~\Csw}%
\itemspace

\author{Skyler J.\ Johnson}						%JOHNSON SJ
\authorspace
\authref  See \hyperlink{Lucas Rusnak}{L.J.~Rusnak}.
\itemspace

\author{Will Johnson}						%JOHNSON W
\authorspace
\bsgitem{2012a}{}  Circular planar resistor networks with nonlinear and signed conductors.  Manuscript, 2012.  
\arXiv{1203.4045}.  
\annot{} Problem:  Recover the conductances of branches of an electrical network, given boundary voltages and currents, when conductances can be nonlinear or negative.  The ``conductances'' are gains.  \S11, ``Applications of negative conductances'':  \S 11.1, ``Removing a mild failure of circular planarity'':  A graphical transform of positive conductances can introduce negative conductances, using an idea from \xref{Schr{\o}der}{1995a}{Konrad Schroder} and \xref{Goff}{2003a}{Michael Goff}.  \S 11.2, ``Knot theory'':  Treating the colors $\pm1$ as conductances, a corollary on pseudoline arrangements and tangles.
\annotctd{} Suggests that the proof in \xref{Goff}{2003a}{Michael Goff} is flawed (see fn.\ 4).
\dated{26 Dec 2012} 
\codes{\Cgn:~\CAdj:~\CAppl}\codesctd{\CSGc:~\CKnot}%
\itemspace

\author{M.\ Jones}							%JONES M
\authorspace
\authref  See \hyperlink{R.\ Crowston, G.\ Gutin, M.\ Jones, \& G.\ Muciaccia}{R.~Crowston}.
\itemspace

\author{[Hidde de Jong]}						%JONG
\authorspace
\authref  See \hyperlink{Hidde de Jong}{H.~de~Jong} (under `D').
\itemspace

\author{Felix Joos}							%JOOS
\authorspace
\authref  See \hyperlink{Tony Huynh}{T.~Huynh}.
\itemspace

\authorx{Mohammadreza Jooyandeh}					%JOOYANDEH
\author{Mohammadreza Jooyandeh, Dariush Kiani, \& Maryam Mirzakhah}
\authorspace
\bsgitem{2009a}{}  Incidence energy of a graph.  \emph{MATCH Commun.\ Math.\ Comput.\ Chem.}\ 62 (2009), no.\ 3, 561--572.
\MR{2568740} (2010j:05238).  
\codes{\Cpar:~\CInc}%
\itemspace

\authorx{Tibor Jord\'an}						%JORDAN
\authorx{Tibor Jordan}
\author{Tibor Jord\'an, Vikt\'oria E.\ Kaszanitzky, \& Shin-ichi Tanigawa}
\authorx{Tibor Jordan, Viktoria E.\ Kaszanitzky, \& Shin-ichi Tanigawa}
\authorspace
\bsgitem{2016a}{}  Gain-sparsity and symmetry-forced rigidity in the plane.  \emph{Discrete Comput.\ Geom.}\ 55 (2016), 314--372.
\MR{3458601}.  \Zbl{1366.52021}.  
\annot{} \S3.2, ``Gain-count matroids'':  Gain-graph generalization of \xref{White--Whiteley}{1983a}{Neil White \& Walter Whiteley}'s count matroids.  
\codes{\CGG:~\CGeom,~\CMatrd}%
\itemspace

\authorx{Heather Jordon}						%JORDON
\author{Heather Jordon [Heather Gavlas]}
\authorspace
\authref  See also \hyperlink{Gary Chartrand}{G.~Chartrand} and \hyperlink{Dean Hoffman}{D.~Hoffman}.
\authrefspace

\author{Heather Jordon, Richard McBride, \& Shailesh Tipnis}
\authorspace
\bsgitem{2009a}{}  The convex hull of degree sequences of signed graphs.  \emph{Discrete Math.}\ 309 (2009), no.\ 19, 5841--5848.
\MR{2551962} (2010k:05120).  \Zbl{1208.05043}.					
\annot{} Consider signed simple graphs of order $n$.  $P_n :=$ polytope determined by the inequalities from \xref{Hoffman and Jordon}{2006a}{Dean Hoffman \& Heather Jordon} that characterize net degree vectors.  Thm.\ 2.7:  $P_n = $conv(net degree vectors).  Thm.\ 2.9:  Each vertex of $P_n$ $\leftrightarrow$ a unique signed graph, which is a signed $K_n$.  \S3:  Comparison with net degree vectors of digraphs.
[As in other papers on net degree sequences, the best viewpoint is that ``signed'' edges are oriented negative edges and ``directed'' edges are oriented positive edges.]
\dated{1 Oct 2009} 
\codes{\CSG:~\Cori:~\CInvar:~\CGeom}%
\itemspace

\author{Gwena\"el Joret}						%JORET
\authorx{Gwenael Joret}
\authorspace
\authref  See \hyperlink{Nancy E.\ Clarke}{N.E.~Clarke}, \hyperlink{Michele Conforti}{M.~Conforti}, and \hyperlink{Samuel Fiorini}{S.~Fiorini}.
\itemspace

\author{Leif Kj{\ae}r J{\o}rgensen}					%JORGENSEN
\authorx{Leif Kjaer Jorgensen}
\authorspace
\bsgitem{1989a}{}  Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs.  \emph{J.\ Graph Theory} 13 (1989), 75--85.  
\MR{0982869} (90d:05186).  \Zbl{672.05070}.
\annot{} Let $\sigma_{\text{op}}(\Gamma)$, or $\sigma_{\text{odd}}(\Gamma)$, be the largest $s$ for which $-\Gamma$ contains a subdivision of $-K_s$ (an ``odd-path-$K_sS$''), or $[-\Gamma]$ contains an antibalanced subdivision of $K_s$ (an ``odd-$K_sS$'').  Thm.\ 4: $\sigma_{\text{op}}(\Gamma), \sigma_{\text{odd}}(\Gamma) \approx \sqrt{n}$.  Thms.~7, 8 (simplified): For $p = 4, 5$ and large enough $n = \#V$, $\sigma_{\text{odd}}(\Gamma) \geq p$ or $\Gamma$ is a specific exceptional graph.  \Conjecture\ 9.  The same holds for all $p \geq 4$.  [\Problem.  Generalize this to signed graphs.]
\codes{\Cpar:~\CXtreml}%
\itemspace

\author{J.\ Paulraj Joseph}						%JOSEPH JP
\authorspace
\authref  See \hyperlink{V.\ Vilfred}{V.~Vilfred}.
\itemspace

\authorx{James Joseph}						%JOSEPH J
\author{James Joseph \& Mayamma Joseph}
\authorspace
\bsgitem{2023a}{}  Roman domination in signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ 8 (2023), no.\ 2, 349--358.
\MR{4570022}.  
\annot{} $D$ dominates if $N^+[D]=V$.  Roman dominating function (RDF): $f\in\{0,1,2\}^V$ such that $(I+A(\Sigma))f \geq \mathbf1$ and $f\inv(0) \subseteq N^+(f\inv(2))$.  Roman dominating number $:= \min_f \sum_u f(u)$.  Lemma 2:  5 forbidden induced paths.  Signed paths, cycles, stars admitting RDF are characterized.
\dated{22 Apr 2023} 
\codes{\CSG}%
\itemspace

\bsgitem{20xxa}{}  Roman domination number of signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ (to appear).
\codes{\CSG}%
\itemspace

\author{Mayamma Joseph}						%JOSEPH M
\authorspace
\authref  See \hyperlink{James Joseph}{J.~Joseph}, \hyperlink{Prajakta Bharat Joshi}{P.B.~Joshi} and \hyperlink{S.R.\ Shreyas}{S.R.~Shreyas}.
\itemspace

\authorx{Prajakta Bharat Joshi}						%JOSHI PB
\author{Prajakta Bharat Joshi \& Mayamma Joseph}
\authorspace
\bsgitem{2017a}{}  Further results on color energy of graphs.  \emph{Acta Univ.\ Sapientiae, Informatica} 9 (2017), no.\ 2, 119--133.
\Zbl{1380.05067}.  
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CAdj:~\CEig}%
\itemspace

\author{Prajakta Bharat Joshi, Mayamma Joseph, \& Mukti Acharya}
\authorspace
\bsgitem{20xxa}{}  Color energy of signed graphs.  Submitted.  \emph{}\ 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Shalini Joshi}						%JOSHI
\authorspace
\authref  See \hyperlink{B.\ Devadas Acharya}{B.D.~Acharya}.
\itemspace

\author{J\"urgen Jost}						%JOST
\authorx{Jurgen Jost}
\authorspace
\authref  See also \hyperlink{Raffaella Mulas}{R.~Mulas}.
\authrefspace

\author{J\"urgen Jost \& Raffaella Mulas}
\authorx{Jurgen Jost \& Raffaella Mulas}
\authorspace
\bsgitem{2019a}{}  Hypergraph Laplace operators for chemical reaction networks.  \emph{Adv.\ Math.}\ 351 (2019), 870--896.
\MR{3955999}.  \Zbl{1415.05107}.  \arXiv{1804.01474}.  
\par
\codes{\CSH(\CGen):~\CLap:~\CEig}%
\itemspace

\author{Simon Joyce}						%JOYCE
\authorspace
\bsgitem{2017a}{}  \emph{Interaction Graphs Derived From Activation Functions and Their Application to Gene Regulation.}  Doctoral dissertation, Binghamton University, 2017.
\MR{3755540} (no rev).  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\authorx{Tadeusz J\'ozefiak} 						%JOZEFIAK
\authorx{Tadeusz Jozefiak}
\author{Tadeusz J\'ozefiak \& Bruce Sagan}
\authorx{Tadeusz Jozefiak \& Bruce Sagan}
\authorspace
\bsgitem{1992a}{}  Free hyperplane arrangements interpolating between rootsystem arrangements.  In:  \emph{S\'eries formelles et combinatoirealg\'ebrique} (Actes du colloque, Montr\'eal, 1992), pp.\ 265--270.Publ.\ Lab.\ Combin.\ Inform.\ Math., Vol.\ 11.  D\'ep.\ de math.\ et d'infor\-ma\-tique, Universit\'e de Qu\'ebec \`a Montr\'eal, 1992.  
\annot{} Summarizes the freeness results in \xrefd{1993a}{Tadeusz Jozefiak \& Bruce Sagan}.
\par
\codes{\Csg,~\Cgg:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\bsgitem{1993a}{}  Basic derivations for subarrangements of Coxeter arrangements.  \emph{J.\ Algebraic Combin.}\ 2 (1993), 291--320.  
\MR{1235282} (94j:52023).  \Zbl{798.05069}.
\annot{} The hyperplane arrangements (over fields with characteristic $\neq 2$) corresponding to certain signed graphs are shown to be ``free''.  Explicit bases and the exponents are given.  The signed graphs are:  $+K_{n-1} \subseteq \Sigma_1 \subseteq +K_n$ (known), $\pm K_n \subseteq \Sigma_2 \subseteq \pm K_n^{\circ}$, $\pm K_n \subseteq \Sigma_3 \subseteq \pm K_n^{\circ}$; also, those obtained from $+K_n$ or $K_n^{\circ}$ by adding all negative links in the order of their larger vertex (assuming ordered vertices) (Thms.~4.1, 4.2) or smaller vertex (Thms.~4.4, 4.5); and those obtained from $\pm K_{n-1}$ by adding positive edges ahead of negative ones (Thm.\ 4.3).  
[For further developments see \xref{Edelman and Reiner}{1994a}{Paul H.\ Edelman \& Victor Reiner}.]  
Similar theorems hold for complex arrangements when the sign group is replaced by the complex $s$-th roots of unity (\S5).  The M\"obius functions of $\Sigma_2$, known from \xref{Hanlon}{1988a}{Phil Hanlon}, are deduced in \S6.
\codes{\Csg,~\Cgg:~\CGeom,~\Cmatrd,~\CInvar}%
\itemspace

\author{[Michael Juenger]}						%JUENGER
\authorspace
\authref  See \hyperlink{Michael Junger}{M.~J\"unger}.
\itemspace

\author{Ji-Hwan Jung}						%JUNG JH
\authorspace
\authref  See \hyperlink{Gi-Sang Cheon}{G.-S.~Cheon}.
\itemspace

\author{Michael J\"unger} 						%JUNGER
\authorx{Michael Junger}
\authorspace
\authref   See \hyperlink{Francisco Barahona}{F.~Barahona}, \hyperlink{C.\ De Simone}{C.~De~Simone}, \hyperlink{Martin Grotschel}{M.~Gr\"otschel}, \hyperlink{Frauke Liers}{F.~Liers}, and \hyperlink{Matteo Palassini}{M.~Palassini}.
\itemspace

\authorx{Mark Jungerman}						%JUNGERMAN
\author{Mark Jungerman \& Gerhard Ringel}
\authorspace
\bsgitem{1978a}{} The genus of the $n$-octahedron:  Regular cases.  \emph{J.\ Graph Theory} 2 (1978), 69--75.  
\MR{0485485} (58 \#5315).  \Zbl{384.05037}.
\annot{} ``Cascades'': see \xref{Youngs}{1968a}{J.W.T.\ Youngs}.
\codes{\Csg:~\COri:~\CAppl}%
\itemspace

\author{K.\ J\"ungling}						%JUNGLING
\authorx{K.\ Jungling}
\authorspace
\bsgitem{1975a}{}  Exact solution of a nonplanar two-dimensional Ising model with short range two-spin interaction.  \emph{J.\ Phys.\ C} 8 (1975), L169--L171.
\annot{} Sequel to \xref{J\"ungling and Obermair}{1974a}{K.\ Jungling \& G.\ Obermair}.  Physics of signed diagonal square lattice with two bond strengths, reduced to Baxter model.  [See \xref{Southern, Chui, and Forgacs}{1980a}{B.W.\ Southern, S.T.\ Chui, \& G.\ Forgacs}, \xref{Garel and J.M.\ Maillard}{1983a}{T.\ Garel \& J.M.\ Maillard}.]
\dated{16 Jun 2012} 
\codes{\CPhys:~\Csg:~\Cwg}%
\itemspace

\author{K.\ J\"ungling \& G.\ Obermair}
\authorx{K.\ Jungling \& G.\ Obermair}
\authorspace
\bsgitem{1974a}{}  Note on universality and the eight-vertex model.  \emph{J.\ Phys.\ C} 7 (1974), L363--L365.
\annot{} More general but less developed predecessor of \xref{J\"ungling}{1975a}{K.\ Jungling}.
\dated{16 Jun 2012} 
\codes{\CPhys:~\Csg:~\Cwg}%
\itemspace

\author{Dieter Jungnickel}						%JUNGNICKEL
\authorspace
\authref  See \hyperlink{Christian Fremuth-Paeger}{C.~Fremuth-Paeger}.
\itemspace

\author{Samuel Jurkiewicz}						%JURKIEWICZ
\authorspace
\authref  See \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de Freitas}.
\itemspace

\author{James Justus}						%JUSTUS
\authorspace
\bsgitem{2005a}{}  Qualitative scientific modeling and loop analysis.  \emph{Philosophy of Science} 72 (2005), 1272--1286.
\MR{2295282} (2007j:00008).  
\annot{} Philosophical discussion of qualitative differential equations with emphasis on \xref{Levins}{1975a}{Richard Levins}.
\dated{9 Sept 2010} 
\par
\codes{\CSD:~\CQM:~\CSta:~\CExp}%
\itemspace


\label{LETTER K} \hypertarget{LETTER K}{}	% LETTER  K


\author{Jerald A.\ Kabell} 						%KABELL JA
\authorspace
\authref  See also \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}.
\authrefspace

\bsgitem{1985a}{}  Co-balance in signed graphs.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 10 (1985), 5--8.  
\MR{0959659} (89i:05232).  \Zbl{635.05028}.					
\annot{} Cobalance means that every cutset has positive sign product.  Thm.:  $\Sigma$ is cobalanced iff every vertex star has evenly many negative edges.  For planar graphs, corollaries of this criterion and Harary's bipartition theorem result from duality.  [The theorem follows easily by looking at the negative subgraph.]
\codes{\CSG:~\CBal(\CDual),~\CBal}%
\itemspace

\bsgitem{1988a}{}  An algorithmic look at cycles in signed graphs. 
250th Anniversary Conf.\ on Graph Theory (Fort Wayne, Ind., 1986).  \emph{Congressus Numer.}\ 63 (1988), 229--230.  
\MR{0988654} (90d:05143).  \Zbl{666.05046}.
\codes{\CSG,~\CSD:~\CBal:~\CAlgor}%
\itemspace

\author{Kasper Kabell}						%KABELL K
\authorspace
\authref  See \xref{Geelen and Kabell}{2009a}{Jim Geelen \& Kasper Kabell}.
\itemspace

\authorx{Navid Kafai}						%KAFAI
\author{Navid Kafai \& Farideh Heydari}
\authorspace
\bsgitem{20xxa}{}  Maximizing the indices of a class of signed complete graphs.  \emph{Commun.\ Combin.\ Optim.}\ (to appear).
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{N.\ Kafai, F.\ Heydari, N.\ Jafari Rad, \& M.\ Maghasedi}
\authorspace
\bsgitem{2021a}{}  On the signed complete graphs with maximum index.  \emph{Iranian J.\ Sci.\ Tech.\ Trans.\ A Sci.}\ (2021), no.\ 6, 2085--2090.  
\MR{4329226}.  
\arXiv{2102.03308}.  
\bsgitem{2021b}{}  Correction.  \emph{Iranian J.\ Sci.\ Tech.\ Trans.\ A Sci.}\ (2021), no.\ 6, 2249.
\MR{4329240}.  
\annot{} Assuming $A$ is unicyclic of order $k$, $\lambda_1(K_n(-A))$ is maximized by $A =$ triangle with pendant star.  [\Cf\ \xref{Ebrahim Ghorbani \& Arezoo Majidi}{2021a}{Ebrahim Ghorbani \& Arezoo Majidi}.]
\dated{26 May 2021} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Jeff Kahn} 							%KAHN
\authorspace
\bsgitem{1980a}{}  Varieties of combinatorial geometries.  
In:  \emph{Report on the XVth Denison-O.S.U.\ Math.\ Conf.}\ (Granville, Ohio, 1980), pp.\ 90--91.  Dept.\ of Math., Ohio State University, Columbus, Ohio, 1980.
\annot{} Brief (and slightly incomplete) announcement of \xref{Kahn and Kung}{1982a}{Jeff Kahn \& Joseph P.S.\ Kung}.
\dated{18 July 2014} 
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Jeff Kahn \& Joseph P.S.\ Kung}
\authorspace
\bsgitem{1980a}{} Varieties and universal models in the theory of combinatorial geometries.  \emph{Bull.\ Amer.\ Math.\ Soc.\ (N.S.)} 3 (1980), 857--858.  
\MR{0578380} (81i:05051).  \Zbl{473.05025}. 					
\annot{} Announcement of \xrefd{1982a}{Jeff Kahn \& Joseph P.S.\ Kung}.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1982a}{\dag\dag} Varieties of combinatorial geometries.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 271 (1982), 485--499.  
\MR{0654846} (84j:05043).  \Zbl{503.05010}.					
Repr.\ in:  Joseph P.S.\ Kung,  \emph{A Source Book in Matroid Theory}, pp.\ 395--409, with commentary, pp.\ 335--338.  Birkh\"auser, Boston, 1986.
\MR{0890330} (88e:05028).  \Zbl{597.05019}.
\annot{} A ``variety'' is a class closed under deletion, contraction, and direct summation and having for each rank a ``universal model'', a single member containing all others.  There are two nontrivial types of variety of finite matroids: matroids representable over $\GF(q)$, and gain-graphic matroids with gains in a finite group $\fG$.  The universal models of the latter are the Dowling geometries $Q_n(\fG)$.  
\annotctd{} It is incidentally proved (\S7, pp.\ 490--492) that Dowling geometries of non-group quasigroups cannot exist in rank $n \geq 4$.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1986a}{}  A classification of modularly complemented geometric lattices.  \emph{European J.\ Combin.}\ 7 (1986), 243--248.  
\MR{0862370} (87i:06026).  \Zbl{614.05018}.					
\annot{} Such a geometric lattice of rank $\geq 4$, if not a projective geometry with few points deleted, is a Dowling lattice.
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Jeff Kahn \& Roy Meshulam}
\authorspace								%@pre
\bsgitem{1998a}{} On the number of group-weighted matchings.  \emph{J.\ Algebraic Combin.}\ 7 (1998), 285--290.
\MR{1616012} (99b:05113).  \Zbl{899.05042}.
\annot{} Continues \xref{Aharoni, Meshulam, and Wajnryb}{1995a}{R.\ Aharoni, R.\ Meshulam, \& B.\ Wajnryb} (\qv\ for definitions), generalizing its Thm.\ 1.3 (the case $k=\#\fK = 2$ of the following).  Let $m = $ number of $0$-weight matchings, $\delta = $ minimum degree.  Thm.\ 1.1:  If $m > 0$ then $m \geq (\delta - k + 1)!$.  \Conjecture\ 1.2.  $k$ can be reduced.  (See the paper for details.)  [\Question.  Is there a generalization to weighted digraphs?  One could have two kinds of arcs: some weighted from $\fK$, and some weighted $0$.  The perfect matching might be replaced by an alternating Hamilton cycle or a spanning union of disjoint alternating cycles.]
\codes{\CWG}%
\annotctd{} Thm.\ 2.1:  In a $\fK$-weighted simple digraph with all outdegrees $> k$, there is a nonempty set of disjoint cycles whose total weight is $0$.
\codes{\CWD}%
\itemspace

\authorx{Tom\'a\v{s} Kaiser}						%KAISER
\authorx{Tomas Kaiser}
\author{Tom\'a\v{s} Kaiser, Robert Lukot'ka, Edita M\'a\v{c}ajov\'a, \& Edita Rollov\'a}
\authorx{Tomas Kaiser, Robert Lukotka, Edita Macajova, \& Edita Rollova}
\authorspace
\bsgitem{2019a}{}  Shorter signed circuit covers of graphs.  \emph{J.\ Graph Theory} 92 (2019), no.\ 1, 39--56.
\MR{3984679}.  \Zbl{1425.05128}.  \arXiv{1706.03808}.				
\codes{\CSG:~\Cflows}%
\itemspace

\author{Tom\'a\v{s} Kaiser, Robert Lukot'ka, \& Edita Rollov\'a}
\authorx{Tomas Kaiser, Robert Lukot'ka, \& Edita Rollova}
\authorspace
\bsgitem{2017a}{}  Nowhere-zero flows in signed graphs: A survey.  In:  Domenico Labbate, ed., \emph{Selected Topics in Graph Theory and Its Applications}, pp.\ 85--104.  Lect.\ Notes Sem.\ Interdiscip\. Mat., 14.  Seminario Interdisciplinare di Matematica (S.I.M.), Potenza, Italy, 2017.
\Zbl{06769482}.  \arXiv{1608.06944}.						
\codes{\CSG:~\CFlows}%
\itemspace

\author{Tom\'a\v{s} Kaiser \& Edita Rollov\'a}
\authorx{Tomas Kaiser \& Edita Rollova}
\authorspace
\bsgitem{2016a}{}  Nowhere-zero flows in signed series-parallel graphs.  \emph{SIAM J.\ Discrete Math.}\ 30 (2016), no.\ 2, 1248--1258.
\MR{3510003}.  \Zbl{1338.05107}.  \arXiv{1411.1788}.				
\codes{\CSG:~\CFlows}%
\itemspace

\author{Naonori Kakimura}						%KAKIMURA
\authorspace
\authref  See also \hyperlink{Charles Carlson}{C.~Carlson}.
\authrefspace

\bsgitem{2010a}{}  Matching structure of symmetric bipartite graphs and a generalization of P\'olya's problem.  \emph{J.\ Combin.\ Theory Ser.\ B} 100 (2010), 650--670.
\MR{2718684} (2011j:\-05265).  \Zbl{1208.05112}.					
\annot{} Symmetric matching theory of a bipartite graph with left-right symmetry, with a symmetric Mendelsohn--Dulmage theorem.   
[A symmetrically bipartite graph $\Gamma'$ is the signed covering graph of an all-negative signed graph $-\Gamma$, possibly with half edges.  A symmetrical matching in $\Gamma'$ corresponds to a subgraph of $-\Gamma$ with maximum degree 1.  
\Problem.  Develop the symmetric matching theory of any graph with an involutory, fixed-point-free automorphism in terms of a matching theory of signed graphs with half edges.]  
\dated{29 Sept 2011} 
\codes{\Csg:~\Ccov:~\CStr}%
\itemspace

\authorx{Naonori Kakimura}						%KAKIMURA
\author{Naonori Kakimura \& Ken-ichi Kawarabayashi}
\authorspace
\bsgitem{2012a}{}  Packing cycles through prescribed vertices under modularity constraints.  \emph{Adv.\ Appl.\ Math.}\ 49 (2012), no.\ 2, 97--110.
\MR{2946427}.  \Zbl{1245.05102}.  
\codes{\Cgg:~\CCircles}%
\itemspace

\bsgitem{2013a}{}  Half-integral packing of odd cycles through prescribed vertices.  \emph{Combinatorica} 33 (2013), no.\ 5, 549--572.
\MR{3132926}.  \Zbl{1324.05148}.  
\codes{\Csg:~\CPar:~\CCircles}%
\itemspace

\author{Debajit Kalita}						%KALITA
\authorspace
\authref  See also \hyperlink{R.B.\ Bapat}{R.B.~Bapat} and \hyperlink{S.\ Barik}{S.~Barik}.
\authrefspace

\bsgitem{2012a}{}  \emph{Spectra of Weighted Directed Graphs.}  Doctoral dissertation, Indian Inst.\ of Technology Guwahati, Guwahati, India, 2012.
\codes{\CSG:~\CCov,~\CAdj:~\CEig}%
\itemspace

\bsgitem{2012b}{}  On 3-colored digraphs with exactly one nonsingular cycle.  \emph{Electronic J.\ LInear Algebra} 23 (2012), 397--421. 
\MR{2928567}.  \Zbl{1252.05122}.  
\annot{} ``3-colored digraph'' = gain graph with gains in $\{\pm1,\pm\mathrm{i}\}$ [not a digraph].
\codes{\CSG:~\CCov,~\CAdj}%
\itemspace

\bsgitem{2013a}{}  Determinant of the Laplacian matrix of a weighted directed graph.  In:  Ravindra B.\ Bapat \etal, eds., \emph{Combinatorial Matrix Theory and Generalized Inverses of Matrices} (Proc.\ Int.\ Workshop and Conf., CMTGIM 2012, Manipal), pp.\ 57--63.  Springer India, New Delhi, 2013.
\MR{3075960}.  \Zbl{1291.05123}.  
\codes{\CSG:~\CCov,~\CLap,~\CEig}%
\itemspace

\bsgitem{2013b}{}  Spectral integral variation and unicyclic 3-colored digraphs with second smallest eigenvalue $1$.  \emph{Linear Algebra Appl.}\ 439 (2013), no.\ 1, 55--65.  
\MR{3045222}.  \Zbl{1282.05123}.  
\codes{\CSG:~\CCov,~\CEig}%
\itemspace

\bsgitem{2015a}{}  Properties of first eigenvectors and first eigenvalues of nonsingular weighted directed graphs.  \emph{Electronic J.\ Linear Algebra} 30 (2015), 227--242.
\MR{3358539}.  \Zbl{1323.05079}.  
\annot{} ``Weighted directed graph'' = complex unit gain graph [not a digraph].
	
\codes{\CGG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2016a}{}  Extremizing first eigenvalue of 3-colored digraphs made with given blocks.  \emph{Linear Algebra Appl.}\ 503 (2016), 83--99.
\MR{3492660}.  \Zbl{1338.05162}.  
\annot{} ``Weighted directed graph'' = complex unit gain graph [not a digraph].
	
\par
\codes{\Cgg:~\CLap:~\CEig}%
\itemspace

\author{Debajit Kalita \& Sukanta Pati}
\authorspace
\bsgitem{2012a}{}  On the spectrum of 3-colored digraphs.  \emph{Linear Multilinear Algebra} 60 (2012), no.\ 6, 743--756.
\MR{2929181}.  \Zbl{1244.05145}.  
\annot{} Reproves the eigenvalue theorem of \xref{Fowler}{2002a}{Patrick W.\ Fowler}.
A ``3-colored digraph'' is a bidirected graph where all negative edges are extraverted.  
\dated{13 Jan 2012, 13 Jan 2015} 
\codes{\CSG:~\CCov,~\CAdj,~\CLap:~\CEig}%
\itemspace

\bsgitem{2014a}{}  A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from $\{\pm1, \pm i\}$.  \emph{Linear Algebra Appl.}\ 449 (2014), 417--434.
\MR{3191876}.  
\annot{} The structure of $\{\pm1,\pm\mathrm{i}\}$-gain graphs with one circle $C$ that have the reciprocal eigenvalue property (every $\lambda$ and $1/\lambda$ have the same multiplicity).  They have a perfect matching of pendant edges, or an exceptional structure depending on $\phi(C)$.
\dated{28 May 2013} 
\par
\codes{\Cgg:~\CAdj:~\CEig,~\CStr}%
\itemspace

\author{Debajit Kalita \& Kuldeep Sarma}
\authorspace
\bsgitem{2022a}{}  Unicyclic 3-colored digraphs with bicyclic inverses.  \emph{Electronic J.\ Linear Algebra} 38 (2022), 519--530.
\MR{4494144}.  \Zbl{1497.05150}.  
\annot{} Solved:  $A(\Phi)\inv = A(\Phi')$ with $\Phi$ as in \xrefd{20xxa}{Debajit Kalita \& Kuldeep Sarma}, where $\Phi'$ is bicyclic.
\dated{19 Sept 2022} 
\codes{\Cgg:~\CAdj}%
\itemspace

\bsgitem{20xxa}{}  On the inverse of unicyclic 3-coloured digraphs.  \emph{Linear Multilinear Algebra} (to appear).
\annot{} Solved:  $A(\Phi)\inv = A(\Phi')$ where $\Phi$ is odd unicyclic with perfect matching and circle gain = $\pm\mathrm{i}$.  Special cases of bipartite or unicyclic $\Phi'$.  $\Lap(\Phi)$ must be nonsingular [because $\Phi$ is connected and unbalanced].  Strong reciprocal eigenvalue property.
\dated{19 Sept 2022} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Meenal M.\ Kaliwal}						%KALIWAL
\authorspace
\authref  See \hyperlink{Prabhakar Ramrao Hampiholi}{P.R.~Hampiholi}.
\itemspace

\author{M.\ Kamaraj}							%KAMARAJ
\authorspace
\authref  See \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\itemspace

\authorx{Hidehiko Kamiya}						%KAMIYA
\author{Hidehiko Kamiya, Akimichi Takemura, \& Hiroaki Terao}
\authorspace
\bsgitem{2009a}{}  The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements.  In:  Fouad El Zein, Alexander I.\ Suciu, \etal, eds., \emph{Arrangements, Local Systems and Singularities} (CIMPA Summer School, Istanbul, 2007), pp.\ 177--190.  Progr.\ Math., Vol.\ 283.  Birkh\"auser Verlag, Basel, 2010.
\MR{3025864}.  \Zbl{1370.32011}.  \arXiv{0707.1381}.				
\codes{\Csg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2011a}{}  Periodicity of non-central integral arrangements modulo positive integers.  \emph{Ann.\ Combin.}\ 15 (2011), no.\ 3, 449--464.
\MR{2836451}.  \Zbl{1233.32018}.  \arXiv{0803.2755}.				
\annot{} \S5, ``Example'':  The affino-signed-graphic arrangement $\hat\cB_m^{[0,a]}$ of \xref{Athanasiadis}{1999a}{Christos A.\ Athanasiadis}.
\dated{26 May 2018} 
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2012a}{}  Arrangements stable under the Coxeter groups.  In:  Anders Bj\"orner \etal, eds., \emph{Configuration Spaces: Geometry, Combinatorics and Topology} (Pisa, 2010), pp.\ 327--354.  Centro di Ricerca Mat.\ Ennio De Giorgi (CRM) Series, No.\ 14. Edizioni della Normale, Pisa, 2012.
\MR{3203646}.  \Zbl{1276.14080}.  \arXiv{1103.5179}.				
\annot{} \S3.5, ``Signed all-subset arrangement''.
\codes{\CSG:~\CGeom,~\CInvar}%
\itemspace

\author{[Axel von Kamp]}						%KAMP
\authorspace
\authref  See \hyperlink{Axel von Kamp}{A.~von~Kamp} (under `V').
\itemspace

\authorx{Daniel Kandel}						%KANDEL
\author{Daniel Kandel, Radel Ben-Av, \& Eytan Domany}
\authorspace
\bsgitem{1990a}{\dag}  Cluster dynamics for fully frustrated systems.  \emph{Phys.\ Rev.\ Lett.}\ 65 (1990), no.\ 8, 941--944.
\annot{} A new probabilistic algorithm for clustering in a ground state (a function $s: V \to \signnos$ such that $\#E^- = l(\Sigma)$) of an all-negative (``fully frustrated'') square lattice $\Sigma$.  A ``cluster'' in $s$ is a partition of $V$ such that switching any part does not change $\#E^-$.  The objective is to join vertices connected by satisfied edges but not those joined by frustrated edges; this cannot be solved uniquely for any unbalanced $\Sigma$, so previous methods (used for balanced $\Sigma$), e.g., nearest-neighbor moves in state space (``single spin flips''), are ineffective (see p.\ 942, col.\ 1; p.\ 943, col.\ 2).  The algorithm depends on the square lattice structure since it works on squares (``plaquettes''); it succeeds because it works through plaquettes instead of edges (p.\ 943, col.\ 2).
[\Problem:  Do state-space algorithms help to approximate signed-graph clustering in the sense of \xref{Davis}{1967a}{James A.\ Davis}?  Finding a ground state is NP-hard in general, though not for planar signed graphs (\cf\ \xref{Katai and Iawi}{1978a}{Osamu Katai \& Sousuke Iwai}, \xref{Barahona}{1982a}{Francisco Barahona}).]
\dated{18 Jun 2012} 
\codes{\CPhys,~\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Lawqueen Kanesh}						%KANESH
\authorspace
\authref  See \hyperlink{Avinandan Das}{A.~Das}.
\itemspace

\author{Qinma Kang}							%KANG QM
\authorspace
\authref  See \hyperlink{Wenqiang Duan}{W.-Q.~Duan}.
\itemspace

\authorx{Rongrong Kang}						%KANG RR
\author{Rongrong Kang \& Xiang Li}
\authorspace
\bsgitem{2022a}{}   Coevolution of opinion dynamics on evolving signed appraisal networks.  \emph{}\ Automatica 137 (2022), art.\ 110138, 10 pp.
\MR{4366235} (no rev).  \Zbl{1482.91164}.  
\codes{\CSG:~\CDyn:~\CBal}%
\itemspace

\author{Yingli Kang}							%KANG YL
\authorspace
\authref  See also \hyperlink{Ligang Jin}{L.-G.~Jin}.
\authrefspace

\bsgitem{2018a}{}  Haj\'os-like theorem for signed graphs.  \emph{European J.\ Combin.}\ 67 (2018), 199--207.
\MR{3707227}.  \Zbl{1371.05114}.  \arXiv{1702.08232}.				
\codes{\CSG:~\CCol:~\CStr}%
\itemspace

\author{Yingli Kang, Xiaoyue Chen, \& Ligang Jin}
\authorspace
\bsgitem{2022a}{}  A study on parity signed graphs: the \emph{rna} number.  \emph{Appl.\ Math.\ Comput.}\ 431 (2022), art.\ 127322, 9 pp.
\MR{4444383}.  
\arXiv{2111.04956}.  
\annot{} Answers questions raised in \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara} and \xref{Acharya, Kureethara, and Zaslavsky}{2021a}{Mukti Acharya, Joseph Varghese Kureethara, \& Thomas Zaslavsky}, and more.  Smart new techniques.
	
\dated{15 Nov 2021} 
\codes{\CLab:~\CSG,~\CSw}%
\itemspace

\author{Yingli Kang \& Eckhard Steffen}
\authorspace
\bsgitem{2016a}{}  The chromatic spectrum of signed graphs.  \emph{Discrete Math.}\ 339 (2016), 2660--2663.
\MR{3518416}.  \Zbl{1339.05169}.  \arXiv{1510.00614}.				
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{2017a}{}  Circular coloring of signed graphs.  \emph{J.\ Graph Theory} 87 (2018), no.\ 2, 135--148.
\MR{3742174}.  \Zbl{1383.05103}.  \arXiv{1509.04488}.				
\codes{\CSG:~\CCol}%
\itemspace

\author{Yunfan Kang}						%KANG YF
\authorspace
\authref  See \hyperlink{Wenqiang Duan}{W.-Q.~Duan}.
\itemspace

\authorx{Mariusz Kaniecki}						%KANIECKI
\author{Mariusz Kaniecki, Justyna Kosakowska, Piotr Malicki, \& Grzegorz Marczak}
\authorspace
\bsgitem{2015a}{}  A horizontal mesh algorithm for a class of edge-bipartite graphs and their matrix morsifications.  \emph{Fundamenta Inform.}\ 136 (2015), no.\ 4, 345--379.
\MR{3320020}.  
\codes{\CSG}%
\itemspace

\author{[M.R.\ Rajesh Kanna]}						%KANNA
\authorspace
\authref  See \hyperlink{M.R.\ Rajesh Kanna}{M.R.~Rajesh Kanna} (under `R').
\itemspace

\author{M.\ Rajesh Kannan}						%KANNAN
\authorspace
\authref  See also \hyperlink{Ranjit Mehatari}{R.~Mehatari} and \hyperlink{Aniruddha Samanta}{A.~Samanta}.
\authrefspace

\author{M.\ Rajesh Kannan, Navish Kumar, \& Shivaramakrishna Pragada}	
\authorspace
\bsgitem{2021a}{}  Bounds for the extremal eigenvalues of gain Laplacian matrices.  \emph{Linear Algebra  Appl.}\ 625 (2021), 212--240.
\MR{4264826}.  \Zbl{1465.05073}.  \arXiv{2102.07560}.  
\codes{\CGG:~\CFr,~\CLap:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Normalized Laplacians for gain graphs.  \emph{Amer.\ J.\ Combin.}\ 1 (2022), 20--39.
\arXiv{2009.13788}.  
\codes{\CGG:~\CFr,~\CLap:~\CEig}%
\itemspace

\author{M.\ Rajesh Kannan \& Shivaramakrishna Pragada}
\authorspace
\bsgitem{20xxa}{}  Signed spectral Tur\'an type theorems.  Submitted.  \emph{}\ 
\arXiv{2204.09870}.  
\annot{} Let $m=\#E$.  Thm.\ 3.3:  $\lambda_1^2 \leq 2(m-l(\Sigma))(1-\omega_\textb\inv)$, where $\lambda_1 =$ max eigenvalue of $A(\Sigma)$ and $\omega_\textb =$ balanced clique number.  Thm.\ 3.7:  $m \leq l(\Sigma) + \frac12 n^2 (1-\omega_\textb\inv)$.  Thm.\ 4.4:  $\lambda_1^2 \leq m + [6(t_+-t_-)]^{2/3}$, where $t_\epsilon =$ \# triangles with sign $\epsilon$.  \S5, ``Signed walks and largest eigenvalue''"  Thm.\ 5.1 connects $\omega_\textb$ with the smallest number of negative $r$-walks in any switching $\Sigma^\zeta$.
\dated{20 Jun 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Sangita Kansal}						%KANSAL
\authorspace
\authref  See also \hyperlink{Mukti Acharya, Rashmi Jain, \& Sangita Kansal}{M.~Acharya}, \hyperlink{Rashmi Jain}{R.~Jain}, and \hyperlink{Payal}{Payal}.
\authrefspace

\author{Sangita Kansal \& Payal Dabas}
\authorspace
\bsgitem{2021a}{}  An introduction to signed Petri net.  \emph{J.\ Math.}\ 2021 (2021), art.\ 5595536, 8 pp. 
\MR{4277469}.  \Zbl{1477.68182}.  
\codes{\CSG:~\CBal(\CGen),~\CAppl}%
\itemspace

\authorx{Konstantinos Kaparis}						%KAPARIS
\author{Konstantinos Kaparis \& Adam N.\ Letchford}
\authorspace
\bsgitem{2018a}{}  A note on the 2-circulant inequalities for the max-cut problem.  \emph{Operations Res.\ Lett.}\ 46 (2018), no.\ 4, 443--447.
\MR{3827655}.  \Zbl{1452.90269}.  
\annot{} Greater depth about \xref{Poljak and Turzik}{1992a}{Svatopluk Poljak \& Daniel Turzik}.
\dated{4 Jun 2018} 
\codes{\CPar:~\CFr,~\CGeom:~\CAlgor}%
\itemspace

\author{Vikram Singh Kapil}						%KAPIL
\authorspace
\authref  See \hyperlink{Ram Parkash Sharma}{R.P.~Sharma}.
\itemspace

\author{Ajai Kapoor}							%KAPOOR
\authorspace
\authref   See \hyperlink{Michele Conforti}{M.~Conforti}.
\itemspace

\author{Roman Kapuscinski}						%KAPUSCINSKI
\authorspace
\authref   See \hyperlink{Patrick Doreian}{P.~Doreian}.
\itemspace

\author{D.\ Karapetyan}						%KARAPETYAN	
\authorspace
\authref  See \hyperlink{Gregory Gutin}{G.~Gutin}.
\itemspace

\author{Mehran Kardar}						%KARDAR
\authorspace
\authref  See \hyperlink{Lawrence Saul}{L.~Saul}.
\itemspace

\authorx{Franti\v{s}ek Kardo\v{s}}					%KARDOS
\authorx{Frantisek Kardos}
\author{Franti\v{s}ek Kardo\v{s} \& Jonathan Narboni}
\authorx{Frantisek Kardos \& Jonathan Narboni}
\authorspace
\bsgitem{2021a}{}  On the 4-color theorem for signed graphs.  \emph{European J.\ Combin.}\ 91 (2021), art.\ 103215, 8 pp.
\MR{4161809}.  \Zbl{1458.05073}.  \arXiv{1906.05638}.				
\annot{} A signed planar graph may have chromatic number $>4$, disproving a conjecture of \xref{M\'a\v{c}ajov\'a, Raspaud, and \v{S}koviera}{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}.
\dated{26 Jul 2019} 
\codes{\CSG:~\CCol}%
\itemspace

\authorx{A.\ Kargaran}						%KARGARAN
\author{A.\ Kargaran, M.\ Ebrahimi, M.\ Riazi, A.\ Hosseiny, \& G.R.\ Jafari}
\authorspace
\bsgitem{2020a}{}  Quartic balance theory: Global minimum with imbalanced triangles.  \emph{Phys.\ Rev.\ E} 102 (2020), art.\ 012310, 8 pp.
\MR{4137484}.  
\arXiv{2001.01719}.						
\annot{} Measuring imbalance by triples and quadruples.	
\dated{18 Dec 2020} 
\codes{\CSG:~\CFr}%
\itemspace

\author{M.M.\ Karkhaneei}						%KARKHANEEI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\authorx{Richard M.\ Karp}						%KARP
\author{Richard M.\ Karp, Raymond E.\ Miller, \& Shmuel Winograd}
\authorspace
\bsgitem{1967a}{}  The organization of computations for uniform recurrence equations.  \emph{J.\ Assoc.\ Computing Machinery} 14 (1967), 563--590.  
\MR{0234604} (38 \#2920).  \Zbl{171.38305} (171, p.\ 383e).
\annot{} Implicitly, concerns the existence of nonpositive directed tours (closed trails) in a $\bbZ^d$-gain graph (the ``dependence graph'' of a system of recurrences).
\codes{\Cgd:~\Ccov}%
\itemspace

\author{K.\ Karuppasamy}						%KARUPPASAMY
\authorspace
\authref  See \hyperlink{P.\ Jeyalakshmi}{P.~Jeyalakshmi}.
\itemspace

\author{Alexander V.\ Karzanov}					%KARZANOV
\authorspace
\authref  See \hyperlink{Maxim A.\ Babenko}{M.A.~Babenko} and \hyperlink{Andrew V.\ Goldberg}{A.V.~Goldberg}.
\itemspace

\authorx{Yasuhiro Kasai}						%KASAI Y
\author{Yasuhiro Kasai, Ayao Okiji, \& Itiro Syozi}
\authorspace
\bsgitem{1981a}{}  Application of real replica method to Syozi model.  \emph{Progress Theor.\ Phys.}\ 65 (1981), no.\ 1, 140--153.
\codes{\CPhys:~\CSG}%
\itemspace

\bsgitem{1981b}{}  The ground state of a replicated Ising system.  \emph{Progress Theor.\ Phys.}\ 65 (1981), no.\ 4, 1439--1442.
\MR{0620472} (82h:82030).  \Zbl{1074.82528}.  
\annot{} Grand partition function $:= \sum_\theta \exp(\#E^--\#E^+)$ over all edge signatures $\theta$ and all switchings of a lattice graph, investigated for a physical phase via multiple replicates and analytic continuation.  [The relevance to signed graphs is obscured by summing over all signatures.]
\dated{17 Aug 2012} 
\codes{\CPhys:~\CSG}%
\itemspace

\bsgitem{1981c}{}  Ising replicated system of $\pm J$ model.  \emph{Progress Theor.\ Phys.}\ 66 (1981), no.\ 5, 1561--1573.
\MR{0642957} (83b:82081).  \Zbl{1074.82547}.  
\annot{} Similar to \xrefd{1981b}{Yasuhiro Kasai, Ayao Okiji, \& Itiro Syozi}, without analytic continuation.  \S2 recapitulates \xrefd{1981b}{Yasuhiro Kasai, Ayao Okiji, \& Itiro Syozi}.  \S3 does calculations for the path graph.  \S4, ``The ground state''.
\dated{17 Aug 2012} 
\codes{\CPhys:~\CSG}%
\itemspace

\author{Yoshi Kashima}						%KASHIMA
\authorspace
\authref  See \hyperlink{Garry Robins}{G.~Robins}.
\itemspace

\authorx{Stanis{\l}aw Kasjan}						%KASJAN
\authorx{Stanislaw Kasjan}
\author{Stanis{\l}aw Kasjan \& Daniel Simson}
\authorspace
\bsgitem{2015a}{}  Mesh algorithms for Coxeter spectral classification of Cox-regular edge-bipartite graphs with loops, I.  Mesh Root Systems.  \emph{Fundamenta Inform.}\ 139 (2015), 153--184.
\MR{3383583}.  
\codes{\CSG}%
\itemspace

\bsgitem{2015b}{}  Mesh algorithms for Coxeter spectral classification of Cox-regular edge-bipartite graphs with loops, II.  Application to Coxeter spectral analysis.  \emph{Fundamenta Inform.}\ 139 (2015), 153--184.
\MR{3383584}.  
\codes{\CSG}%
\itemspace

\bsgitem{2015c}{}  Algorithms for isotropy groups of Cox-regular edge-bipartite graphs.  \emph{Fundamenta Inform.}\ 139 (2015), 249--275.
\MR{3383587}.  
\codes{\CSG}%
\itemspace

\authorx{Eva Kaslik}							%KASLIK
\author{Eva Kaslik \& Ileana Rodica R\u{a}dulescu}
\authorx{Eva Kaslik \& Ileana Rodica Radulescu}
\authorspace
\bsgitem{2022a}{}  Stability and bifurcations in fractional-order gene regulatory networks.  \emph{Appl.\ Math.\ Comput.}\ 421 (2022), art.\ 126916, 15 pp.
\MR{4364751} (no rev).  
\annot{} From abstract:  ``oscillatory behavior \dots\ is expected only in the case of an odd number of repressive genes.''
\dated{6 Feb 2022} 
\codes{\CBiol:~\Cvs:~\CBal}%
\itemspace

\authorx{Adrien Kassel}						%KASSEL
\author{Adrien Kassel \& Thierry L\'evy}
\authorspace
\bsgitem{2020a}{}  A colourful path to matrix-tree theorems.  \emph{Algebraic Combin.}\ 3 (2020), no.\ 2, p. 471--482.
\MR{4099004}.  \Zbl{1436.05049}.  \arXiv{1903.02491}.  
\annot{} Broad generalization of the matrix-tree theorems of \xref{Zaslavsky}{1982a}{Thomas Zaslavsky} and \xref{Chaiken}{1982a}{Seth Chaiken}.
\dated{11 Feb 2022} 
\codes{\CSG,~\CGG:~\CLap}%
\itemspace

\author{P.W.\ Kasteleyn}						%KASTELEYN*
\authorspace
\authref  See also \hyperlink{C.M.\ Fortuin}{C.M.~Fortuin}.
\authrefspace

\bsgitem{1963a}{}  Dimer statistics and phase transitions.  \emph{J.\ Math.\ Phys.}\ 4 (1963), 287--293.
\MR{0153427} (27 \#3394).  
\annot{} \S V, ``The Ising problem'':  The ferromagnetic Ising model can be converted into a dimer-covering problem.  The method has since been applied to signed graphs (the general Ising problem); \cf\ \xref{Thomas and Middleton}{2009a}{Creighton K.\ Thomas \& A.\ Alan Middleton}, \xrefd{2013a}{Creighton K.\ Thomas \& A.\ Alan Middleton}, and references therein.
\dated{10 Jan 2015} 
\codes{\CPhys,~\CAlgor}%
\itemspace

\author{P.W.\ Kasteleyn \& C.M.\ Fortuin}
\authorspace
\bsgitem{1969a}{}  Phase transitions in lattice systems with random local properties.  
In:  \emph{International Conference on Statistical Mechanics} (Proc., Kyoto, 1968), pp.\ 11--14.  Supplement to \emph{J.\ Physical Soc.\ Japan}, Vol.~26, 1969.  Physical Society of Japan, [Tokyo?], 1969.
\annot{} A specialization of the parametrized dichromatic polynomial of a graph: $Q_\Gamma(q,p; x,1)$ where $q_e = 1 - p_e$.  [Essentially, announcing \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn}.]
\codes{\Csgc:~\CGen:~\CInvar,~\CPhys}%
\itemspace

\author{Vikt\'oria E.\ Kaszanitzky}					%KASZANITZKY
\authorx{Viktoria E.\ Kaszanitzky}
\authorspace
\authref  See \hyperlink{Tibor Jordan}{T.~Jord\'an}.
\itemspace

\authorx{E.\ Kaszkurwicz}						%KASZKURWICZ
\author{E.\ Kaszkurwicz \& L.\ Hsu}
\authorspace
\bsgitem{1982a}{}  On qualitative equilibria in Lotka--Volterra models.  In:  \emph{Proceedings of the 1982 American Control Conference} (Rio de Janeiro, 1982), pp.\ 481--483.  IEEE, 1982.
\codes{\Csd:~\CQM:~\CSol}%
\itemspace

\author{Osamu Katai} 						%KATAI O
\authorspace
\authref  See also \hyperlink{Sousuke Iwai}{S.~Iwai}.
\authrefspace

\bsgitem{1979a}{} Studies on aggregation of group structures and group attributes through quantification methods.  D.Eng.\ dissertation, Kyoto University, 1979.						
\itemspace

\author{Osamu Katai \& Sousuke Iwai} 
\authorspace
\bsgitem{1978a}{\dag} Studies on the balancing, the minimal balancing, and the minimum balancing processes for social groups with planar and nonplanar graph structures.  \emph{J.\ Math.\ Psychology} 18 (1978), 140--176.  
\MR{0515232} (83m:92072).  \Zbl{394.92027}.					
\annot{} Balance and detecting balance are discussed at length.  Finding the frustration index $l(\Sigma)$ is solved for planar graphs by converting it into a matching problem in the dual graph with signed vertices.  This applies also when edges are weighted by positive reals.  
[\xref{Barahona}{1982a}{Francisco Barahona} and \xref{Barahona, Maynard, Rammal, and Uhry}{1982a}{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry} have a similar, later, but independent solution for the planar frustration index.  \xref{Barahona}{1981a}{Francisco Barahona}, \xrefd{1990a}{Francisco Barahona} solves toroidal graphs.]  
\annotctd{} The nonplanar problem is treated via $A(\Sigma)$, but amounts to finding $\min_\zeta (\#E^+(\Sigma^\zeta) - \#E^-(\Sigma^\zeta))$ [which is NP-hard].  This suggests an iterative procedure which consists of switching $v \in V$ that minimizes $d^{\pm}(v)$, and repeating; it may not attain the true minimum.  [\xref{Mitra}{1962a}{S.\ Mitra} also proposed this.]
\dated{22 Jun 2012} 
\par
\codes{\CSG,~\CVS,~\CWG,~\CPsS:~\CBal,~\CFr,~\CAlgor,~\CAdj,~\Csw}%
\itemspace

\bsgitem{1978b}{} On the characterization of balancing processes of social systems and the derivation of the minimal balancing processes.  \emph{IEEE Trans.\ Systems Man Cybernetics} SMC-8 (1978), 337--348.  
\MR{0479461} (57 \#18886) (\qv).  \Zbl{383.92025}.					
\annot{} A shorter version of \xrefd{1978a}{Osamu Katai \& Sousuke Iwai}.  Lem.\ 1 [restated]:  $\Sigma$ is balanced iff it switches to all positive.
\dated{22 Jun 2012} 
\par
\codes{\CSG,~\CVS,~\CWG,~\CPsS:~\CBal,~\CFr,~\CAlgor,~\CAdj,~\Csw}%
\itemspace

\bsgitem{1978c}{} Characterization of social balance by statistical and finite-state systems theoretical analysis.  In:  \emph{Proceedings of the International Conference on Cybernetics and Society} (Tokyo, 1978).  IEEE, 1978.
\codes{\CSG,~\CWG,~\CPsS:~\CBal,~\CFr}
\itemspace

\author{Priya Kataria}						%KATARIA
\authorspace
\authref  See \hyperlink{Deepa Sinha}{D.~Sinha}.
\itemspace

\author{M.\ [Moshe] Katz}						%KATZ M
\authorx{Moshe Katz}
\authorspace
\authref  See also \hyperlink{George Converse}{G.~Converse}.
\authrefspace

\bsgitem{1970a}{}  On the extreme points of a certain convex polytope.  \emph{J.\ Combin.\ Theory} 8 (1970), 417--423.
\MR{0255582} (41 \#243).  \Zbl{194.34102}.					
\annot{} The doubly stochastic case (all line sums $= 1$) of Thm.\ 8.2.1 in \xref{Brualdi}{2006a}{Richard A.\ Brualdi}.
\dated{13 Oct 2012} 
\codes{\Csg:~\Cpar:~\CAdj}%
\itemspace

\author{Louis H.\ Kauffman}						%KAUFFMAN LH
\authorspace
\authref  See also \hyperlink{Jay R.\ Goldman}{J.R.~Goldman}.
\authrefspace

\bsgitem{1986a}{} Signed graphs.  Abstract 828-57-12, \emph{Abstracts Amer.\ Math.\ Soc.}\ 7 (1986), no.\ 5, p.\ 307.
\annot{} Announcement of \xrefd{1989a}{Louis H.\ Kauffman}.
\codes{\CSGc:~\CKnot:~\CInvar}%
\itemspace

\bsgitem{1988a}{} New invariants in the theory of knots.  \emph{Amer.\ Math.\ Monthly} 95 (1988), 195--242.  
\MR{0935433} (89d:57005).  \Zbl{657.57001}.						
\annot{} A leisurely development of Kauffman's combinatorial bracket polynomial of a link diagram and the Jones and other knot polynomials, including the basics of \xrefd{1989a}{Louis H.\ Kauffman}.
\codes{\CKnot,~\CSGc:~\CInvar:~\CExp}%
\itemspace

\bsgitem{1989a}{\dag} A Tutte polynomial for signed graphs.  \emph{Discrete Appl.\ Math.}\ 25 (1989), 105--127.  
\MR{1031266} (91c:05082).  \Zbl{698.05026}.						
\annot{} The Tutte polynomial, also called ``Kauffman's bracket of a signed graph'' and equivalent to his bracket of a link diagram, is of an edge-sign-colored graph.  It is defined by a sum over spanning trees of terms that depend on the signs and activities of the edges and nonedges of the tree.  The point is that the deletion-contraction recurrence over an edge has parameters dependent on the color of the edge; also, the parameters of the two colors are related.  The purpose is to develop the bracket of a link diagram combinatorially.
\S3.2, ``Link diagrams'': how link diagrams correspond to signed plane graphs.  \S4, ``A polynomial for signed graphs'', defines the general sign-colored graph polynomial $Q[\Sigma](A,B,d)$ by deletion-contraction, modified multiplication on components, and evaluation on graphs of loops and isthmi.
\S5, ``A spanning tree expansion for $Q[G]$'' [$G$ means $\Sigma$], proves  $Q[\Sigma]$ exists by producing a spanning-tree expansion, shown independent of the edge ordering by a direct argument.  [No dichromatic form of $Q[\Sigma]$ appears; but see successor articles.]  
\S6, ``Conclusion'', remarks that $Q[\Sigma]$ is invariant under signed-graphic Reidemeister moves II and III.
[This significant work, inspired by \xref{Thistlethwaite}{1988a}{Morwen B.\ Thistlethwaite}, led to independent but related generalizations by \xref{Przytycka and Przytycki}{1988a}{Teresa M.\ Przytycka \& Jozef H.\ Przytycki}, \xref{Schw\"arzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh}, \xref{Traldi}{1989a}{Lorenzo Traldi}, and \xref{Zaslavsky}{1992b}{Thomas Zaslavsky} that were partially anticipated by \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn}.  Also see \xrefd{1997a}{Louis H.\ Kauffman}.]
\par
\codes{\CSGc:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{1991a}{}  \emph{Knots and Physics}.  Ser.\ Knots Everything, Vol.\ 1. World Scientific, River Edge, N.J., 1991.
\MR{1141156} (93b:57010).  \Zbl{733.57004}.  
\annot{} 
\bsgitem{}{}  2nd ed., 1993, with added appendices not related to sign-colored graphs.
\MR{1306280} (95i:57010).  \Zbl{868.57001}.  
\bsgitem{}{}  3rd ed., 2001, with more appendices not related to sign-colored graphs.
\MR{1858113} (2002h:57012).  \Zbl{1057.57001}.  
\codes{\CSGc:~\CInvar:~\CExp}%
\itemspace

\bsgitem{1997a}{}  Knots and electricity.  
In: S.~Suzuki, ed., \emph{Knots '96} (Proc.\ Fifth Int.\ Research Inst., Math.\ Soc.\ Japan, Tokyo, 1996), pp.\ 213--230.  World Scientific, Singapore, 1997.
\MR{1664963} (99m:57006).  \Zbl{967.57007}.						
\annot{} \S2, ``A state summation for classical electrical networks'', uses a form of the parametrized dichromatic polynomial $Q_\Gamma(B,A; 1,1)$ [as in \xref{Zaslavsky}{1992b}{Thomas Zaslavsky} \etal; \cf], where $A(e), B(e) \in \bbC^\times$, to compute conductances as in \xref{Goldman and Kauffman}{1993a}{Jay R.\ Goldman \& Louis H.\ Kauffman}.  
\codes{\Csgc:~\CGen:~\CInvar:~\CExp}%
\annotctd{} \S3, ``The bracket polynomial'', discusses the connections with sign-colored graphs and electricity.  \Problem:  Is there a signed graph, not reducible by signed-graphic Reidemeister moves (see \xrefd{1989a}{Louis H.\ Kauffman}) to a tree with loops, whose sign-colored dichromatic polynomial is trivial?  If not, the Jones polynomial detects the unknot.
\codes{\CSGc:~\CInvar:~\CExp}\codesctd{\CSGc:~\CInvar}%
\itemspace


\author{Marcelle Kaufman}						%KAUFMAN M
\authorspace
\authref  See also \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, \hyperlink{Jacques Demongeot}{J.~Demongeot} and \hyperlink{Rene Thomas}{R.~Thomas}.
\authrefspace

\author{M.\ Kaufman, C.\ Soul\'e, \& R.\ Thomas}
\authorx{M.\ Kaufman, C.\ Soule, \& R.\ Thomas}
\authorspace
\bsgitem{2007a}{}  A new necessary condition on interaction graphs for multistationarity.  \emph{J.\ Theor.\ Biol.}\ 248 (2007), 675--685.
\MR{2899089} (no rev).  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Marcelle Kaufman \& Ren\'e Thomas}
\authorx{Marcelle Kaufman \& Rene Thomas}
\authorspace
\bsgitem{2003a}{}  Emergence of complex behaviour from simple circuit structures.  \'Emergence de comportements complexes \`a partir de structures de circuits simples.  \emph{C.R.\ Biologies} 326 (2003), 205--214.
\codes{\CSD:~\CDyn}%
\itemspace

\author{M.\ Kaufman, J.\ Urbain, \& R.\ Thomas}
\authorspace
\bsgitem{1985a}{}  Towards a logical analysis of the immune response.  \emph{J.\ Theor.\ Biol.}\ 114 (1985), no.\ 4, 527--561.
\MR{0796984} (87d:92013).  
\codes{\CBiol,~\Csd:~\CDyn}%
\itemspace

\authorx{Tali Kaufman}						%KAUFMAN T
\author{Tali Kaufman \& Alexander Lubotzky}
\authorspace
\bsgitem{2014a}{}  High dimensional expanders and property testing.  In:  \emph{Proceedings of the 5th Conference on Innovations in Theoretical Computer Science} (ITCS '14, Princeton, N.J., 2014), pp.\ 501--506.  ACM, 2014.
\MR{3359501}.  \Zbl{1365.68462}.  \arXiv{1312.2367}.  
\annot{} The focus is on ``testability''.  \S3.1, ``Sum functions on graphs'':  ``Sum function'' = balanced sign function.  Prop.\ 9:  Balance is not testable on graphs whose girth $\rightarrow \infty$.  Prop.\ 10:  Balance is testable on complete graphs.  Rem.\ 3:  Balance of $(K_n,\sigma)$ is determined by triangles.  [Long known.]  Prop.\ 11 connects triangle testing on the 1-skeleton of a simplicial complex to the first expansion constant of the complex.  [\Question.  What does that mean for flag complexes?]  \S3.3: ``Seidel switching'':  Switching equivalence of signed complete graphs is testable.  [An easy corollary of Prop.\ 10.]
\dated{1 Jul 2022} 
\codes{\Csg:~\CBal,~\CKG}%
\itemspace

\author{Bableen Kaur}						%KAUR B
\authorspace
\authref  See \hyperlink{Deepa Sinha, Deepakshi Sharma, \& Bableen Kaur}{D.~Sinha}.
\itemspace

\author{Ken-ichi Kawarabayashi}					%KAWARABAYASHI
\authorspace
\authref  See also \hyperlink{Shuya Chiba}{Chiba}, \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}, \hyperlink{Erik D.\ Demaine}{E.D.~Demaine}, \hyperlink{Shinya Fujita}{S.~Fujita}, and \hyperlink{Naonori Kakimura}{N.~Kakimura}.
\authrefspace

\bsgitem{2008a}{}  A weakening of the odd Hadwiger's conjecture.  \emph{Combin.\ Probab.\ Computing} 17 (2008), no.\ 6, 815--821.
\MR{2463413} (2009j:05083).  \Zbl{1188.05062}.  
\annot{} $\exists\ f(k)$ such that, if $-\Gamma$ has no $-K_k$ minor, then $V$ partitions into $496k$ parts of order $\leq f(k)$.  Based on \xref{Geelen, Gerards, Reed, Seymour, and Vetta}{2009a}{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}.
[\Problem.  Generalize to signed graphs.]
\dated{4 Feb 2021} 
\codes{\Csg:~\CPar:~\CCol(\CGen)}%
\itemspace

\bsgitem{2009a}{}  Note on coloring graphs without odd-$K_k$-minors..  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), no.\ 4, 728--731.
\MR{2518204} (2010e:05109).  \Zbl{1198.05058}.  
\annot{} Short proof of \xref{Geelen, Gerards, Reed, Seymour, and Vetta}{2009a}{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}: If $-\Gamma$ has no $-K_k$ minor, then $\Gamma$ is $O(k\sqrt{\log k})$-colorable.
\dated{4 Feb 2021} 
\codes{\CPar:~\CCol}%
\itemspace

\bsgitem{2013a}{}  Totally odd subdivisions and parity subdivisions: Structures and coloring.  In: Sanjeev Khanna, ed., \emph{Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '13, New Orleans, 2013), pp.\ 1013--1029.  Soc.\ for Industrial and Appl.\ Math., Philadelphia, 2013.
\MR{3202965}.  \Zbl{1423.05175}.  
\codes{\Cpar:~\CStr,~\CAlgor}%
\itemspace

\author{Ken-ichi Kawarabayashi \& Yusuke Kobayashi}
\authorspace
\bsgitem{2012a}{}  Edge-disjoint odd cycles in 4-edge-connected graphs.  In: Christoph D\"urr and Thomas Wilke, eds., \emph{29th International Symposium on Theoretical Aspects of Computer Science} (Proc.\ STACS'12, Paris, 2012), pp.\ 206--217.  Leibniz Int.\ Proc.\ Informatics, Vol.\ 14.  Leibniz-Zentrum f\"ur Informatik, Wadern, 2012.
\MR{2909315}.  \Zbl{1244.05129}.  
\annot{} Conference version of \xrefd{2016a}{Ken-ichi Kawarabayashi \& Yusuke Kobayashi}.
\codes{\CPar:~\CStr:~\CCycles}%
\itemspace

\bsgitem{2016a}{}  Edge-disjoint odd cycles in 4-edge-connected graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 119 (2016), 12--27. 
\MR{3486335}.  \Zbl{1334.05072}.  
\codes{\CPar:~\CStr:~\CCycles}%
\itemspace

\author{Ken-ichi Kawarabayashi, Orlando Lee, \& Bruce Reed}
\authorspace
\bsgitem{2014a}{}  Removable paths and cycles with parity constraints.  \emph{J.\ Combin.\ Theory Ser.\ B}\ 106 (2014), 115--133.
\MR{3194198}.  \Zbl{1297.05126}.  
\codes{\CPar:~\CStr}%
\itemspace

\author{Ken-ichi Kawarabayashi, Zhentao Li, \& Bruce Reed}
\authorspace
\bsgitem{2010a}{}  Recognizing a totally odd $K_4$-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements.  
In:  Moses Charikar, ed., \emph{Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '10, Austin, Tex., 2010), pp.\ 318--328.  
Society for Industrial and Appl.\ Math.\, Philadelphia, 2010.
 \Zbl{1288.05279}.  
\annot{} Extended abstract.  A totally odd subdivision of $\Gamma$ is a signed graph that is a (signed-graphic) subdivision of $-\Gamma$ and is itself all negative.  
	
Dictionary:  ``parity'' = sign in $-\Gamma$; ``parity path/circle'' = path or circle of specified sign.
\codes{\Csg:~\Cpar:~\CStr,~\CAlgor}%
\itemspace

\author{Ken-ichi Kawarabayashi \& Bojan Mohar}
\authorspace
\bsgitem{2006a}{}  Approximating the list-chromatic number and the chromatic number in minor-closed and odd-minor-closed classes of graphs.  In:  \emph{STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing} (Seattle, 2006), pp.\ 401--416.  ACM, New York, 2006.
\MR{2277166} (2007m:68309).  
\annot{} Odd minor: specifically, a $-K_k$ minor of $-\Gamma$.
\dated{4 Feb 2021} 
\codes{\CPar:~\CCol:~\CAlgor}%
\itemspace

\author{Ken-Ichi Kawarabayashi \& Atsuhiro Nakamoto}
\authorspace
\bsgitem{2007a}{}  The Erd{\H o}s--P\'osa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces.  \emph{Discrete Math.}\ 307 (2007), no.\ 6, 764--768.
\MR{2291454} (2007h:05084).  \Zbl{1112.05056}.					
\codes{\Csg:~\CPar:~\CCircles,~\CTop}%
\itemspace

\author{Ken-ichi Kawarabayashi \& Kenta Ozeki}
\authorspace
\bsgitem{2013a}{}  A simpler proof for the two disjoint odd cycles theorem.  \emph{J.\ Combin.\ Theory Ser.\ B} 103 (2013), 313--319.
\MR{3048156}.  
\annot{} Thm.\ 1 states the characterization for internally 4-connected graphs.  For the generalization to disjoint negative circles in signed graphs of any connectivity see the earlier paper by \xref{Slilaty}{2007a}{Daniel C.\ Slilaty}.
\dated{25 Jun 2013} 
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\author{Ken-Ichi Kawarabayashi \& Bruce Reed}
\authorspace
\bsgitem{2008a}{}  Fractional coloring and the odd Hadwiger's conjecture.  \emph{European J.\ Combin.}\ 29 (2008), no.\ 2, 411--417.
\MR{2388377} (2008k:05192.  \Zbl{1142.05027}.  
\annot{} Thm.:  If $-\Gamma$ has no $-K_p$ minor, then $\Gamma$ has a fractional $2p$-coloring.
\Cf\ \xref{Geelen, Gerards, Reed, Seymour, and Vetta}{2009a}{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}.
[\Problem.  Generalize to signed graphs.]
\dated{4 Feb 2021} 
\codes{\Csg:~\CPar:~\CCol((\CGen)}%
\itemspace

\bsgitem{2009a}{}  A nearly linear time algorithm for the half integral parity disjoint paths packing problem.  In:  Claire Mathieu, ed., \emph{Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms} (New York, 2009), pp.\ 1183--1192.  SIAM, Philadelphia, 2009.
\MR{2807561} (no rev).  \Zbl{1423.68349}.  
\codes{\Csg:~\CPar:~\CAlgor}%
\itemspace

\bsgitem{2009b}{}  Highly parity linked graphs.  \emph{Combinatorica} 29 (2009), no.\ 2, 215--225.
\MR{2520281} (2010k:05157).  \Zbl{1212.05143}.					
	
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\bsgitem{2010a}{}  An (almost) linear time algorithm for odd cycles transversal.  
In:  Moses Charikar, ed., \emph{Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '10, Austin, Tex., 2010), pp.\ 365--378.  
Society for Industrial and Appl.\ Math.\, Philadelphia, 2010.
\MR{2809682} (2012j:68132).  \Zbl{1288.05280}.  
\annot{} ``Odd cycles transversal problem'':  Is $l_0(-\Gamma) \leq k$?  An $O(m\alpha(m,n))$ algorithm for fixed $k$, where $\alpha$ = inverse Ackermann function; also solves edge form:  Is $l(-\Gamma) \leq k$?  Also, an $O(m\alpha(m,n) + n\log n)$ algorithm for 2-packing of $k$ odd circles.  
Also, simplified proof of \xref{Bruce Reed}{1999a}{Bruce Reed}.
\dated{29 Dec 2020} 
\codes{\Csg:~\Cpar:~\Cfr,~\CAlgor}%
\itemspace

\bsgitem{2010b}{}  Odd cycle packing.  In:  \emph{STOC'10---Proceedings of the 2010 ACM International Symposium on Theory of Computing} (Cambridge, Mass., 2010), pp.\ 695--704.  Association for Computing Machinery (ACM), New York, 2010.
\MR{2743319} (no rev).  \Zbl{1293.05177}.  
\annot{} Extended abstract.
\codes{\Csg:~\CPar:~\CCircles}%
\itemspace

\author{Ken-ichi Kawarabayashi, Bruce Reed, \& Orlando Lee}
\authorspace
\bsgitem{2009a}{}  Removable cycles in non-bipartite graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), no.\ 1, 30--38.
\MR{2467816} (2009k:05108).  \Zbl{1176.05041}.  
\annot{} $\Gamma$ 3-connected, $s,t \in V$, $\forall$ 3-cut $X$ each component of $\Gamma \setminus X$ contains $s$ or $t$.  Then (1) $\exists$ induced $st$-path $P$ such that $-\Gamma \setminus V(P)$ is unbalanced, or (2) $\Gamma$ is planar such that every negative face contains one of $s, t$; but not both (1) and (2).  
[\Problem.  Generalize to signed graphs.]
\dated{4 Feb 2021} 
\codes{\CPar:~\CStr}%
\itemspace

\author{Ken-ichi Kawarabayashi, Bruce Reed, \& Paul Wollan}
\authorspace
\bsgitem{2011a}{}  The graph minor algorithm with parity conditions.  In:  Rafail Ostrovsky, ed., \emph{2011 IEEE 52nd Annual Symposium on Foundations of Computer Science---FOCS 2011} (Palm Springs, Calif., 2011), pp.\ 27--36.  IEEE Computer Soc., Los Alamitos, Calif., 2011.
\MR{2932677}.  \Zbl{1293.00031} (book).  
\annot{} The graph minors algorithm for a $-\Gamma$ minor, specifically $-K_n$ (``odd $K_n$ minor'').  Disjoint paths with parity conditions.  Related problems.
\dated{4 Feb 2021} 
\codes{\Csg:~\CPar:~\CAlgor}%
\itemspace

\author{Ken-ichi Kawarabayashi \& Zi-Xia Song}
\authorspace
\bsgitem{2007a}{}  Some remarks on the odd Hadwiger's conjecture.  \emph{Combinatorica} 27 (2007), no.\ 4, 429--438.
\MR{2359825} (2008h:05046).  \Zbl{1164.05025}.  
\annot{} Thm.:  $\exists\ f(k)$: If $\Gamma$ is $(496k+13)$-connected of order $\geq f(k)$, then $-\Gamma$ has a $-K_k$ minor or $-\Gamma\ \setminus$ $8k$ vertices is balanced.
Dictionary:  ``odd complete minor of $\Gamma$'' = $-K_k$ minor of $-\Gamma$.  
\Cf\ \xref{Geelen, Gerards, Reed, Seymour, and Vetta}{2009a}{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}.
[\Problem.  Generalize to signed graphs.]
\dated{4 Feb 2021} 
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\author{Ken-ichi Kawarabayashi \& Paul Wollan}
\authorspace
\bsgitem{2006a}{}  Non-zero disjoint cycles in highly connected group labelled graphs.  \emph{J.\ Combin.\ Theory Ser.\ B}\ 96 (2006), no.\ 2, 296--301.
\MR{2208356} (2006j:05108).  \Zbl{1090.05038}.  
\codes{\CGG:~\CCircles}%
\itemspace

\author{Ken-ichi Kawarabayashi \& David R.\ Wood}
\authorspace
\bsgitem{2012a}{}  Cliques in odd-minor-free graphs.  
In:  \emph{Proceedings of the Eighteenth Computing: The Australasian Theory Symposium} (CATS '12 , Melbourne, 2012), pp.\ 133--138.  Australasian Theory Symp., Vol.\ 128.  Australian Computer Soc., Darlinghurst, Australia, 2012.
\codes{\Csg:~\Cpar:~\CStr,~\CAlgor}%
\itemspace

\authorx{Yasushi Kawase}						%KAWASE
\author{Yasushi Kawase, Yusuke Kobayashi, \& Yutaro Yamaguchi}
\authorspace
\bsgitem{2015a}{}  Finding a path with two labels forbidden in group-labeled graphs.  In:  Magn\'us M.\ Halld\'orsson \etal, eds., \emph{Automata, Languages, and Programming}, Part I (Proc.\ 42nd Int.\ Colloq., ICALP 2015, Kyoto), pp.\ 797--809.  Lect.\ Notes Computer Sci., Vol.\ 9134.  Springer, Heidelberg, 2015.
\MR{3382488}.  \Zbl{1440.05176}.  
\annot{} Extended abstract of \xrefd{2020a}{Yasushi Kawase, Yusuke Kobayashi, \& Yutaro Yamaguchi}.
\codes{\CGG:~\CPaths:~\CAlgor}%
\itemspace

\bsgitem{2020a}{}  Finding a path with two labels forbidden in group-labeled graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 143 (2020), 65--122.
\MR{4089576}.  \Zbl{1437.05208}.  \arXiv{1807.00109}.  
\annot{} Gain graphs with two forbidden gain values.
\codes{\CGG:~\CPaths:~\CAlgor}%
\itemspace

\authorx{N.\ Kawashima}						%KAWASHIMA
\author{N.\ Kawashima and H.\ Rieger}
\authorspace
\bsgitem{1997a}{}  Finite-size scaling analysis of exact ground states for $\pm J$ spin glass models in two dimensions.  \emph{Europhys.\ Lett.}\ 39 (1997), no.\ 1, 85--90.
\arXiv{cond-mat/9612116}.  
\codes{\CPhys,~\CSG:~\CFr,~\CState}%
\itemspace

\bsgitem{2004a}{}  Recent progress in spin glasses.  In:  H.T.\ Diep, ed., \emph{Frustrated Spin Systems}, Ch.\ 9, pp.\ 491--596.  World Scientific, Hackensack, N.J., 2004.
\Zbl{1104.82002} (book).  
\annot{} Many subsections throughout on open problems about $\pm J$ models (signed graphs; ground state $\leftrightarrow$ switching to fewest negative edges $\leftrightarrow$ frustration index $l(\Sigma)$) and continuous models (stochastic edge signs and weights); esp., ground state computations, mostly on excessively small graphs, looking for phase transitions and critical points.  For $\pm J$, e.g: \S\S 9.2.1, 9.2.4, 9.3.3, 9.6.3.  \S9.6:  XY, Heisenberg, Potts spins (in $S^2$, $S^3$, $[q]$).  \S9.7, ``Weak disorder'':  ``gauge invariance'' (switching invariance) implies some properties.  Ample references.
\dated{15 Aug 2018}
\par
\codes{\CSG,~\CPhys:~\CFr,~\CSw:~\CExp,~\CRef}%
\itemspace

\author{B.\ Kawecka-Magiera}						%KAWECKA-MAGIERA
\authorspace
\authref  See \hyperlink{M.J.\ Krawczyk}{M.J.~Krawczyk}.
\itemspace

\author{K.\ Kazemian}						%KAZEMIAN
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Peter Keevash}						%KEEVASH
\authorspace
\authref  See \hyperlink{Igor Balla}{I.~Balla}.
\itemspace

\authorx{Nata\v{s}a Kej\v{z}ar}						%KEJZAR
\authorx{Natasa Kejzar}
\author{Nata\v{s}a Kej\v{z}ar, Zoran Nikoloski, \& Vladimir Batagelj}
\authorx{Natasa Kejzar, Zoran Nikoloski, \& Vladimir Batagelj}
\authorspace
\bsgitem{2008a}{}  Probabilistic inductive classes of graphs.  \emph{J.\ Math.\ Sociology} 32 (2008), no.\ 2, 85--109.
\Zbl{1135.91424}.  
\annot{} \S3.4, ``Dynamic of social balance processes'':  Probabilistic rules for dynamics of balance processes.
\dated{9 Jan 2022} 
\codes{\CSG:~\CDyn:~\CRand}%
\itemspace

\authorx{Christine A.\ Kelley}						%KELLEY
\author{Christine A.\ Kelley \& Joerg Kliewer}
\authorspace
\bsgitem{2010a}{}  Algebraic constructions of graph-based nested codes from protographs.  In:  \emph{2010 IEEE International Symposium on Information Theory} (Proc., Austin, Tex., 2010), pp.\ 829--833.  
\arXiv{1006.2977}.  
\codes{\CGG:~\CCov:~\CAppl}%
\itemspace

\author{Alexander Kelmans}						%KELMANS
\authorspace
\authref  See \hyperlink{Jose F.\ De Jesus}{J.F.~De~Jes\'us}.
\itemspace

\author{Dzh.\ Kemeni \& Dzh.\ Snell}					%[KEMENI]*
\authorspace
\authref  See \hyperlink{John G.\ Kemeny \& J.\ Laurie Snell}{J.G.~Kemeny and J.L.~Snell}.
\itemspace

\author{John G.\ Kemeny}						%KEMENY*
\authorspace
\bsgitem{1959a}{}  Mathematics without numbers.  Quantity and Quality.  \emph{Daedelus} 88 (1959), no.\ 4, 577--591.
\annot{} Model No.\ 1 expounds signed graphs in social psychology from \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}.
\dated{27 Dec 2012} 
\par
\codes{\CSG:~\CPsS,~\CBal:~\CExp}%
\itemspace

\author{John G.\ Kemeny \& J.\ Laurie Snell}
\authorspace
\bsgitem{1962a}{} \emph{Mathematical Models in the Social Sciences}.  Introductions to Higher Mathematics.  Ginn, Boston [Blaisdell, Waltham, Mass.], 1962.  
Repr.: MIT Press, Cambridge, Mass., 1972.  
\MR{0140375} (25 \#3797), repr.\ \MR{0363521} (50 \-\#15959), repr.\ \MR{0519512} (80a:92060).  \Zbl{256.92003} (256, p.\ 92c) (no rev).
\annot{} Chapter VIII:  ``Organization theory:  Applications of graph theory.''  See pp.\ 97--101 and 105--107. 
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\bsgitem{1972a}{}  (As ``Dzh.\ Kemeni \& Dzh.\ Snell'')  \emph{Kiberneticheskoe Modelriovanie.  Nekotorye Prilozheniya}.  Transl.\ B.G. Mirkin.  Preface by I.B.\ Gutchin.  ``Sovetskoe Radio'', Moscow, 1972.  
\Zbl{256.92002}.  
\annot{} Russian translation of \xrefd{1962a}{John G.\ Kemeny \& J.\ Laurie Snell}.
\codes{\CSG:~\CBal:~\CExp}%
\itemspace

\authorx{Arnfried Kemnitz}						%KEMNITZ
\author{Arnfried Kemnitz \& Margit Voigt}
\authorspace
\bsgitem{2021a}{}  A note on complex-4-colorability of signed planar graphs.  \emph{Electronic J.\ Combin.}\ 
\MR{4257780}.  \Zbl{1464.05154}.  
\annot{} For some signed planar graphs, proper coloration by $\pm1, \pm\mathrm{i}$ is not possible.
\dated{6 May 2021} 
\codes{\CSG:~\CCol(\CGen)}%
\itemspace

\author{B.K.\ Kempegowda}						%KEMPEGOWDA
\authorspace
\authref  See \hyperlink{M.R.\ Rajesh Kanna}{M.R.~Rajesh Kanna}.
\itemspace

\author{Mark Kempton}						%KEMPTON
\authorspace
\authref  See \hyperlink{Fan Chung}{F.~Chung}.
\itemspace

\author{A.\ Joseph Kennedy}						%KENNEDY AJ
\authorspace
\authref  See also \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\authrefspace

\bsgitem{2007a}{}  Class partition algebras as centralizer algebras of wreath products.  \emph{Commun.\ Algebra} 35 (2007), no.\ 1, 145--170.
\MR{2287557} (2008j:16072).  \Zbl{1151.20006}.					
\par
\codes{\Cgg:~\Cmatrd:~\CAlgeb}%
\itemspace

\author{John W.\ Kennedy}						%KENNEDY JW
\authorspace
\authref  See \hyperlink{Michael L.\ Gargano}{M.L.~Gargano}.
\itemspace

\authorx{Jeff L.\ Kennington}						%KENNINGTON
\author{Jeff L.\ Kennington \& Richard V.\ Helgason}
\authorspace
\bsgitem{1980a}{}  \emph{Algorithms for Network Programming}.  Wiley, New York, 1980.  
\MR{0581251} (82a:90173).  \Zbl{502.90056}.
\annot{} Ch.~5:  ``The simplex method for the generalized network problem.''
\par
\codes{\CGN:~\CMatrd(\CBases):~\CExp}%
\itemspace

\author{Richard Kenyon}						%KENYON
\authorspace
\bsgitem{2011a}{}  Spanning forests and the vector bundle Laplacian.  \emph{Ann.\ Prob.}\ 39 (2011), no.\ 5, 1983--2017.
\MR{2884879} (2012k:82011).  
\codes{\Cgg:~\CLap}%
\itemspace

\authorx{Anne-Marie Kermarrec}					%KERMARREC
\author{Anne-Marie Kermarrec \& Afshin Moin}
\authorspace
\bsgitem{2011a}{}  Energy models for drawing signed graphs.  Res.\ rep., 2011, 29 pp.
\HAL{inria-00605924}.  
\codes{\CSG:~\CClu:~\CGeom}%
\itemspace

\author{Anne-Marie Kermarrec \& Christopher Thraves}
\authorspace
\bsgitem{2011a}{}  Can everybody sit closer to their friends than their enemies?  In: Filip Murlak and Piotr Sankowski, eds., \emph{Mathematical Foundations of Computer Science 2011} (36th Int.\ Symp., Warsaw), pp.\ 388--399.  \LNCS 6907.  Springer, Heidelberg, 2011.
\MR{2881711}.  \Zbl{1343.68184}.  
\annot{} Can $(K_n,\sigma)$ be drawn in $\bbR^l$ so every positive neighbor is closer than every negative neighbor, for each vertex?  Polynomial-time algorithm for $l=1$.  [Continued by \xref{Cygan, Pilipczuk, \etal{}}{2012a}{Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, \& Jakub Onufry Wojtaszczyk}.]
\dated{26 Apr 2012} 
\codes{\CSG:~\CGeom:~\CKG:~\CBal,~\CAlgor,~\CClu}%
\itemspace

\bsgitem{2014a}{}  Signed graph embedding: when everybody can sit closer to friends than enemies.  Manuscript, 2014.
\arXiv{1405.5023}.  
\annot{} \Cf\ \xrefd{2011a}{Anne-Marie Kermarrec \& Christopher Thraves}.
\codes{\CSG:~\CGeom:~\CKG:~\CBal,~\CAlgor,~\CClu}%
\itemspace

\author{Julie Kerr}							%KERR
\authorspace
\bsgitem{1999a}{}  A basis for the top homology of a generalized partition lattice.  \emph{J.\ Algebraic Combin.}\ 9 (1999), 47--60. 
\MR{1676728} (2000k:05265).  \Zbl{921.05063}.
\annot{} The lattice is isomorphic to the semilattice of $k$-composed partitions of a set with a top element adjoined.  (See \xref{R.\ Gill}{1998b}{Robert Gill}.)
\par
\codes{\Cgg:~\Cmatrd:~\CGeom,~\CTop}
\itemspace

\author{Mehtab Khan}						%KHAN M
\authorspace
\authref  See also \hyperlink{Rashid Farooq}{R.~Farooq}, \hyperlink{Sumaira Hafeez}{S.~Hafeez}, and \hyperlink{Fareeha Jamal}{F.~Jamal}.
\authrefspace

\author{Mehtab Khan \& Rashid Farooq}
\authorspace
\bsgitem{2017a}{}  On the energy of bicyclic signed digraphs.  \emph{J.\ Math.\ Inequalities} 11 (2017), no.\ 3, 845--862.
\MR{3732818}.  \Zbl{1371.05164}.  
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{Muhammad Ali Khan}						%KHAN MA
\authorspace
\authref  See \hyperlink{S.\ Pirzada}{S.~Pirzada}.
\itemspace

\author{H.\ Kharaghani}						%KHARAGHANI
\authorspace
\bsgitem{2003a}{} On a class of symmetric balanced generalized weighing matrices.  \emph{Designs Codes Cryptogr.}\ 30 (2003), no.\ 2, 139--149.
\MR{2006485} (2004j:05027).  \Zbl{1036.05016}.
\annot{} A ``balanced generalized weighing matrix'' is the group-ring adjacency matrix $\hat A$ of a gain digraph $\vec\Phi$, with finite gain group $\fG$, such that $\hat A \hat A^* = kI + ls(J-I)$ where $s := \sum_{g\in\fG} g$.  Constructs examples of $\hat A$ where $\fG$ is cyclic and $\vec\Phi$ is symmetric with no loops.  [The article does not mention gain digraphs.]	
\codes{\Cgg:~\CAdj}%
\itemspace

\authorx{F.\ Kharari}							%KHARARI
\author{F.\ Kharari \& \`E.\ Palmer [Frank Harary \& Edgar M.\ Palmer]}
\authorx{F.\ Kharari \& E.\ Palmer [Frank Harary \& Edgar M.\ Palmer]}
\authorspace
\authref  See \xref{Harary and Palmer}{1977a}{Frank Harary \& Edgar M.\ Palmer}.
\itemspace

\author{Abdelkader Khelladi}						%KHELLADI*
\authorspace
\authref  See also \hyperlink{Ouahiba Bessouf}{O.~Bessouf}.  
\authrefspace

\bsgitem{1985a}{}  \emph{Propri\'et\'es alg\'ebriques des structures combinatoires.}  Th\`ese de doctorate des sciences math\'ematiques, Universit\'e des Sciences et de la Technologie Houari Boumediene, Alger, 1985.
\annot{} Partie I,  ``Flots, potentiels et tensions dans les graphes biorient\'es'':  Bidirected (and signed) graphs.  \S2, ``Graphes biorient\'es -- Graphes sign\'es''.  \S3, ``Flots dans les graphes biorient\'es, Conjecture de Bouchet''.  \S4, ``Potentiels et tensions, Facteurs dans les graphs biorient\'es''.
	
\annotctd{} Introduces quasibalance (called ``m-balance''): Two negative circles have at least 2 common vertices.  
\dated{10 Jan 2019} 
\par
\codes{\CSG:~\COri:~\CStr,~\CFlows}%
\itemspace

\bsgitem{1987a}{}  Nowhere-zero integral chains and flows in bidirected graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 43 (1987), 95--115.  
\MR{0897242} (88h:05045).  \Zbl{617.90026}.					
\annot{} Improves the result of \xref{Bouchet}{1983a}{Andre Bouchet} about nowhere-zero integral flows on a signed graph.  $\Sigma$ has such an 18-flow if 4-connected, a 30-flow if 3-connected and without a positive triangle, and if quasibalanced (``m-balanced''; \cf\ \xrefd{1985a}{Abdelkader Khelladi}) a 6-flow (proving Bouchet's conjecture in that case).
\dated{rev 10 Jan 2019} 
\codes{\CSG:~\CMatrd,~\CFlows,~\COri}%
\itemspace

\bsgitem{1999a}{}  Colorations g\'en\'eralis\'ees, graphes biorient\'es et deux ou trois choses sur Fran{\c c}ois.  Symposium \`a la M\'emoire de Fran{\c c}ois Jaeger (Grenoble, 1998).  \emph{Ann.\ Inst.\ Fourier (Grenoble)} 49 (1999), 955--971.
\\ \MR{1703433} (2000h:05083).  \Zbl{917.05026}.					
\annot{} Comments on the results of \xref{Bouchet}{1983a}{Andre Bouchet} and \xref{Khelladi}{1987a}{Abdelkader Khelladi}.
\par
\codes{\CSG:~\CMatrd,~\CFlows,~\COri}%
\itemspace

\authorx{Boris N.\ Kholodenko}						%KHOLODENKO
\author{Boris N.\ Kholodenko, Anatoly Kiyatkin, Frank J.\ Bruggeman, Eduardo Sontag, Hans V.\ Westerhoff, \& Jan B.\ Hoek}
\authorspace
\bsgitem{2002a}{}  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks.  \emph{Proc.\ Nat.\ Acad.\ Sci.}\ 99 (2002), no.\ 20, 12841--12846.
\annot{} A matrix-based method to infer the signs (and magnitudes) of an interaction digraph from measurement of the interactions between modules of the digraph.
\dated{25 Jan 2015} 
\codes{\CSD:~\CAlgor:~\CBiol}%
\itemspace

\author{Antonina P.\ Khramova}						%KHRAMOVA
\authorspace
\authref  See \hyperlink{Aida Abiad}{A.~Abiad}.
\itemspace

\author{Dariush Kiani}						%KIANI
\authorspace
\authref  See \hyperlink{Ivan Gutman}{I.~Gutman}, \hyperlink{Hasti Hamidzade}{H.~Hamidzade}, \hyperlink{Mohammadreza Jooyandeh}{M.~Jooyandeh}, and \hyperlink{Maryam Mirzakhah}{M.~Mirzakhah}.
\itemspace

\author{Arnott Kidner}						%KIDNER
\authorspace
\authref  See \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}.
\itemspace

\author{Kathleen P.\ Kiernan}						%KIERNAN
\authorspace
\authref  See \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}.
\itemspace

\author{Dae San Kim}						%KIM DS
\authorspace
\authref  See \hyperlink{Taekyun Kim}{T.K.~Kim}.
\itemspace

\authorx{Dongseok Kim}						%KIM DS
\author{Dongseok Kim \& Jaeun Lee}
\authorspace
\bsgitem{2008a}{}  The chromatic numbers of double coverings of a graph.  \emph{Discrete Math.}\ 308 (2008), no.\ 22, 5078--5086.
\MR{2450445} (2009k:05082).  \Zbl{1158.05026}.					
\par
\codes{\CSG:~\CCov:~\CCol}%
\itemspace

\author{Eun Jung Kim}						%KIM EJ
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\authorx{Jeong-Rae Kim}						%KIM JR
\author{Jeong-Rae Kim, Yeoin Yoon, \& Kwang-Hyun Cho}
\authorspace
\bsgitem{2008a}{}  Coupled feedback loops form dynamic motifs of cellular networks.  \emph{Biophys.\ J.}\ 94 (2008), 359--365.
\annot{} Collects the effects of the three types of coupled cycles (signed $++$, $+-$, $--$) in an interaction signed digraph in biological examples modelled by differential equations.  Observes that $++$ cycle pairs ``enhance signal amplification and'' bistability, $--$ enhance homeostasis, and $+-$ ``enable reliable decision-making'' by the biological system.  [\Cf\ \xref{Sriram, Soliman, and Fages}{2009a}{K.\ Sriram, Sylvain Soliman, \& Francois Fages}.]
\dated{16 Jan 2015} 
\codes{\CBiol:~\CSD:~\CDyn}%
\itemspace

\author{Jong-Jae Kim}						%KIM JJ
\authorspace
\authref  See \hyperlink{Ojiro Nagai}{O.~Nagai}.
\itemspace

\authorx{Ringi Kim}							%KIM RG
\author{Ringi Kim, Seog-Jin Kim, \& Xuding Zhu}
\authorspace
\bsgitem{2022a}{}  Signed colouring and list colouring of $k$-chromatic graphs.  \emph{J.\ Graph Theory}\ 99 (2022), no.\ 4, 637--650. 
\MR{4429172}.  
\par
\codes{\CSG:~\CCol}%
\itemspace

\author{Sangwook Kim}						%KIM SW
\authorspace
\authref  See \hyperlink{JiSun Huh}{J.-S.~Huh}.
\itemspace

\author{Seog-Jin Kim}						%KIM SJ
\authorspace
\authref  See also \hyperlink{Ringi Kim}{R.-G.~Kim}.
\authrefspace

\author{Seog-Jin Kim \& Kenta Ozeki}
\authorspace
\bsgitem{2019a}{}  A note on a Brooks' type theorem for DP-coloring.  \emph{J.\ Graph Theory} 91 (2019), no\ 2, 148--161.
\MR{3948125}.  \Zbl{1419.05076}.  \arXiv{1709.09807}.  
\codes{\CSG:~\CCol}%
\itemspace

\authorx{Taekyun Kim}						%KIM TK
\author{Taekyun Kim \& Dae San Kim}
\authorspace
\bsgitem{20xxa}{}  A study on degenerate Whitney numbers of the first and second kinds of Dowling lattices.  Submitted.  \emph{}\ 
\arXiv{2103.08904}.  
\annot{} These ``degenerate Whitney numbers'' are a further generalization of ``$r$-Whitney numbers'', \cf\ \xref{Mez\H{o}}{2010a}{Istvan Mezo}.  [Their meaning is puzzling as they appear to be closely related to Dowling's characteristic polynomials.] 
\dated{21 Jul 2022} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Andrew D.\ King}						%KING
\authorspace
\authref  See \hyperlink{Tony Huynh}{T.~Huynh}.
\itemspace

\author{Harunobu Kinoshita}						%KINOSHITA H
\authorspace
\authref  See \hyperlink{Takeo Yamada}{T.~Yamada}.
\itemspace

\author{Shin'ichi Kinoshita}						%KINOSHITA S
\authorspace
\authref  See also \hyperlink{Takeshi Yajima}{T.~Yajima}.
\authrefspace

\author{Shin'ichi Kinoshita \& Hidetaka Terasaka} 
\authorspace
\bsgitem{1957a}{}  On unions of knots.  \emph{Osaka Math J.}\ 9 (1957), 131--153.
\MR{0098386} (20 \#4846).  \Zbl{080.17001}.					
\annot{} Employs the sign-colored graph of a link diagram from \xref{Bankwitz}{1930a}{C.\ Bankwitz} to form certain combinations of links.
\codes{\CSGc:~\CKnot}%
\itemspace

\author{M.\ Kirby}							%KIRBY
\authorspace
\authref  See \hyperlink{A.\ Charnes}{A.~Charnes}.
\itemspace

\author{Steve Kirkland}						%KIRKLAND
\authorspace
\authref  See also \hyperlink{Michael S.\ Cavers}{M.~Cavers}, \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de Freitas}, \hyperlink{Felix Goldberg}{F.~Goldberg}, \hyperlink{Carla Silva Oliveira}{C.S.~Oliveira}, and \hyperlink{Jeffrey Stuart}{J.~Stuart}.
\authrefspace

\bsgitem{2011a}{}  Sign patterns for eigenmatrices of nonnegative matrices.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 9, 999--1018.
\MR{2826068} (2012j:15049).  \Zbl{1239.15011}.  
\codes{\CQM,~\CSD}%
\itemspace

\author{Steve Kirkland, J.J.\ McDonald, \& M.J.\ Tsatsomeros}
\authorspace
\bsgitem{1996a}{}  Sign-patterns which require a positive eigenvalue.  \emph{Linear Multilinear Algebra} 41 (1996), no.\ 3, 199--210.
\MR{1430028} (97j:15009).  \Zbl{871.15009}.  
\codes{\CQM,~\CSD}%
\itemspace

\author{Steve Kirkland \& Debdas Paul}
\authorspace
\bsgitem{2011a}{}  Bipartite subgraphs and the signless Laplacian matrix.  \emph{Appl.\ Anal.\ Discrete Math.}\ 5 (2011), no.\ 1, 1--13.
\MR{2809028} (2012c:05191).  
\par
\codes{\CPar:~\CEig,~\Cincid}%
\itemspace

\author{Scott Kirkpatrick}						%KIRKPATRICK
\authorspace
\authref  See also \hyperlink{David Sherrington}{D.~Sherrington} and \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\authrefspace

\bsgitem{1977a}{}  Frustration and ground-state degeneracy in spin glasses.  \emph{Phys.\ Rev.\ B} 16 (1977), no.~10, 4630--4641.  
\annot{} Estimates the number of ground states of signed $d$-dimensional hypercubic lattices, $d\geq2$.  With random signs, of which the proportion $x$ is negative, the expected proportion of negative (``frustrated'') squares is computed to be $4x(1-x)[x^2+(1-x)^2] \leq 0.5$, $\approx$ for $.2<x<.8$.  [This assumes the squares' signs are independent, which is only true when $d=2$.]  Certain ice models are equivalent to signed graphs (p.\ 4632).  
\S III, ``Exact results'':  In $d=2$ the strings pairing negative squares in a ground state are short on average.  In $d=3$ ground states are more difficult to find [a conclusion essentially proved by \xref{Barahona}{1982a}{Francisco Barahona}] but there are interesting remarks on strings pairing negative squares.  
\S IV, ``Monte Carlo results'': ``carried out on fairly large samples'' in $d=2,3$ for Ising spins ($\pm1$) [with 1977 computing power].  Are there many ground states or only one (up to global spin reversal)?  Evidence in $d=2$ suggests signed graphs (``$\pm1$'') are quite different from randomly weighted signed graphs (``Gaussian'').  Signed-graph behavior differs for very low vs.\ middling density of negative edges; there seem to be more ground states at middling density.  There seem to be fewer ground states in $d=3$ than $d=2$ (p.\ 4637).  Discussion of expected behavior of low-frustration states; a remarkable planar example in Fig.\ 14.
Dictionary:  ``bond'' = edge; ``state'' = $\zeta:V\to\signnos$; frustrated bond = $\sigma^\zeta(e)=-1$; frustration = $\#(\sigma^\zeta)\inv(-1)$; ``ground state'' = switching with min frustration = $l(\Sigma)$; ``degeneracy'' = multiple states with same amount of frustration.
\dated{22 Jan 2015}
\codes{\CPhys:~\CSG,~\CState(\Cfr),~\CSw}%
\itemspace

\author{Scott Kirkpatrick \& David Sherrington}
\authorspace
\bsgitem{1978a}{\dag}  Infinite-ranged models of spin-glasses.  \emph{Phys.\ Rev.\ B} 17 (1978), no.\ 11, 4384--4403.
Repr.\ in M.~M\'ezard, G.~Parisi, and M.A.~Virasoro, \emph{Spin Glass Theory and Beyond}, pp.\ 109--128.  World Scientific Lect.\ Notes in Physics, Vol.\ 9.  World Scientific, Singapore, 1987.
\annot{} Random edge weights and signs on $K_n$ with vertex signs $\pm1$.  Most interesting:  \S~VI, ``Statics for $T \neq 0$'', where the ``energy'' (frustration index $l(\Sigma)$) landscape of random signs is described, based on computer experiments, as consisting of deep valleys, each having several local minima of $l$ separated by slightly higher ridges, and with high-$l$ barriers separating the valleys.  [Presumably, the distance function is Hamming distance between reduced sign functions, i.e., those with $E^- = l$.]  [A seminal successor to \xref{Edwards and Anderson}{1975a}{S.F.\ Edwards \& P.W.\ Anderson}.  This picture, while convincing, has never been proved; it remains an object of intense curiosity.  \Cf\ \xref{Marvel, Kleinberg, Kleinberg, and Strogatz}{2011a}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}, \xrefd{2011b}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}.]
\dated{22 Aug 2012, 23 Jan 2015} 
\codes{\CPhys:~\Csg:~\CFr(\CState)}%
\itemspace

\author{Nanao Kita}							%KITA
\authorspace
\bsgitem{2020a}{}  Signed analogue of general Kotzig--Lov\'asz decomposition.  \emph{Discrete Appl.\ Math.}\ 284 (2020), 61--70.
\MR{4115457}.  \Zbl{1443.05085}.  \arXiv{1709.07414}.				
\codes{\CSG:~\COri:~\CStr}%
\itemspace

\bsgitem{20xxb}{}  Bidirected graph II: Extension of basilica order.  In preparation.  
\codes{\CSG:~\COri:~\CStr}%
\itemspace

\bsgitem{20xxc}{}  Bidirected graph III: Algorithms for basilica decomposition.  In preparation.  
\codes{\CSG:~\COri:~\CStr}%
\itemspace

\authorx{Ouail Kitouni}						%KITOUNI
\author{Ouail Kitouni \& Nathan Reff}
\authorspace
\bsgitem{2019a}{}  Lower bounds for the Laplacian spectral radius of an oriented hypergraph.  \emph{Australasian J.\ Combin.}\ 74 (2019), no.\ 3, 408--422.
\MR{3969739}.  \Zbl{1419.05127}.  
\codes{\CSH:~\CLap:~\CEig}%
\itemspace

\authorx{Teeradej Kittipassorn}						%KITTIPASSORN
\author{Teeradej Kittipassorn \& G\'abor M\'esz\'aros}
\authorx{Teeradej Kittipassorn \& Gabor Meszaros}
\authorspace
\bsgitem{2015a}{\dag}  Frustrated triangles.  \emph{Discrete Math.}\ 338 (2015), no.\ 12, 2363--2373.
\MR{3373339}.  \Zbl{1318.05019}.  \arXiv{1411.1749}.				
\annot{} Thorough study of the number $c_3^-$ of negative triangles in a signed $K_n$.  Two-thirds of the numbers from $0$ to $\binom{n}{3}$ cannot be $c_3^-(K_n,\sigma)$.  Some numbers that are, are $0=a_0\leq b_0 \leq a_1 \leq \cdots \leq a_m \leq b_m \approx n^{3/2}$ where $b_i=a_i+i(i-1)$ and $a_{i+1}=b_i+(n-2)-i(i+1)$.  For $i\leq m$, $c_3^-(K_n,\sigma)\in[a_i,b_i]$ iff $l(K_n,\sigma)=i$.  $\exists\, f(n)$ such that if $n\gg0$, all $j\in[f(n),\binom{n}{3}-f(n)]$ are $c_3^-(K_n,\sigma)$'s.  Etc.
[For other studies of negative circles \cf\ \xref{Tomescu}{1976a}{Ioan Tomescu}, \xref{Popescu and Tomescu}{1996a}{Dragos-Radu Popescu \& Ioan Tomescu}, \xref{Antal, Krapivsky, and Redner}{2005a}{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner}, \xref{Schaefer and Zaslavsky}{2019a}{Alex Schaefer \& Thomas Zaslavsky}.]
\dated{26 Sept 2015, 6 Jan 2017} 
\codes{\Csg:~\CFr:~\CCircles}%
\itemspace

\author{Anatoly Kiyatkin}						%KIYATKIN
\authorspace
\authref  See \hyperlink{Boris N.\ Kholodenko}{B.N.~Kholodenko}.
\itemspace

\author{Ralf Klamma}						%KLAMMA
\authorspace
\authref  See \hyperlink{Mohsen Shahriari}{M.~Shahriari}.
\itemspace

\author{Steffen Klamt}						%KLAMT
\authorspace
\authref  See also \hyperlink{Ioannis N.\ Melas}{I.N.~Melas}.
\authrefspace

\author{Steffen Klamt, Julio Saez-Rodriguez, Jonathan A.\ Lindquist, Luca Simeoni, \& Ernst D.\ Gilles}
\authorspace
\bsgitem{2006a}{}  A methodology for the structural and functional analysis of signaling and regulatory networks.  \emph{BMC Bioinformatics}\ 7 (2006), art.\ 56, 26 pp.
\par
\codes{\CBiol,~\CSD,~\CSH:~\CAlgor}%
\itemspace

\author{Steffen Klamt \& Axel von Kamp}
\authorspace
\bsgitem{2009a}{}  Computing paths and cycles in biological interaction graphs.  \emph{BMC Bioinformatics} 10 (2009), art.\ 181, 11 pp.\ + 2 supplements.
\annot{} Interaction graph: a signed digraph.
\par
\codes{\CSD~\CDyn:~\CAlgor(\CPaths,~\CCycles),~\CBiol}%
\itemspace

\author{Victor Klee}							%KLEE*
\authorspace
\authref  See also \hyperlink{Clark Jeffries}{C.~Jeffries}.
\authrefspace

\bsgitem{1971a}{} The greedy algorithm for finitary and cofinitary matroids.
In:  Theodore S.\ Motzkin, ed., \emph{Combinatorics}, pp.\ 137--152.  Proc.\ Sympos.\ Pure Math., Vol.~19.  Amer.\ Math.\ Soc., Providence, R.I., 1971.
\MR{0332538} (48 \#10865).  \Zbl{229.05031}.					
\annot{} Along with \xref{Sim\~oes-Pereira}{1972a}{J.M.S.\ Simoes-Pereira}, invents the bicircular matroid (here, for infinite graphs).
\codes{\CBic}%
\itemspace

\bsgitem{1989a}{}  Sign-patterns and stability.  
In: Fred Roberts, ed., \emph{Applications of Combinatorics and Graph Theory to the Biological and Social Sciences}, pp.\ 203--219.  IMA Vols.\ Math.\ Appl., Vol.~17.  Springer-Verlag, New York, 1989.  
\MR{1009377} (90h:34081).  \Zbl{747.05057}.					
\annot{} When are various forms of stability of a linear differential equation $\dot x = Ax$ determined solely by the sign pattern of $A$?  A survey of elegant combinatorial criteria.  Signed digraphs [alas] play but a minor role.
\par
\codes{\CQM:~\CSta,~\CSD:~\CExp,~\CRef}%
\itemspace

\bsgitem{1993a}{}  Open Problem 2.  
In: Richard A.\ Brualdi, Shmuel Friedland, and Victor Klee, eds., \emph{Combinatorial and Graph-Theoretical Problems in Linear Algebra}, p.\ 257.  IMA Vols.\ Math.\ Appl., Vol.~50.  Springer-Verlag, New York, 1993. 
\MR{1240954} (94d:00012) (book).  \Zbl{780.00017} (book).
\annot{} A question about sign solvability that generalizes ``the infamous even cycle problem.''
\dated{13 Apr 2009}
\codes{\CQM:~\Csd:~\CSol,~\CSta}%
\itemspace

\author{Victor Klee, Richard Ladner, \& Rachel Manber} 
\authorspace
\bsgitem{1984a}{}  Signsolvability revisited.  \emph{Linear Algebra Appl.}\ 59 (1984), 131--157.  
\MR{0743051} (86a:15004).  \Zbl{543.15016}.					
\codes{\CSD,~\CQM:~\CSol,~\CAlgor}%
\itemspace

\author{Victor Klee \& Pauline van den Driessche} 
\authorspace
\bsgitem{1977a}{} Linear algorithms for testing the sign stability of a matrix and for finding $Z$-maximum matchings in acyclic graphs.  \emph{Numer.\ Math} 28 (1977), 273--285.  
\MR{1553991} (no rev).  \Zbl{348.65032}, (\Zbl{352.65020}).  				
\codes{\CSD:~\CQM,~\CSta,~\CAlgor}%
\itemspace

\author{Sulamita Klein}						%KLEIN
\authorspace
\authref  See \hyperlink{Luerbio Faria}{L.~Faria}.
\itemspace

\author{Jon M.\ Kleinberg}						%KLEINBERG J
\authorspace
\authref  See \hyperlink{David Easley}{D.~Easley}, \hyperlink{Jure Leskovec}{J.~Leskovec}, and \hyperlink{Seth A.\ Marvel}{S.A.~Marvel}.
\itemspace

\author{Robert D.\ Kleinberg}						%KLEINBERG R
\authorspace
\authref  See \hyperlink{Seth A.\ Marvel}{S.A.~Marvel}.
\itemspace

\authorx{Peter Kleinschmidt}						%KLEINSCHMIDT
\author{Peter Kleinschmidt \& Shmuel Onn}
\authorspace
\bsgitem{1995a}{}  Oriented matroid polytopes and polyhedral fans are signable.  
In: Egon Balas and Jens Clausen, eds., \emph{Integer Programming and Combinatorial Optimization} (4th Int.\ IPCO Conf., Copenhagen, 1995, Proc.), pp.\ 198--211.  \LNCS 920.  Springer, Berlin, 1995.  
\MR{1367982} (97b:05040).  
\annot{} In a graded partially ordered set with $0$ and $1$, assign a sign to each covering pair $(x, y)$ where $y$ is covered by $1$.  This is an ``exact signing'' if in every upper interval there is just one $y$ whose coverings are all positive.  Then the poset is ``signable''.
\codes{\CS:~\CGeom}%
\itemspace

\bsgitem{1996a}{}  Signable posets and partitionable simplicial complexes.  \emph{Discrete Comput.\ Geom.}\ 15 (1996), 443--466.  
\MR{1384886} (97a:52014).  \Zbl{853.52010}.						
\annot{} See \xrefd{1995a}{Peter Kleinschmidt \& Shmuel Onn} for definition.  Signability is a generalization to posets of partitionability of a simplicial complex (Prop.\ 3.1).  Shellable posets, and face lattices of spherical polytopes and oriented matroid polytopes, are signable.  A stronger property of a simplicial complex, ``total signability'', which applies for instance to simplicial oriented matroid polytopes (Thm.\ 5.12), implies the upper bound property (Thm.\ 4.4).  Computational complexity of face counting and of deciding shellability and  partitionability are discussed in \S6.
\codes{\CS:~\CGeom,~\CAlgor}%
\itemspace


\author{Joseph B.\ Klerlein} 						%KLERLEIN
\authorspace
\authref  See also \hyperlink{Robert L.\ Hemminger}{R.L.~Hemminger}.
\authrefspace

\bsgitem{1975a}{}  Characterizing line dipseudographs.  
In:  F.~Hoffman \etal, eds., \emph{Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing} (Boca Raton, 1975), pp.\ 429--442.  Congressus Numerantium, XIV.  Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1975.  
\MR{0396322} (53 \#190).  \Zbl{325.05106}.					
\annot{} Continues the topic of \xref{Hemminger and Kerlein}{1979a}{Robert L.\ Hemminger \& Joseph B.\ Klerlein}.
\codes{\Csg:~\CLG,~\Cori}%
\itemspace

\author{Joerg Kliewer}						%KLIEWER
\authorspace
\authref  See \hyperlink{Christine A.\ Kelley}{C.A.~Kelley}.
\itemspace

\author{Darwin Klingman} 						%KLINGMAN*
\authorspace
\authref  See \hyperlink{Joyce Elam}{J.~Elam}, \hyperlink{Fred Glover}{F.~Glover}, and \hyperlink{John Hultz}{J.~Hultz}.
\itemspace

\author{Elizabeth Klipsch}						%KLIPSCH
\authorspace
\bsgitem{20xxa}{}  Some signed graphs that are forbidden link minors for orientation embedding.  Manuscript.
\annot{} For each $n \geq 5$, either $-K_n$ or its 1-edge deletion, but not both, is a forbidden link minor.  Which one it is, is controlled by Euler's polyhedral formula, provided $n \geq 7$.  [A long version with excruciating detail is available.]
\codes{\CSG:~\CTop,~\CPar}%
\itemspace

\authorx{Ton Kloks}							%KLOKS
\author{Ton Kloks, Haiko M\"uller, \& Kristina Vu\v{s}kovi\'c}
\authorx{Ton Kloks, Haiko Muller, \& Kristina Vuskovic}
\authorspace
\bsgitem{2009a}{}  Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), 733--800.
\MR{2522592} (2010j:05345).  \Zbl{1218.05160}.					
\annot{} A decomposition theorem for graphs without induced even circles and $K_4 \setminus e$'s.  [\Question.  Does it make sense to generalize to signed graphs without chordless balanced circles (longer than 3?) or $[K_4 \setminus e]$'s?]
\dated{10 Mar 2011} 
\codes{\Cpar:~\CStr}%
\itemspace

\author{T.\ Klotz}							%KLOTZ
\authorspace
\authref  See also \hyperlink{J.F.\ Valdes}{J.F.~Vald\'es} and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\authrefspace

\author{T.\ Klotz \& S.\ Kobe}
\authorspace
\bsgitem{1994a}{}  `Valley structures' in the phase space of a finite {\sc 3D} Ising spin glass with $\pm I$ interactions.  \emph{J.\ Phys.\ A} 27 (1994), L95--L100.
\MR{1269034} (no rev).  
\annot{} The energy (i.e., $\#E^-(\Sigma^\zeta)$) landscape of switchings of a signed graph, the underlying graph being a cubic lattice.
	
\dated{4 Jan 2015} 
\par
\codes{\CSG:~\CState(\Cfr),~\CSw,~\CPhys}%
\itemspace

\author{Kolja Knauer}						%KNAUER
\authorspace
\authref  See \hyperlink{Stefan Felsner}{S.~Felsner}.
\itemspace

\author{Oliver Knill}							%
\authorspace
\bsgitem{2017a}{}  On the arithmetic of graphs.  Manuscript, 2017.  
\arXiv{1706.05767}.  
\annot{} ``Signed graph'' = formal difference $G_1 - G_2$, roughly equivalent to defining $\Sigma = \Sigma^+-\Sigma^-$.
\codes{\CSG:~\CAlgeb}%
\itemspace

\bsgitem{2019a}{}  More on numbers and graphs.  Manuscript, 2019.  
\arXiv{1905.13387}.  
\par
\codes{\CSG:~\CAlgeb}%
\itemspace

\bsgitem{2021a}{}  Remarks about the arithmetic of graphs.  Manuscript, 2021.  
\arXiv{2106.10093}.  
\par
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{Klaus Knorr}							%KNORR
\authorspace
\authref  See \hyperlink{J.D.\ Noh, H.\ Rieger, M.\ Enderle, \& K.\ Knorr}{J.D.~Noh}.
\itemspace

\author{Andrew Knyazev}						%KNYAZEV
\authorspace
\bsgitem{2018a}{}  On spectral partitioning of signed graphs.  In:  Fredrik Manne, Peter Sanders, and Sivan Toledo, eds.,\emph{2018 Proceedings of the SIAM Workshop on Combinatorial Scientific Computing} (CSC18, Bergen, Norway, 2018), pp.\ 11--22.  SIAM ePub, 2018.
\arXiv{1701.01394}.  
\annot{} Spectral clustering bipartitions $V$ according to signs in an eigenvector of min eigenvalue of Laplacian.  Argues that for $\Sigma$, Laplacian $\Lap(\Sigma) = \diag(\deg) - A(\Sigma)$ is better than ``signed Laplacian'' $D - A(\Sigma)$ where $D$ has large positive diagonal as in \xref{Kunegis, Schmidt, \etal\!}{2010a}{Jerome Kunegis, Stephan Schmidt, Andreas Lommatzsch, Jurgen Lerner, Ernesto W.\ De Luca, \& Sahin Albayrak}.
\dated{8 Feb 2021} 
\codes{\CSG:~\CClu:~\CLap}%
\itemspace

\author{Lori Koban [Lori Fern]}						%KOBAN
\authorx{Lori Koban}	
\authorspace
\authref  See also \hyperlink{Lori Fern}{L.~Fern}.
\authrefspace

\bsgitem{2004a}{}  Comments on ``Supersolvable frame-matroid and graphic-lift lattices'' by T.\ Zaslavsky.  \emph{European J.\ Combin.}\ 25 (2004), 141--144.
\MR{2031808} (2004k:05054).  \Zbl{1031.05032}.
\annot{} Correction to Thm.\ 2.1; improved (and corrected) proof of Thm.\ 2.2 of \xref{Zaslavsky}{2001a}{Thomas Zaslavsky}.
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{2004b}{}  \emph{Two Generalizations of Biased Graph Theory: Circuit Signatures and Modular Triples of Matroids, and Biased Expansions of Biased Graphs}.  Doctoral dissertation, State University of New York at Binghamton, 2004. 
\MR{2706325} (no rev).  
\annot{} Ch.\ 1:  ``Circuit signatures and modular triples.''  When can gains be applied to matroids, as they are to graphs in \xref{Zaslavsky}{1991a}{Thomas Zaslavsky}, to produce a linear class of circuits and hence a lift matroid?  Theorem 1.4.1:  When the group has exponent $>2$, one needs a ternary circuit signature, thus a ternary matroid.  Theorem 1.4.5:  When the group has exponent 2 the matroid must be binary (no circuit signature is required).
\par
\codes{\CMatrd:~\CGG:~\CGen}%
\annotctd{} Ch.\ 2:  ``Biased expansions of biased graphs.''  Generalizes group and biased expansions of a graph and the chromatic (and bias-matroid characteristic) polynomial formulas (\xref{Zaslavsky}{1995b}{Thomas Zaslavsky}, \xrefd{20xxj}{Thomas Zaslavsky}) to expansions of a biased graph.  Ch.\ 3:  ``When are biased expansions actually group expansions?''  Partial results about characterizing biased expansions of biased graphs that are group expansions; counterexamples to several plausible conjectures.
\codes{\CGG:~\CMatrd,~\CInvar,~\CGeom}%
\itemspace

\bsgitem{2008a}{}  A modular triple characterization of circuit signatures.  \emph{European J.\ Combin.}\ 29 (2008), no.\ 1, 159--170.
\MR{2368623} (2008k:05040).  \Zbl{1127.05021}.					
\annot{} Four kinds of circuit signatures of a matroid can be characterized through modular triples of copoints or circuits.  They are lift signatures as well as the previously known weak orientations, orientations, and ternary signatures.  Lifting signatures are needed to define a matroid with gains and thereby a lift matroid determined by the gains.
\par
\codes{\CGG:~\CGen,~\CMatrd}%
\itemspace

\bsgitem{2012a}{}  Biased expansions of biased graphs and their chromatic polynomials.  \emph{Ann.\ Combin.}\ 16 (2012), no.\ 4, 781--788.
\MR{3000445}.  
\annot{} Generalizes group and biased expansions of a graph (\xref{Zaslavsky}{1995b}{Thomas Zaslavsky}, Ex.\ 3.8; \xref{Zaslavsky}{2001a}{Thomas Zaslavsky}, Ex.\ 4.1; \xref{Zaslavsky}{20xxj}{Thomas Zaslavsky}) to biased expansions of a biased graph.  The definition is similar but tricky.  The chromatic polynomials follow similar formulas.
\dated{20 Oct 2012} 
\par
\codes{\CGG:~\CInvar,~\CMatrd}%
\itemspace

\author{Yusuke Kobayashi}						%KOBAYASHI
\authorspace
\authref  See \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi} and \hyperlink{Yasushi Kawase}{Y.~Kawase}.
\itemspace

\author{S.\ Kobe}							%KOBE
\authorspace
\authref  See \hyperlink{T.\ Klotz}{T.~Klotz}, \hyperlink{J.F.\ Valdes}{J.F.~Vald\'es}, and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\authorx{William Kocay}						%KOCAY
\author{William Kocay \& Douglas Stone}
\authorspace
\bsgitem{1993a}{}  Balanced network flows.  \emph{Bull.\ Inst.\ Combin.\ Appl.}\ 7 (1993), 17--32.
\MR{1206759} (93j:05148).  \Zbl{804.05057}.					
\annot{} Balanced network = signed covering graph of $-\Gamma$ with edges $vw$ lifted to arcs $(+v,-w)$, and with added source and sink.
\dated{8 Mar 2011} 
\par
\codes{\Csg:~\Ccov}%
\itemspace

\bsgitem{1995a}{}  An algorithm for balanced flows.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 19 (1995), 3--31.
\MR{1358494} (96j:90087).  \Zbl{841.68098}.					
\annot{} Continuation of \xrefd{1993a}{William Kocay \& Douglas Stone}.
\dated{8 Mar 2011} 
\codes{\Csg:~\Ccov:~\CAlgor}%
\itemspace

\authorx{Muralidharan Kodialam}					%KODIALAM M
\author{Muralidharan Kodialam \& James B.\ Orlin}
\authorspace
\bsgitem{1991a}{}  Recognizing strong connectivity in (dynamic) periodic graphs and its relation to integer programming.  
In: \emph{Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms} (San Francisco, 1991), pp.\ 131--135.  Assoc.\ for Computing Machinery, New York, 1991.
\Zbl{800.68639}.  
\annot{} Linear programming methods to find the strongly connected components of a periodic digraph from the static graph: i.e., of the covering digraph of a gain digraph $\Phi$ with gains in $\bbQ^d$ by looking at $\Phi$.  \Cf\ \xref{Cohen and Megiddo}{1993a}{Edith Cohen \& Nimrod Megiddo}, whose goals are similar but algorithms differ.
\par
\codes{\CGD(\CCov):~\CBal,~\CCircles:~\CAlgor}%
\itemspace

\authorx{Vijay Kodiyalam}						%KODIYALAM V
\author{Vijay Kodiyalam, R.\ Srinivasan, \& V.S.\ Sunder}
\authorspace
\bsgitem{2000a}{}  The algebra of $G$-relations.  \emph{Proc.\ Indian Acad.\ Sci., Math.\ Sci.}\ 110 (2000), no.\ 3, 263--292.
\MR{1781906} (2001k:16019) (\qv).  \Zbl{992.16015}.  
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\authorx{Yoshitaka Koga}						%KOGA
\author{Yoshitaka Koga, Tatsuya Maruta, \& Keisuke Shiromoto}
\authorspace
\bsgitem{2018a}{}  On critical exponents of Dowling matroids.  \emph{Designs Codes Cryptogr.}\ 86 (2018), 1947--1962.
\MR{3816208}.  \Zbl{1394.05014}.  
\annot{} Higher-weight Dowling matroids, \cf\ \xref{Kung}{1996b}{Joseph P.S.\ Kung}, [\xref{Ravagni}{2022a}{Alberto Ravagnani}].  Connections to linear codes.  Thm.\ 17 and \S4: Bounds for certain cases.
\dated{17 Nov 2022} 
\codes{\CMatrd:~\CInvar,~\Cgg(\CGen)}%
\itemspace

\author{Shungo Koichi}						%KOICHI
\authorspace
\bsgitem{2014a}{}  The Buneman index via polyhedral split decomposition.  \emph{Adv.\ Appl.\ Math.}\ 60 (2014), 1--24.
\MR{3256746}.  \Zbl{1300.05053}.  
\annot{} \S\S4 2--3: Signed partial partitions (treated as sign-symmetric partitions of $\pm[n] \cup\{0\}$).  Two sets of signed partial partitions are equivalent if one is converted to the other by switching in $\pm K_n\full$.  A ``signed bipartition'' is a signed partial partition with one block (that is, ignoring the 0 block).
	
\dated{28 Jan 2015} 
\codes{\Csg:~\CMatrd}%
\itemspace

\author{Johan Kok}							%KOK
\authorspace
\authref  See \hyperlink{Naduvath K.\ Sudev}{N.K.~Sudev}.
\itemspace

\author{Tamara Koledin}						%KOLEDIN
\authorspace
\authref  See also \hyperlink{Milica Andelic}{M.~An{\dj}eli\'c}.
\authrefspace

\author{Tamara Koledin \& Zoran Stani\'c}
\authorx{Tamara Koledin \& Zoran Stanic}
\authorspace
\bsgitem{2017a}{}  Connected signed graphs of fixed order, size, and number of negative edges with maximal index.  \emph{Linear Multilinear Algebra}\ 65 (2017), no.\ 11, 2187--2198.
\MR{3740690}.  
\annot{} Consider all $\Sigma$ with simple, connected $|\Sigma|$ and fixed values $|V|=n$, $|E|=m$, $|E^-|=k$.  Thm.\ 4.1:  With some exceptions, if $\lambda_{\max}(A(\Sigma_0)) = \max\{ \lambda_{\max}(A(\Sigma)): \forall\ \Sigma \}$, then $\Sigma_0^-$ is a bipartite chain graph and each $\Sigma_0^+{:}U_i$ is a threshold graph.
\dated{6 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig:~\CStr}%
\itemspace

\bsgitem{2020a}{}  On a class of strongly regular signed graphs.  \emph{Publ.\ Math.\ Debrecen} 97 (2020), no.\ 3-4, 353--365.
\MR{4194066}.  
\annot{} \Cf\ \xref{Stani\'c}{2019d}{Zoran Stanic}.  All SRSG are sorted into 5 classes; Class 3 is where $c = \frac12(a+b) \neq 0$.  Thms.\ 4.1--2:  $\Sigma$ in Class 3 is net regular.  $A(\Sigma)$ has exactly 3 eigenvalues.  The net-regularity degree $d^\pm$ is a simple eigenvalue.  For the other eigenvalues, $\lambda^2 + \frac12(b-a)\lambda + \frac12(a+b) = r$.  Similarities to strongly regular graphs, e.g., Thm.\ 4.4:  A connected, regular, net-regular, inhomogeneous signed graph with 3 eigenvalues, where $d^\pm$ is a simple eigenvalue, is strongly regular.  Thm.\ 4.8:  $\Sigma$ in Class 3 with $d^\pm=0$ is complete with $a+b=-2$.  Thm.\ 4.9:  If $\Sigma$ is in Class 3 and the eigenvalues $\neq d^\pm$ have equal multiplicity, then $\Sigma$ is complete, $d^\pm=0$, and $a=b=-1$.  Thm.\ 5.2.  $\Sigma$ in Class 3 with $r\leq10$ is complete, with one possible exception.  Two incomplete examples are constructed from 3-class Johnson schemes.
\dated{4 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Notes on Johnson and Hamming signed graphs.  \emph{Bull.\ Math.\ Soc.\ Sci.\ Math.\ Roumanie} 65 (113) (2022), no.\ 3, 303--315.
\MR{4488335}.  
\annot{} Strongly regular $\Sigma$ (\cf\ \xref{Stani\'c}{2019d}{Zoran Stanic}) related to association schemes.  $\Sigma$ based on Johnson and Hamming graphs.  NASC for strong regularity.  The first known strongly regular $\Sigma$ with $>4$ eigenvalues.
\dated{19 Sept 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Alexandra Kolla}						%KOLLA
\authorspace
\authref  See \hyperlink{Charles Carlson}{C.~Carlson}.
\itemspace

\author{J\'anos Koml\'os}						%KOMLOS
\authorx{Janos Komlos}
\authorspace
\bsgitem{1997a}{}  Covering odd cycles.  \emph{Combinatorica} 17 (1997), 393--400.
\MR{1606044} (99b:\-05114).  \Zbl{902.05036}.
\annot{} Sharp asymptotic upper bounds on frustration index and vertex frustration number for all-negative signed graphs with fixed negative girth.  Improves \xref{Bollob\'as, Erd\H{o}s, Simonovits, \& Szemer\'edi}{1978a}{B.\ Bollobas, P.\ Erdos, M.\ Simonovits, \& E.\ Szemeredi}.  [\Problem.  Generalize to arbitrary signed graphs or signed simple graphs.]
\par
\codes{\CPar:~\CFr}%
\itemspace

\author{Helene J.\ Kommel} 						%KOMMEL
\authorspace
\authref  See \xref{Harary and Kommel}{1978a}{Frank Harary \& Helene J.\ Kommel}.
\itemspace

\authorx{A.\ Kompi\v{s}ov\'a}						%KOMPISOVA
\authorx{A.\ Kompisova}
\author{A.\ Kompi\v{s}ov\'a \& E.\ M\'a\v{c}ajov\'a}
\authorx{A.\ Kompisova \& E.\ Macajova}
\authorspace
\bsgitem{2019a}{}  Flow number and circular flow number of signed cubic graphs.  \emph{Acta Math.\ Univ.\ Comenianae (N.S.)}\ 88 (2019), no.\ 3, 877--883.
\MR{4014147}.  
\par
\codes{\CSG:~\CFlows:~\CInvar}%
\itemspace

\bsgitem{2022a}{}  Flow number and circular flow number of signed cubic graphs.  \emph{Discrete Math.}\ 345 (2022), no.\ 8, art.\ 112917, 12 pp.
\MR{4409974}.  \Zbl{1490.05104}.  
\codes{\CSG:~\CFlows:~\CInvar}%
\itemspace

\author{Christian Komusiewicz}						%KOMUSIEWICZ
\authorspace
\authref  See \hyperlink{Falk Huffner}{F.~H\"uffner}.
\itemspace

\author{D\'enes K\"onig} 						%KONIG*
\authorx{Denes Konig}
\authorspace
\bsgitem{1936a}{} \emph{Theorie der endlichen und unendlichen Graphen: Kombinatorische Topologie der Streckenkomplexe}.
Mathematik und ihre Anwendungen, Band 16.  Akademische Verlagsges., Leipzig, 1936.
Repr.: Chelsea, New York, 1950. 
\MR{0036989} (12, 195).  \Zbl{013.22803} (13, p.\ 228c).				
\annot{} \S~X.3, ``Komposition von B\"usheln'', contains Thms.~9--16 of Ch.~X.  I restate them in terms of a signature on the edge set; K\"onig says subgraph or $p$-subgraph (``$p$-Teilgraph'') to mean what we would call the negative edge set of a signature or a balanced signature.  Instead of signed switching, K\"onig speaks of set summation (``composition'') with a vertex star (``B\"uschel'').  His theorems apply to finite and infinite graphs except where stated otherwise.  Thm.\ 9: The edgewise product of balanced signatures is balanced.  Thm.\ 10:  Every balanced signing of a finite graph is a switching of the all-positive signature.  Thm.\ 11: A signature is balanced iff it has a Harary bipartition [see \xref{Harary}{1953a}{Frank Harary}].  Thm.\ 12 (cor.\ of 11):  A graph is bicolorable iff every circle has even length.  [K\"onig makes this fundamental theorem a corollary of a signed-graph theorem!]  Thm.\ 13:  A signature is balanced if (not ``only if'', but that is obvious) every circle of a fundamental system is positive.  Thm.\ 14:  A graph with $n$ vertices (a finite number) and $c$ components has $2^{n-c}$ balanced signings.  Thm.\ 16:  The set of all vertex switchings, except for one in each finite component of $\Gamma$, forms a basis for the space of all finitely generated switchings.
\annotctd{} [Switching is born here but not recognized until reinvented.  K\"onig's one failing was not to see the role of edge and circle signs; thus, I regard \xref{Harary}{1953a}{Frank Harary} as the true invention of signed graphs.]
\dated{rev.\ 26 Aug 2018}
\codes{\Csg:~\CBal,~\Csw,~\CEnum}%
\itemspace

\bsgitem{1986a}{} \emph{Theorie der endlichen und unendlichen Graphen.  Mit einer Abhandlung von L.~Euler} 
Ed.\ and introd.\ by H.~Sachs, introd.\ by P.\ Erdos, biographical essay by T.~Gallai [in English].  
Teubner-Archiv zur Math., 6.  BSB B.\ G.\ Teubner, Leipzig, 1986.
\MR{0886676} (88i:01168).  \Zbl{608.05002}.
\annot{} Reprint of \xrefd{1936a}{Denes Konig} together with Euler's paper (in Latin and German) on the K\"onigsberg bridges and supplementary material.
\par
\codes{\Csg:~\CBal,~\Csw,~\CEnum}%
\itemspace

\bsgitem{1990a}{} \emph{Theory of Finite and Infinite Graphs}.  
Transl.\ Richard McCoart, commentary by W.T.\ Tutte, biographical sketch by T.\ Gallai.  Birkh\"auser, Boston, 1990.
\MR{1035708} (91f:01026).  \Zbl{695.05015}.						
\annot{} English translation of \xrefd{1936a}{Denes Konig}.  \S~X.3: ``Composition of stars''.  [``Kreis'' (circle, meaning circle) is unfortunately translated as ``cycle''---one of the innumerable meanings of ``cycle''.]
\codes{\Csg:~\CBal,~\Csw,~\CEnum}%
\itemspace

\author{Alexander Kononov}						%KONONOV
\authorspace
\authref  See \hyperlink{Victor Il'ev}{V.~Il'ev}.
\itemspace

\author{Jack [Jacobus] H.\ Koolen}					%KOOLEN
\authorx{Jack H.\ Koolen}
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Meng-Yue Cao}{M.-Y.~Cao}, \hyperlink{Taeyoung Chung}{T.Y.~Chung}, \hyperlink{Gary Greaves}{G.~Greaves}, and \hyperlink{Hye Jin Jang}{H.J.~Jang}.
\itemspace

\author{Justin Koonin}						%KOONIN
\authorspace
\bsgitem{2014a}{}  Topology of eigenspace posets for imprimitive reflection groups.  \emph{J.\ Combin.\ Theory Ser.\ A}  127 (2014), 121--148.
\MR{3252658}.  \Zbl{1301.06010}.  \arXiv{1208.4435}.				
\annot{} An eigenspace poset is described in terms of ``$d$-divisible, $k$-evenly colored Dowling lattices'', which are subposets of Dowling lattices.
\dated{12 Jul 2016} 
\codes{\Cgg:~\CMatrd,~\CGeom,~\CGen}%
\itemspace

\author{Evgeniia Korchevskaia}						%KORCHEVSKAIA
\authorspace
\authref  See \hyperlink{Tianran Chen}{T.-R.~Chen}.
\itemspace

\author{Hideo Kosako}						%KOSAKO
\authorspace
\authref  See also \hyperlink{Suck Joong Moon}{S.J.~Moon}.
\authrefspace

\author{Hideo Kosako, Suck Joong Moon, Katsumi Harashima, \& Takeo Ikai}
\authorspace
\bsgitem{1993a}{}  Variable-signed graph.  \emph{Bull.\ Univ.\ Osaka Pref.\ Ser.\ A} 42 (1993), 37--49.  
\MR{1287466} (96e:05167).  \Zbl{798.05070}.
\annot{} ``Variable-signed graph'' = signed simple (di)graph $\Sigma$ with switching function $p$ and switched graph $\Sigma^{p}$.  Known basic properties of switching are established.  More interesting: planar duality when $|\Sigma|$ is planar.  The planar dual $|\Sigma|*$ inherits the same edge signs; a dual vertex has sign of the surrounding primal face boundary.  Property 9 is in effect the statements:  (1)  If a signed plane graph has $f$ negative face boundaries, then $l(\Sigma) \geq f/2$.  (2)  If the negative faces fall into two connected groups with oddly many faces in each, (1) can be improved to $\geq f/2 + 1$.  Finally, incidence matrices are studied that are only superficially related to signs.  [The paper is hard to interpret due to mathematical imprecision and language difficulty.]
\codes{\CSG:~\CSw,~\Cfr,~\CDual,~\CInc}%
\itemspace

\author{Justyna Kosakowska}						%KOSAKOWSKA
\authorspace
\authref  See also \hyperlink{Mariusz Kaniecki}{M.~Kaniecki}.
\authrefspace

\bsgitem{2012a}{}  Inflation algorithms for positive and principal edge-bipartite graphs and unit quadratic forms.  \emph{Fundamenta Inform.}\ 119 (2012), no.\ 2, 149--162.
\MR{2977486}.  
\codes{\CSG}%
\itemspace

\author{George E.\ Kostakis}						%KOSTAKIS
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\author{Alexandr V.\ Kostochka}					%KOSTOCHKA
\authorspace
\authref   See \hyperlink{A.A.\ Ageev}{A.A.~Ageev} and \hyperlink{Ervin Gyori}{E.~Gy\"ori}.
\itemspace

\author{Sven Kosub}							%KOSUB
\authorspace
\authref  See \hyperlink{Tatsuya Akutsu}{T.~Akutsu}.
\itemspace

\author{Bal\'azs Kotnyek}						%KOTNYEK
\authorx{Balazs Kotnyek}
\authorspace
\authref  See also \hyperlink{Gautam Appa}{G.~Appa} and \hyperlink{Leonidas S.\ Pitsoulis}{L.S.~Pitsoulis}.
\authrefspace

\bsgitem{2002a}{}  \emph{A Generalization of Totally Unimodular and Network Matrices}.  Doctoral thesis, London School of Economics, 2002.
\annot{} Introducing binet matrices; \cf\ \xref{Appa and Kotnyek}{2006a}{Gautam Appa \& Balazs Kotnyek}.  A binet matrix is $A = \incid{B}\inv\incid(\Beta)$ where $\Beta$ is a bidirected graph (which may be assumed to have no balanced components) and $B$ is a basis for $\bfF(\Sigma(\Beta))$.  \Problem:  To recognize a binet matrix.  Thm.:  If an $n\times m$ matrix $A$ is an indecomposable binet matrix, then at most one component of $B$ has no half edge (and the remaining component has a negative circle).  
[Further work in \xref{Appa and Kotnyek}{2006a}{Gautam Appa \& Balazs Kotnyek}, \xref{Musitelli}{2007a}{Antoine Musitelli}, \xrefd{2007a}{Antoine Musitelli}.]
\dated{15 January 2013} 
\codes{\CSG:~\COri:~\CInc,~\CAlgor}%
\itemspace

\author{Manoshi Kotoky}						%KOTOKY
\authorspace
\authref  See \hyperlink{Arun Kumar Baruah}{A.K.~Baruah}.
\itemspace

\author{A.\ Kotzig} 							%KOTZIG*
\authorspace
\bsgitem{1968a}{}  Moves without forbidden transitions in a graph.  \emph{Mat.\ \v{C}asopis} 18 (1968), 76--80.  
\MR{0242709} (39 \#4038).  \Zbl{155.31901} (155, p.\ 319a).
\codes{\Cpar:~\Cori}%
\itemspace

\author{Paulina Koutsaki}						%KOUTSAKI
\authorspace
\authref  See \hyperlink{Jared C.\ Bronski}{J.C.~Bronski}.
\itemspace

\authorx{Istv\'an Kov\'acs}						%KOVACS
\authorx{Istvan Kovacs}
\author{Istv\'an Kov\'acs, Aleksander Malni\v{c}, Dragan Maru\v{s}i\v{c}, \& \v{S}tefko Miklavi\v{c}}
\authorx{Istvan Kovacs, Aleksander Malnic, Dragan Marusic, \& Stefko Miklavic}
\authorspace
\bsgitem{2009a}{}  One-matching bi-Cayley graphs over abelian groups.  \emph{European J.\ Combin.}\ 30 (2009), 602--616.
\MR{2489254} (2010e:05247).  \Zbl{1204.05079}.  
\annot{} Uses gain graphs (``voltage graphs'') to construct graphs with certain kinds of automorphisms.
\dated{28 Mar 2017} 
\codes{\CGG:~\CCov:~\CAlgeb}%
\itemspace

\author{Vladislav B.\ Kovchegov}					%KOVCHEGOV
\authorspace
\bsgitem{1984a}{}  Markov's model of relations in small group and analysis of group structures.  (In Russian.)  In:  \emph{Mathematical Methods in Sociological Researches}.  Inst.\ of Sociology, Moscow, 1984.
\annot{} See \xrefd{1994a}{Vladislav B.\ Kovchegov}.
\codes{\CSG:~\CWG,~\CAdj,~\CBal,~\CClu}%
\itemspace

\bsgitem{1989a}{}  Balance and maximum unergodicity hypothesis for institutions.  (In Russian.)  In:  \emph{Mathematical Modelling of Social Processes}.  Acad.\ of Social Sciences, Moscow, 1989.
\annot{} See \xrefd{1994a}{Vladislav B.\ Kovchegov}.
\codes{\CSG:~\CWG,~\CAdj,~\CBal,~\CClu}%
\itemspace

\bsgitem{1992a}{}  Modeling of human institutions by network of automatons with relations.  In:  \emph{Proceedings of the Eleventh European Meeting of Cybernetics and Systems Research} (Vienna, 1992), pp.\ 989--995. 
\annot{} See \xrefd{1994a}{Vladislav B.\ Kovchegov}.
\codes{\CSG:~\CWG,~\CAdj,~\CBal,~\CClu}%
\itemspace

\bsgitem{1994a}{}  A model of dynamics of group structures of human institutions.  \emph{J.\ Math.\ Sociology} 18 (1994), no.\ 4, 315--332.
\MR{1262516} (no rev).  \Zbl{829.92028}.  
\annot{} A ``model of the institution with relations'' consists of a loopless digraph $D=(V,A)$ with $V = \{1,2,\ldots,n\}$, a group $\fG$, sets $X$ and $Y$, and functions $f: A \to Y$, $z_i: Y \to \cP(\fG) \ \forall i\in V$.  We consider $r: A \to \fG$ such that $r(i,j) \in z_i(f(i,j))$.  [That is, $(D,r)$ is a gain digraph with gain group $\fG$.  The edges are colored by $f$ and the gains are constrained by the list $z_i(y)$ for each vertex $i$ and color $y$.]
\dated{24 Nov 2012} 
\par
\codes{\CSG:~\CWG,~\CAdj,~\CBal,~\CClu,~\CGeom}%
\itemspace

\author{Robin Koytcheff}						%KOYTCHEFF
\authorspace
\authref  See \hyperlink{Etay Ziv}{E.~Ziv}.
\itemspace

\author{David Krackhardt} 						%KRACKHARDT
\authorspace
\authref   See \hyperlink{Patrick Doreian}{P.~Doreian}.
\itemspace

\authorx{Thomas Krajewski}						%KRAJEWSKI
\author{Thomas Krajewski, Iain Moffatt, \& Adrian Tanasa}
\authorspace
\bsgitem{2015a}{}   An extension of the Bollob\'as-Riordan polynomial for vertex partitioned ribbon graphs: definition and universality.  EuroComb 2015 (Bergen, 2015).  \emph{Electronic Notes Discrete Math.}\ 49 (2015), 285--292.
 \Zbl{1346.05124}.  
\codes{\Csg:~\CTop:~\CInvar}%
\itemspace

\bsgitem{2018a}{}  Hopf algebras and Tutte polynomials.  \emph{Adv.\ Appl.\ Math.}\ 95 (2018), 271--330.
\MR{3759218}.  \Zbl{1379.05057}.  \arXiv{1508.00814}.				
\annot{} Surface-embedded signed graphs (``ribbon graphs'') are a main example.
\dated{17 Apr 2019} 
\codes{\Csg:~\CTop:~\CInvar}%
\itemspace

\author{Thomas Krajewski, Vincent Rivasseau, \& Fabien Vignes-Tourneret}
\authorspace
\bsgitem{2011a}{}  Topological graph polynomial and quantum field theory.  Part II: Mehler kernel theories.  \emph{Ann.\ Henri Poincar\'e} 12 (2011), 483--545.
\MR{2785137} (2012m:81060).  \Zbl{1217.81128}.  \arXiv{0912.5438}.			
\codes{\Csg:~\CTop:~\CInvar}%
\itemspace

\authorx{Daniel Kr\'al'}						%KRAL
\authorx{Daniel Kral'}
\author{Daniel Kr\'al', Jean-S\'ebastien Sereni, \& Ladislav Stacho}
\authorx{Daniel Kral', Jean-Sebastien Sereni, \& Ladislav Stacho}
\authorspace
\bsgitem{2012a}{}  Min-max relations for odd cycles in planar graphs.  \emph{SIAM J.\ Discrete Math.}\ 26 (2012), no.\ 3, 884--895.
\MR{3022112}.  \Zbl{1256.05119}.  \arXiv{1108.4281}.				
\annot{} $\nu :=$ max number of vertex-disjoint negative circles.  The vertex frustration number $l_0(\Sigma) \leq 6\nu(\Sigma)$ for planar $|\Sigma|$, improving on \xref{Fiorini, Hardy, Reed, and Vetta}{2005a}{Samuel Fiorini, Nadia Hardy, Bruce Reed, \& Adrian Vetta}, \xrefd{2007a}{Samuel Fiorini, Nadia Hardy, Bruce Reed, \& Adrian Vetta}.   
Dictionary:  ``odd'' = negative, ``even'' = positive, ``transversal'' = $X \subseteq V$ such that $\Sigma \setminus X$ is balanced.
\dated{1 Oct 2012, rev 14 Jan 2017} 
\codes{\Csg:~\Cpar:~\CFr,~\CCircles}%
\itemspace

\author{Daniel Kr\'al' \& Heinz-J\"urgen Voss}
\authorx{Daniel Kral' \& Heinz-Jurgen Voss}
\authorspace
\bsgitem{2004a}{}  Edge-disjoint odd cycles in planar graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 90 (2004), 107--120.
\MR{2041320} (2005d:05089).  \Zbl{1033.05064}.					
\annot{} Thm.\ 1:  For a signed plane graph $\Gamma$, the frustration index $l(\Sigma) \leq 2 \nu'$, where $\nu' :=$ maximum number of edge-disjoint negative circles.  
Dictionary:  ``odd'' = negative, ``even'' = positive.
[Continued in \xref{Fiorini, Hardy, Reed, and Vetta}{2007a}{Samuel Fiorini, Nadia Hardy, Bruce Reed, \& Adrian Vetta}, Thm.\ 3.]
\dated{6 Feb 2011} 
\par
\codes{\Csg:~\CPar:~\CFr,~\CCircles}%
\itemspace

\author{Mark A.\ Kramer}						%KRAMER
\authorspace
\authref  See also \hyperlink{Olayiwola O.\ Oyeleye}{O.O.~Oyeleye}.
\authrefspace

\author{M.A.\ Kramer \& B.L.\ Palowitch, Jr.}
\authorspace
\bsgitem{1987a}{}  A rule-based approach to fault diagnosis using the signed directed graph.  \emph{AIChE J.}\ 33 (1987), 1067--1078.
\MR{0895873} (88j:94060).  
\annot{} Vertex signs indicate directions of change in vertex variables; signed directed edges describe relations among these directions.
\annotctd{} Truth tables for a signed edge as a function of endpoint signs.  Algorithms for deducing logical rules about states (assignments of vertex signs) from the signed digraph.  Has a useful discussion of previous literature, e.g., \xref{Iri, Aoki, O'Shima, and Matsuyama}{1979a}{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama}.
\par
\codes{\CSD,~\CVS:~\CAppl,~\CAlgor,~\CRef}%
\itemspace

\author{P.L.\ Krapivsky}						%KRAPIVSKY
\authorspace
\authref  See \hyperlink{T.\ Antal}{T.~Antal}.
\itemspace

\author{I.\ Krasikov}							%KRASIKOV
\authorspace
\bsgitem{1988a}{}  A note on the vertex-switching reconstruction.  \emph{Int.\ J.\ Math.\ Math.\ Sci.}\ 11 (1988), 825--827.  
\MR{0959466} (89i:05204).  \Zbl{663.05046}.						
\annot{} Following up \xref{Stanley}{1985a}{Richard P.\ Stanley}, a signed $K_n$ is reconstructible from its single-vertex switching deck if its negative subgraph is disconnected [therefore also if its positive subgraph is disconnected] or if the minimum degree of its positive or negative subgraph is sufficiently small.  All done in terms of Seidel switching of unsigned simple graphs.
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1994a}{}  Applications of balance equations to vertex switching reconstruction.  \emph{J.\ Graph Theory} 18 (1994), 217--225.  
\MR{1268771} (95d:05091).  \Zbl{798.05039}.						
\annot{} Following up \xref{Krasikov and Roditty}{1987a}{I.\ Krasikov \& Y.\ Roditty}, $(K_n, \sigma)$ is reconstructible from its $s$-vertex switching deck if $s = \frac{1}{2}{n-r}$ where $r \in \{0, 2\}$ and $r \equiv n \pmod 4$, or $r = 1 \equiv n \pmod 2$; also, if $s = 2$ and the minimum degree of the positive or negative subgraph is sufficiently small.  Also, bounds on $\#E^-$ if $(K_n, \sigma)$ is not reconstructible.  Negative-subgraph degree sequence: reconstructible when $s = 2$ and $n \geq 10$.  Done in terms of Seidel switching of unsigned simple graphs.
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1996a}{}  Degree conditions for vertex switching reconstruction.  \emph{Discrete Math.}\ 160 (1996), 273--278.  
\MR{1417580} (97f:05137).  \Zbl{863.05056}.						
\annot{} If the minimum degrees of its positive and negative subgraphs obey certain bounds, a signed $K_n$ is reconstructible from its $s$-switching deck.  The main bound involves the least and greatest even zeros of the Krawtchouk polynomial $K_s^n(x)$.  Done in terms of Seidel switching of unsigned simple graphs.  [More details in Zbl.]
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\author{Ilia Krasikov \& Simon Litsyn}
\authorspace
\bsgitem{1996a}{}  On integral zeros of Krawtchouk polynomials.  \emph{J.\ Combin.\ Theory Ser.\ A} 74 (1996), 71--99.  
\MR{1383506} (97i:33005).  \Zbl{853.33008}.						
\annot{} Among the applications mentioned (pp.\ 72--73):  2.  ``Switching reconstruction problem'', i.e., graph-switching reconstruction as in \xref{Stanley}{1985a}{Richard P.\ Stanley} etc.  4.  ``Sign reconstruction problem'', i.e., reconstructing a signed graph from its $s$-edge negation deck, which is the multiset of signed graphs obtained by separately negating each subset of $s$ edges (here called ``switching signs'', but it is not signed-graph switching); this is a new problem.
\codes{\Ckg:~\Csw,~\CTG}\codesctd{\CSG}%
\itemspace

\author{I.\ Krasikov \& Y.\ Roditty}
\authorspace
\bsgitem{1987a}{}  Balance equations for reconstruction problems.  \emph{Arch.\ Math.\ (Basel)} 48 (1987), 458--464.  
\MR{0888875} (88g:05096).  \Zbl{594.05049}.					
\annot{} \S2: ``Reconstruction of graphs from vertex switching''.  Corollary 2.3.  If a signed $K_n$ is not reconstructible from its $s$-vertex switching deck, a certain linear Diophantine system (the ``balance equations'') has a certain kind of solution.  For $s = 1$ the balance equations are equivalent to \xref{Stanley's}{1985a}{Richard P.\ Stanley} theorem; for larger $s$ they may or may not be.  All is done in terms of Seidel switching of unsigned simple graphs.  [\xref{Ellingham and Royle}{1992a}{M.N.\ Ellingham \& Gordon F.\ Royle} note a gap in the proof of Lemma 2.5.]
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1992a}{}  Switching reconstruction and Diophantine equations.  \emph{J.\ Combin.\ Theory Ser.\ B} 54 (1992), 189--195.  
\MR{1152446} (93e:05072).  \Zbl{702.05062}. 					
\annot{} Main Theorem.  Fix $s \geq 4$.  If $n$ is large and (for odd $s$) not evenly even, every signed $K_n$ is reconstructible from its $s$-vertex switching deck.  Different results hold for $s = 2, 3$.  (This is based on and strengthens \xref{Stanley}{1985a}{Richard P.\ Stanley}.)  Theorems 5 and 6 concern reconstructing subgraph numbers.  All done in terms of Seidel switching of unsigned simple graphs.  
\par
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1994a}{}  More on vertex-switching reconstruction.  \emph{J.\ Combin.\ Theory Ser.\ B} 60 (1994), 40--55.  
\MR{1256582} (94j:05090).  \Zbl{794.05092}. 						
\annot{} Based on \xrefd{1987a}{I.\ Krasikov \& Y.\ Roditty} and strengthening \xref{Stanley}{1985a}{Richard P.\ Stanley}:  Theorem 7.  A signed $K_n$ is reconstructible if the Krawtchouk polynomial $K_s^n(x)$ ``has one or two even roots [lying] far from $n/2$'' (the precise statement is complicated).  Numerous other partial results, e.g., a signed $K_n$ is reconstructible if $s = \frac{1}{2}(n - r)$ where $r = 0, 1, 3$, or $2, 4, 5, 6$ with side conditions.  All is done in terms of Seidel switching of unsigned simple graphs.
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\author{Jan Kratochv{\'\i}l}						%KRATOCHVIL
\authorx{Jan Kratochvil}
\authorspace
\authref  See also \hyperlink{Eva Jelinkova}{E.~Jel{\'\i}nkov\'a}.
\authrefspace

\bsgitem{1989a}{}  Perfect codes and two-graphs.  \emph{Comment.\ Math.\ Univ.\ Carolin.}\ 30 (1989), no.\ 4, 755--760.
\MR{1045906} (91a:05080).  \Zbl{693.05060}.					
\annot{} A two-graph $\cT$ has a perfect code if every graph in its switching class has a 1-perfect vertex code (a perfect dominating set).  Thm.\  $\cT$ has a perfect code iff one of its graphs is the union of up to 3 disjoint cliques iff $\cT$ has no sub-pentagons and no sub-4-cocliques.
\dated{21 Mar 2011} 
\codes{\CTG:~\CSw,~\Cdom}%
\itemspace

\bsgitem{2003a}{}  Complexity of hypergraph coloring and Seidel's switching.  
In:  Hans L.\ Bodlaender, ed., \emph{Graph-Theoretic Concepts in Computer Science} (29th Int.\ Workshop, WG 2003, Elspeet, Neth., 2003), pp.\ 297--308.  \LNCS 2880.  Springer-Verlag, Berlin, 2003. 
\MR{2080089} (no rev).  \Zbl{1255.68082}.						
\annot{} Results about properties as in \xref{Kratochv{\'\i}l, Ne\v{s}et\v{r}il, \& Z\'yka}{1992a}{Jan Kratochvil, Jaroslav Nesetril, \& Ondrej Zyka}.  E.g., switchability to a regular graph is NP-complete.
\dated{21 Mar 2011} 
\codes{\CTG:~\CSw}%
\itemspace

\author{Jan Kratochv{\'\i}l, Jaroslav Ne\v{s}et\v{r}il, \& Ond\v{r}ej Z\'yka}
\authorx{Jan Kratochvil, Jaroslav Nesetril, \& Ondrej Zyka}
\authorspace
\bsgitem{1992a}{}  On the computational complexity of Seidel's switching.  
In:  Jaroslav Ne\v{s}et\v{r}il and Miroslav Fiedler, eds., \emph{Fourth Czechoslovak Symposium on Combinatorics, Graphs and Complexity} (Prachatice, 1990), pp.\ 161--166.  
Ann.\ Discrete Math., Vol.\ 51.  North-Holland, Amsterdam, 1992.  
\MR{1206260} (93j:05156).  \Zbl{768.68047}.					
\annot{} Is a given graph switching isomorphic to a graph with a specified property?  (This is Seidel switching of simple graphs.)  Depending on the property, this question may be in P or be NP-complete, whether the original property is in P or is NP-complete.  Properties: containing a Hamilton path; containing a Hamilton circle; no induced $P_2$; regularity; etc.  Thm.\ 4.1:  Switching isomorphism and graph isomorphism are polynomially equivalent.
\codes{\CTG:~\CSw:~\CAlgor}%
\itemspace

\authorx{Stefan Kratsch}						%KRATSCH
\author{Stefan Kratsch \& Magnus Wahlstr\"om}
\authorx{Stefan Kratsch \& Magnus Wahlstrom}
\authorspace
\bsgitem{2012a}{}  Compression via matroids: a randomized polynomial kernel for odd cycle transversal.  
In:  \emph{Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '12), pp.\ 94--103.  Soc.\ for Industrial and Appl.\ Math., Philadelphia, 2012.
\MR{3205199}.  
\codes{\Csg:~\Cpar:~\Cfr:~\CAlgor}%
\itemspace

\bsgitem{2014a}{}  Compression via matroids: a randomized polynomial kernel for odd cycle transversal.  \emph{ACM Trans.\ Algorithms} 10 (2014), no.\ 4, art.\ 20, 15 pp.
\MR{3254508}.  
\par
\codes{\Csg:~\Cpar:~\Cfr:~\CAlgor}%
\itemspace

\authorx{M.J.\ Krawczyk}						%KRAWCZYK
\author{M.J.\ Krawczyk, K.\ Malarz, B.\ Kawecka-Magiera, A.Z.\ Maksymowicz, \& K.\ Ku{\l}akowski}
\authorx{M.J.\ Krawczyk, K.\ Malarz, B.\ Kawecka-Magiera, A.Z.\ Maksymowicz, \& K.\ Kulakowski}
\authorspace
\bsgitem{2005a}{}  Spin-glass properties of an Ising antiferromagnet on the Archimedean $(3,12^2$) lattice.  \emph{Phys.\ Rev.\ B} 72 (2005), art.\ 24445, 5 pp.
\codes{\Cpar:~\CState(\Cfr)}%
\itemspace

\authorx{A.\ Krieger}							%KRIEGER
\author{A.\ Krieger \& B.\ O'Connor}
\authorspace
\bsgitem{2013a}{}  Tutte polynomial of signed graphs and its categorification.  Slides from a lecture at Ohio State University, 2013.  \url{https://people.math.osu.edu/chmutov.1/wor-gr-su13/pres.pdf}
\annot{} Introduces trivariate Tutte polynomial.  See \xref{Goodall, Litjens, Regts, and Vena}{2017a}{Andrew Goodall, Bart Litjens, Guus Regts, \& Lluis Vena} \etseq\ for further developments.
\codes{\CSG:~\CInvar}%
\itemspace

\author{Matthias Kriesell}						%KRIESELL
\authorspace
\authref  See \hyperlink{Jorgen Bang-Jensen}{J.~Bang-Jensen}.
\itemspace

\author{D.S.\ Krotov}							%KROTOV
\authorspace
\authref  See also \hyperlink{Evgeny Bespalov}{E.~Bespalov}.
\authrefspace

\bsgitem{2010a}{}  On connection between the switching separability of a graph and its subgraphs.  (In Russian.)  \emph{Diskretn.\ Anal.\ Issled.\ Oper.}\ 17 (2010), no.\ 2, 46--56, 101.
\bsgitem{2010b}{}  On a connection between the switching separability of a graph and its subgraphs.  (English trans.)  \emph{ J.\ Appl.\ Industrial Math.}\ 5 (2011), no.\ 2, 240--246.
\MR{2682089} (2011h:05119).  \Zbl{1249.05184}.  \arXiv{1104.0003}.			
\annot{} $\Gamma$ is ``switching separable'' if $\exists$ Seidel switching that is nontrivially disconnected.  (Trivially:  all vertices but one are connected.)  Thm.: If all $\Gamma\setminus v$ and $\Gamma \setminus \{u,v\}$ are, then $\Gamma$ is.  Deleting only single vertices is insufficient, for odd $n>4$.
\dated{31 Jul 2018} 
\codes{\Ctg:~\CSw:~\CStr}%
\itemspace

\authorx{Uffe Krusenstjerna-Hastr{\o}m}					%KRUSENSTJERNA-HASTROM
\authorx{Uffe Krusenstjerna-Hastrom}
\author{Uffe Krusenstjerna-Hastr{\o}m \& Bjarne Toft}
\authorx{Uffe Krusenstjerna-Hastrom \& Bjarne Toft}
\authorspace
\bsgitem{1980a}{}  Special subdivisions of $K_4$ and 4-chromatic graphs.  \emph{Monatsh.\ Math.}\ 89 (1980), no.\ 2, 101--109.
\MR{0572886} (81g:05058).  \Zbl{0421.05025}.  
\annot{} Special case of \xref{Toft's}{1975a}{Bjarne Toft} conjecture.
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\author{Vyacheslav Krushkal}						%KRUSHKAL
\authorspace
\authref  See also \hyperlink{Paul Fendley}{P.~Fendley}.
\authrefspace

\bsgitem{2011a}{}  Graphs, links, and duality on surfaces.  \emph{Combin.\ Prob.\ Computing} 20 (2011), 267--287.
\MR{2769192} (2012d:05190).  \Zbl{1211.05029}.  
\annot{} \S7, ``A multivariate graph polynomial'':  A partially parametrized rank-generating polynomial (``multivariate Tutte polynomial'') for graphs embedded in surfaces, with the somewhat awkward duality relation (7.3).  \Cf\ \xref{Chmutov and Pak}{2007a}{Sergei Chmutov \& Igor Pak} and \xref{Chmutov}{2009a}{Sergei Chmutov}.
\dated{12 Jan 2012} 
\codes{\CGGw:~\CInvar}%
\itemspace

\author{F.\ Krzakala}							%KRZAKALA
\authorspace
\authref  See \hyperlink{J.-P.\ Bouchaud}{J.-P.~Bouchaud}.
\itemspace

\author{Ying-Qiang Kuang}						%KUANG
\authorspace
\authref  See \hyperlink{Zhi-Hong Chen}{Z.H.~Chen}.
\itemspace

\author{Boris D.\ Kudryashov}						%KUDRYASHOV
\authorspace
\authref  See \hyperlink{Irina E.\ Bocharova}{I.E.~Bocharova}.
\itemspace

\author{Christian Kuehn}						%KUEHN
\authorspace
\authref  See \hyperlink{Raffaella Mulas}{R.~Mulas}.
\itemspace

\authorx{Lukas K\"uhne}						%KUHNE
\authorx{Lukas Kuhne}
\author{Lukas K\"uhne \& Geva Yashfe}
\authorx{Lukas Kuhne \& Geva Yashfe}
\authorspace
\bsgitem{2019a}{}  Representability of matroids by $c$-arrangements is undecidable.  Manuscript, 2019, 2020.
\MR{4526828}.  
\arXiv{1912.06123}.  
\annot{} \S4, ``Generalized Dowling geometries'':  Rank 3 full frame matroids derived from finite group presentations.  Representation by matrix gains.
Dictionary:  ``triangle matroid'' = rank-3 full frame matroid.
\Cf\ \xrefd{2020a}{Lukas Kuhne \& Geva Yashfe}.
\dated{20 Oct 2020} 
\par
\codes{\Cgg:~\CMatrd:~\CGeom}%
\itemspace

\bsgitem{2020a}{}  Undecidability of $c$-arrangement matroid representations.  Proc.\ 32nd Int.\ Conf.\ Formal Power Seri.\ Algebraic Combin.\ (Online, 2020).  \emph{S\'em.\ Lotharingien Combin.}\ 84B (2020), art.\ 87, 12 pp. 
\MR{4138714}.  \Zbl{1447.05049}.  
\codes{\Cgg:~\CMatrd:~\CGeom}%
\itemspace

\authorx{Bernard Kujawski}						%KUJAWSKI
\author{Bernard Kujawski, Mark Ludwig, \& Peter Abell}
\authorspace
\bsgitem{2010a}{}  Structural balance dynamics and group formation: An exploratory study.  Manuscript, 2010.
\codes{\CSG:~\CBal}%
\itemspace

\author{Krzysztof Ku{\l}akowski}					%KULAKOWSKI
\authorx{Krzysztof Kulakowski}
\authorspace
\authref  See also \hyperlink{Premiyslaw Gawronski}{P.~Gawro\'nski}, \hyperlink{Forough Hassanibesheli}{F.~Hassanibesheli}, \hyperlink{Anna Manka-Krason}{A.~Ma\'nka-Kraso\'n}, \hyperlink{Bosiljka Tadic}{B.~Tadi\'c}, and \hyperlink{Joanna Tomkowicz}{J.~Tomkowicz}.
\authrefspace

\bsgitem{2007a}{}  Some recent attempts to simulate the Heider balance problem.  \emph{Computing in Science and Engineering} 9 (July/Aug.\ 2007), no.\ 4, 86--91.  
\\ \arXiv{physics/0612197}.  
\codes{\CSG,~\CWG:~\CBal,~\CDyn:~\CExp}%
\itemspace

\author{Krzysztof Ku{\l}akowski, Premiys{\l}aw Gawro\'nski, \& Piotr Gronek}
\authorx{Krzysztof Kulakowski, Premiyslaw Gawronski, \& Piotr Gronek}
\authorspace
\bsgitem{2005a}{}  The Heider balance: a continuous approach.  \emph{Int.\ J.\ Mod.\ Phys.\ C} 16 (2005), no.\ 5, 707--716.
\Zbl{1103.91405}.  \arXiv{physics/0501073}.  
\codes{\CSG,~\CWG:~\CBal,~\CDyn}%
\itemspace

\author{Devadatta M.\ Kulkarni}						%KULKARNI
\authorspace
\authref  See \hyperlink{Jerrold W.\ Grossman}{J.W.~Grossman}.
\itemspace

\author{Samir Khuller}						%KULLER S
\authorspace
\authref  See \hyperlink{Saba Ahmadi}{S.~Ahmadi}.
\itemspace

\author{Navish Kumar}						%KUMAR N
\authorspace
\authref  See \hyperlink{M.\ Rajesh Kannan}{M.R.~Kannan}.
\itemspace

\author{R.\ Pradeep Kumar}						%KUMAR RP
\authorspace
\authref  See \hyperlink{M.R.\ Rajesh Kanna}{M.R.~Rajesh Kanna}.
\itemspace

\author{Sandeep Kumar}						%KUMAR S
\authorspace
\authref  See \hyperlink{Deepa Sinha}{D.~Sinha}.
\itemspace

\author{T.R.\ Vasanth Kumar}						%KUMAR TRV
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{[Vijaya Kumar]} 						%KUMAR V
\authorspace
\authref  See \hyperlink{G.R.\ Vijayakumar}{G.R.~Vijayakumar}.
\itemspace

\author{[Anita Kumari Rao]}						%KUMARI RAO
\authorspace
\authref  See \hyperlink{Anita Kumari Rao}{A.K.~Rao} (under `R').
\itemspace

\author{J\'er\^ome Kunegis}						%KUNEGIS
\authorx{Jerome Kunegis}
\authorspace
\bsgitem{2011a}{}  \emph{On the Spectral Evolution of Large Networks}.  Doctoral dissertation, Univ.\ Koblenz-Landau, 2011.
\annot{} Ch.\ 5, ``Signed networks'', pp.\ 79--106.
\par
\codes{\CSG,~\CSD:~\CLap:~\CBal,~\CClu,~\CFr,~\CEig,~\CWG,~\CPred,~\CPsS}%
\itemspace

\bsgitem{2014a}{}  Applications of structural balance in signed social networks.  Manuscript, 2014.
\arXiv{1402.6865}.  
\annot{} \S3, ``Measuring structural balance: The signed clustering coefficient'':  A new definition; the coefficient for $\Sigma$ is $3 \big(\sum_{C_3\subseteq\Sigma} \sigma(C_3)\big) / \#\{(e,f): e\sim f\}.$  Also defined for signed digraphs.
\dated{8 Jan 2016} \S4, ``Visualizing structural balance: Signed graph drawing'':  Applies $\Lap(\Sigma)$ to signed-graph drawing.  
\S5, ``Capturing structural balance: The signed Laplacian'':  I.e., $\Lap(\Sigma)$.  \S5.3, ``Balanced graphs'':  Then $\Spec \Lap(\Sigma) = \Spec \Lap(|\Sigma|)$ and the eigenvectors of $\Sigma$ are switched (componentwise) from those of $|\Sigma|$.  
\S6, ``Measuring structural balance 2: Algebraic conflict'':  The smallest eigenvalue of $\Lap(\Sigma)$ is dubbed ``algebraic conflict'' since it is $>0$ iff $\Sigma$ is unbalanced.  \Cf\ \xref{Hou}{2005a}{Yaoping Hou}.  
\S7, `Maximizing structural balance: Signed spectral clustering'':  Uses $\Lap(\Sigma)$, alternatively $D\inv A(\Sigma)$ ($D=$ diagonal degree matrix).  
\S8, ``Predicting structural balance: Signed resistance distance'':  A way to compute resistance distance for ``signed resistances'' = weighted signed edges, with an adapted Kirchhoff's current law.  Used for edge prediction.
\annotctd{} Partly expository.
\par
\codes{\CSG,~\CSD:~\CLap:~\CBal,~\CClu,~\CFr,~\CEig,~\CWG,~\CPred,~\CPsS}%
\itemspace

\author{J\'er\^ome Kunegis, Andreas Lommatzsch, \& Christian Bauckhage}
\authorx{Jerome Kunegis, Andreas Lommatzsch, \& Christian Bauckhage}
\authorspace
\bsgitem{2009a}{}  The slashdot zoo: mining a social network with negative edges. 
In: \emph{Proceedings of the 18th International Conference on the World Wide Web} (Madrid, 2009), pp.\ 741--750.  Assoc.\ for Computing Machinery, New York, 2009.
\par
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\author{J\'er\^ome Kunegis, Julia Preusse, \& Felix Schwagereit}
\authorx{Jerome Kunegis, Julia Preusse, \& Felix Schwagereit}
\authorspace
\bsgitem{2013a}{}  What is the added value of negative links in online social networks?  In:  \emph{WWW '13: Proceedings of the 22nd International Conference on World Wide Web} (Rio de Janeiro, 2013), pp.\ 727--736.  ACM, 2013.
\annot{} A fairly large partial positive subgraph tends to predict negative edges but only by using a combination of centrality and proximity predictors.
\dated{29 Dec 2020} 
\codes{\CPsS:~\CSG,~\CPred:~\CAlgor}%
\itemspace

\author{J\'er\^ome Kunegis \& Stephan Schmidt}
\authorx{Jerome Kunegis \& Stephan Schmidt}
\authorspace
\bsgitem{2007a}{}  Collaborative filtering using electrical resistance network models with negative edges.  In:  Petra Perner, ed., \emph{Advances in Data Mining: Theoretical Aspects and Applications} (Proc.\ 7th Industrial Conf., ICDM 2007, Leipzig), pp.\ 269--282.  \LNCS 4597.  Springer, Berlin, 2007.
\par
\codes{\Csg,~\CWG:~\CLap}%
\itemspace

\author{J\'er\^ome Kunegis, Stephan Schmidt, \c{S}ahin Albayrak, Christian Bauckhage, \& Martin Mehlitz}
\authorx{Jerome Kunegis, Stephan Schmidt, Sahin Albayrak, Christian Bauckhage, \& Martin Mehlitz}
\authorspace
\bsgitem{2008a}{}  Modeling collaborative similarity with the signed resistance distance kernel.  
In:  Malik Ghallab \etal, eds., \emph{ECAI 2008 -- 18th European Conference on Artificial Intelligence}, pp.\ 261--265.  
Frontiers in Artificial Intelligence and Applications, Vol.\ 178.  IOS Press, Amsterdam, 2008.
\codes{\CSG:~\CAdj,~\CAlgor}%
\itemspace

\author{J\'er\^ome Kunegis, Stephan Schmidt, Andreas Lommatzsch, J\"urgen Lerner, Ernesto W.\ De Luca, \& Sahin Albayrak}
\authorx{Jerome Kunegis, Stephan Schmidt, Andreas Lommatzsch, Jurgen Lerner, Ernesto W.\ De Luca, \& Sahin Albayrak}
\authorspace
\bsgitem{2010a}{}  Spectral analysis of signed graphs for clustering, prediction and visualization.  
In:  Srinivasan Parthasarathy \etal, eds., \emph{Proceedings of the Tenth SIAM International Conference on Data Mining} (Columbus, Ohio, 2010), pp.\ 559--570.  Soc.\ for Industrial and Appl.\ Math., 2010.
\codes{\CSG:~\CEig,~\CClu,~\CGeom,~\CPred,~\CAlgor}%
\itemspace

\author{Joseph P.S.\ Kung} 						%KUNG
\authorspace
\authref  See also \hyperlink{Joseph E.\ Bonin}{J.E.~Bonin} and \hyperlink{Jeff Kahn}{J.~Kahn}.
\authrefspace

\bsgitem{1986a}{}  Numerically regular hereditary classes of combinatorial geometries.  \emph{Geom.\ Dedicata} 21 (1986), 85--105.  
\MR{0850567} (87m:05056).  \Zbl{591.05019}.
\annot{} Examples include Dowling geometries, Ex.\ (6.2), and the frame matroids of full group expansions of graphs in certain classes; see pp.\ 98--99.
\par
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{1986b}{}  Radon transforms in combinatorics and lattice theory.  In: Ivan Rival, ed., \emph{Combinatorics and Ordered Sets} (Proc., Arcata, Calif., 1985), pp.\ 33--74.  Contemp.\ Math., Vol.\ 57.  Amer.\ Math.\ Soc., Providence, R.I., 1986.
\MR{0856232} (88d:05024).  \Zbl{595.05006}.
\annot{} P.\ 41: Exposition of \xref{Stanley}{1985a}{Richard P.\ Stanley} from the viewpoint of the finite Radon transform.
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1987a}{}  Research Problem 87.  \emph{Discrete Math.}\ 65 (1987), 105--106.
\annot{} Conjecture:  For every group $\fG$, $\exists\ k=k_\fG$ such that if $M$ is a rank-$n$ matroid ($n>k$) where every rank-$k$ interval $[x,\hat1] \cong Q_k(\fG)$, then $M \subseteq Q_n(\fG)$.  [This should be provable.  $k$ should be small.]
\dated{9 Apr 1987} 
\codes{\Cgg:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{1990a}{}  Combinatorial geometries representable over $\GF(3)$ and $\GF(q)$.  I.  The number of points.  \emph{Discrete Comput.\ Geom.}\ 5 (1990), 83--95.  
\MR{1018017} (90i:05028).  \Zbl{697.51007}.
\annot{} The Dowling geometry over the sign group is the largest simple ternary matroid not containing the ``Reid matroid''.
\codes{\Csg:~\CMatrd:~\CXtreml}%
\itemspace

\bsgitem{1990b}{}  The long-line graph of a combinatorial geometry.  II.  Geometries representable over two fields of different characteristic.  
\emph{J.\ Combin.\ Theory Ser.\ B} 50 (1990), 41--53.  
\MR{1070464} (91m:51007).  \Zbl{645.05026}.					
\annot{} Dowling geometries used in the proof of Prop.\ (1.2).
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1993a}{}  Extremal matroid theory.  In: Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 21--61.  
Contemp.\ Math., Vol.\ 147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.  
\MR{1224696} (94i:05022).  \Zbl{791.05018}.					
\annot{} Survey with new results; largely on size bounds and extremal matroids for certain minor-closed classes.   \S2.7:  ``Gain-graphic matroids,'' i.e., frame matroids of gain graphs.  P.~30, top and fn.~9 on extremal gain-graph theory.  
\S4.3:  ``Varieties.''  Conj.\ (4.9)(c) on growth rates.  \S4.5, ``Framed gain-graphic matroids,'' i.e., cones over (``framed'') frame matroids in projective space.  \S6.1: ``Cones,'' i.e., unions of long lines on a common point: p.\ 47.  
Thm.\ (6.15) is a quadratic bound on matroids whose minors exclude (approximately) $q+2$-point lines and non-frame planes.  Conj.\ (7.1) on directions in $\bbC^n$-matroids proposes that cyclic Dowling matroids are extremal.  
\S8:  ``Concluding remarks,'' on a possible ternary analog of Seymour's decomposition theorem.
\par
\codes{\CGG:~\CMatrd:~\CXtreml,~\CStr,~\CExp,~\CRef}%
\itemspace

\bsgitem{1993b}{}  The Radon transforms of a combinatorial geometry.  II.  Partition lattices.  \emph{Adv.\ Math.}\ 101 (1993), 114--132.  
\MR{1239455} (95b:05051).  \Zbl{786.05018}.
\annot{} Dowling lattices are lower-half Sperner.  The proof is given only for partition lattices.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1996a}{}  Matroids.  In:  M.\ Hazewinkel, ed., \emph{Handbook of Algebra}, Vol.\ 1, pp.\ 157--184.  North-Holland (Elsevier), Amsterdam, 1996.  
\MR{1421801} (98c:05040).  \Zbl{856.05001}.					
\annot{} \S6.2:  ``Gain-graphic matroids,'' i.e., frame matroids of gain graphs.
\par
\codes{\CGG:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{1996b}{\dag}  Critical problems.  In:  Joseph E.\ Bonin, James G.\ Oxley, and Brigitte Servatius, eds., \emph{Matroid Theory} (Proc., Seattle, 1995), pp.\ 1--127.  
Contemp.\ Math., Vol.\ 197.  Amer.\ Math.\ Soc., Providence, R.I., 1996.
\MR{1411690} (97k:05049).  \Zbl{862.05019}. 					
\annot{} A remarkable more-than-survey with numerous new results and open problems.  \S4.5:  ``Abstract linear functionals in Dowling group geometries''.  \S6: ``Dowling geometries and linear codes'', concentrates on higher-weight Dowling geometries, extending \xref{Bonin}{1993b}{Joseph E.\ Bonin}.  \S7.4: ``Critical exponents of classes of gain-graphic geometries''.  \S7.5:  ``Growth rates of classes of gain-graphic geometries''.  \S8.5: ``Jointless Dowling group geometries''.  Cor.\ 8.30.  \S8.11: ``Tangential blocks in $\cZ(A)$''.  Also see pp.\ 56, 61, 88, 92, 114.  
Dictionary:  ``Gain-graphic matroids'' = frame matroids of gain graphs.
\codes{\CGG,~\CGen:~\CMatrd}%
\itemspace

\bsgitem{1998a}{}  A geometric condition for a hyperplane arrangement to be free.  \emph{Adv.\ Math.}\ 135 (1998), 303--329.
\MR{1620842} (2000f:05023).  \Zbl{905.05017}.					
\annot{} Delete from a Dowling geometry a subset $S$ that contains no whole plane.  Found: necessary and sufficient conditions for the characteristic polynomial to factor completely over the integers.  When the geometry corresponds to a  hyperplane arrangement, many more of the arrangements are not free than are free; however, if $S$ contains no whole line, all are free (so the characteristic polynomial factors completely over $\bbZ$) while many are not supersolvable.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2000a}{}  Critical exponents, colines, and projective geometries.  \emph{Combin.\ Probab.\ Comput.}\ 9 (2000), 355--362.
\MR{1786924} (2002f:05048).  \Zbl{974.51008}.						
\annot{} Higher-weight Dowling geometries yield counterexamples to a conjecture.
\codes{\Cgg:~\CGen:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2001a}{}  Twelve views of matroid theory.  
In: Sungpyo Hong \etal, eds., \emph{Combinatorial \& Computational Mathematics} (Proc., Pohang, 2000), pp.\ 56--96.  World Scientific, Singapore, 2001. 
\MR{1868420} (2002i:05028).  \Zbl{1001.05038}.						
\annot{} \S5: ``Graph theory and lean linear algebra''.  ``Lean'' means at most 2 nonzero coordinates, hence gain graphs.  \S6, ``Varieties of finite matroids'', summarizes \xref{Kahn and Kung}{1982a}{Jeff Kahn \& Joseph P.S.\ Kung}.  \S7, ``Secret-sharing matroids'':  \Question.  Is the Dowling matroid $Q_n(\fG)$ a secret-sharing matroid?
\codes{\CGG:~\CMatrd}%
\annotctd{} \S11, ``Generic rank-generating polynomials'':  The ``Tugger polynomial'' is a partially parametrized rank-generating polynomial (\cf\ \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}).
\codes{\CSc(\CMatrd):~\CGen:~\CInvar}%
\itemspace

\bsgitem{2002a}{}  Curious characterizations of projective and affine geometries.  Special issue in memory of Rodica Simion.  \emph{Adv.\ Appl.\ Math.}\ 28 (2002), 523--543.
\MR{1900006} (2003c:51008).  \Zbl{1007.51001}.
\annot{} Dowling geometries $\bfF(\fG K_n\full)$ (if $\#\fG>2$) and jointless Dowling geometries $\bfF(\fG K_n)$ (if $\#\fG>4$) exemplify Lemma 3.4, which says that 5 numbers characterize the line sizes in a simple matroid with all lines of size 2, 3, or $l$.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2006a}{}  Minimal blocks of binary even-weight vectors.  \emph{Linear Algebra Appl.}\ 416 (2006), 288--297.
\MR{2242730} (2008d:05038).  \Zbl{1115.05012}.					
\annot{} \S4, ``Minimal blocks from graphs'':  $\GF(q)^\times\cdot \Gamma$ is a minimal $k$-block over $\GF(q)$ if $\Gamma$ is minimally $j$-chromatic for a certain $j = f(k)$, and is a minimal 1-block if $\Gamma$ is an odd circle.
\dated{20 Jun 2011} 
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Joseph P.S.\ Kung \& James G.\ Oxley}
\authorspace
\bsgitem{1988a}{}  Combinatorial geometries representable over $\GF(3)$ and $\GF(q)$.  II.  Dowling geometries.  \emph{Graphs Combin.}\ 4 (1988), 323--332.  
\MR{0965387} (90i:05029).  \Zbl{702.51004}.  
\annot{} For $n \geq 4$, the Dowling geometry of rank $n$ over the sign group is the unique largest simple matroid of rank $n$ that is representable over $\GF(3)$ and $\GF(q)$.
\codes{\Csg:~\CMatrd:~\CXtreml}%
\itemspace

\authorx{H.\ Kunze}							%KUNZE
\author{H.\ Kunze \& D.\ Siegel}
\authorspace
\bsgitem{1994a}{}  A graph theoretical approach to monotonicity with respect to initial conditions.  In:  Xinzhi Liu and David Siegel, eds., \emph{Comparison Methods and Stability Theory} (Proc., Waterloo, Ont., 1993).  Lect.\ Notes Pure Appl.\ Math.,  Vol.\ 162.  Marcel Dekker, New York, 1994.
\MR{1291622} (95g:34065).  
\codes{\CSD:~\CBal,~\CDyn}%
\itemspace

\bsgitem{1999a}{}  A graph theoretical approach to monotonicity with respect to initial conditions II.  \emph{Nonlinear Analysis} 35 (1999), 1--20.
\MR{1634009} (99g:34032).  
\par
\codes{\CSD:~\CBal,~\CDyn}%
\itemspace

\author{David Kuo}							%KUO D
\authorspace
\authref   See \hyperlink{Jing-Ho Yan}{J.H.~Yan}.
\itemspace

\author{Y.S.\ Kuo}							%KUO YS
\authorspace
\authref  See also \hyperlink{Wei-Kuan Shih}{W.-S.~Shih}.
\authrefspace

\author{Y.S.\ Kuo, T.C.\ Chern, \& Wei-kuan Shih}
\authorspace
\bsgitem{1988a}{}  Fast algorithm for optimal layer assignment.  In:  \emph{Proceedings of the 25th ACM/IEEE Design Automation Conference} (Anaheim, Calif., 1988), pp.\ 554--559.
\annot{} Algorithm, by minimum perfect matching, for $l(\Sigma)$ for a weighted signed graph that is cubic and planar.  See \xref{Kuo--Chern--Shih}{1988a}{Y.S.\ Kuo, T.C.\ Chern, \& Wei-kuan Shih}.  [Authors are unaware of \xref{Katai and Iwai}{1978a}{Osamu Katai \& Sousuke Iwai} or \xref{Barahona}{1982a}{Francisco Barahona} etc.]
\dated{21 Dec 2014} 
\codes{\CWG,~\Csg:~\Cfr:~\CAlgor}%
\itemspace

\author{Yueh-Er Kuo}						%KUO YE
\authorspace
\authref  See \hyperlink{Gloria de Leon-Calio}{G.~de~Leon-Calio}.
\itemspace

\author{Ranan D.\ Kuperman}						%KUPERMAN
\authorspace
\authref  See \hyperlink{Zeev Maoz}{Z.~Maoz}.
\itemspace

\author{Joseph Varghese Kureethara}					%KUREETHARA
\authorspace
\authref  See also \hyperlink{Athira P Ranjith} and \hyperlink{Mukti Acharya}{M.~Acharya}.
\authrefspace

\bsgitem{2020a}{}  New developments in the study of parity signed graphs.  In:  Samayan Narayanamoorthy, ed., \emph{Advances In Applicable Mathematics-- ICAAM2020} (Coimbatore, India, 2020).  AIP Conf.\ Proc., Vol.\ 2261, art.\ 020001, 4 pp.
\annot{} \Cf\ \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara}.
\dated{13 Dec 2020} 
\codes{\CSG:~\CLab:~\CExp}%
\itemspace

\author{Jin Ho Kwak}							%KWAK JH
\authorspace
\authref  See also \hyperlink{Ian P.\ Goulden}{I.P.~Goulden}.
\authrefspace

\author{Jin Ho Kwak, Sungpyo Hong, Jaeun Lee, \& Moo Young Sohn}
\authorspace
\bsgitem{2000a}{}  Isoperimetric numbers and bisection widths of double coverings of a complete graph.  \emph{Ars Combin.}\ 57 (2000), 49--64.
\MR{1796626} (2001h:05083).  \Zbl{1064.05076}.
\codes{\Csg:~\CKG:~\CCov}%
\itemspace

\author{Jin Ho Kwak \& Jaeun Lee}
\authorspace
\bsgitem{2001a}{}  Enumeration of graph coverings, surface branched coverings and related group theory.  
In: Sungpyo Hong \etal, eds., \emph{Combinatorial \& Computational Mathematics} (Proc., Pohang, 2000), pp.\ 97--161.  World Scientific, Singapore, 2001. 
\MR{1868421} (2003b:05083).  \Zbl{1001.05092}.
\annot{} Voltage graphs (i.e., gain graphs) and their covering graphs (``derived graphs'') defined in \S1; emphasis on groups and counting group covering graphs of a graph.
\codes{\Cgg:~\CCov,~\CTop}%
\itemspace

\bsgitem{2009a}{}  Enumerating coverings.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 9, pp.\ 181--198.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581546} (no rev).  \Zbl{1225.05202}.  
\annot{} Counting of various kinds of covering graphs via gain graphs (``voltage graphs'').  \S1, ``Introduction'': Definition of voltage graphs.  \S2  ``Graph coverings'' via voltage graphs.  Then counting:  \S3, ``Regular coverings'';  \S4, ``Surface branched coverings'';.  \S5, ``Regular surface branched coverings''.  \S6, ``Distribution of surface branched coverings''.
\dated{12 Jun 2013} 
\codes{\CTop:~\Cgg,~\CEnum,~\CCov:~\CExp}%
\itemspace

\author{J.H.\ Kwak, Jaeun Lee, \& Young-hee Shin}
\authorspace
\bsgitem{2004a}{}  Balanced regular coverings of a signed graph and regular branched orientable surface coverings over a non-orientable surface.  \emph{Discrete Math.}\ 275 (2004), 177--193.
\MR{2026284} (2004i:05036).  \Zbl{1030.05034}.					
\annot{} The number of isomorphism types of regular balanced coverings of a signed graph.  A covering is a sign-preserving covering projection from one signed graph to another.
\codes{\CSG:~\CTop:~\CEnum}%
\itemspace

\authorx{Yung-Keun Kwon}						%KWON YK
\author{Yung-Keun Kwon \& Kwang-Hyun Cho}
\authorspace
\bsgitem{2007a}{}  Boolean dynamics of biological networks with multiple coupled feedback loops.  \emph{Biophys.\ J.}\ 92 (2007), 2975--2981 + suppl.\ 2 pp.
\annot{} Simulations suggest that more positive cycles lead to more fixed points and more negative cycles lead to more non-fixed-point attractors (with a fixed number of variables [or genes]).
\dated{16 Jan 2015} 
\par
\codes{\CSD:~\CDyn:~\CStr}%
\itemspace


\label{LETTER L} \hypertarget{LETTER L}{}	% LETTER  L


\author{Vincent Labatut}						%LABATUT
\authorspace
\authref  See \hyperlink{Nejat Arinik}{N.~Ar{\i}n{\i}k}, \hyperlink{Rosa Figueiredo}{R.~Figueiredo}, and \hyperlink{Israel Mendonca}{I.~Mendon\c{c}a}.
\itemspace

\author{Domenico Labbate}						%LABBATE
\authorspace
\authref  See \hyperlink{Marien Abreu}{M.~Abreu}.
\itemspace

\author{Martine Labb\'e}						%LABBE
\authorspace
\authref  See \hyperlink{Vincent Devloo}{V.~Devloo} and \hyperlink{Rosa M.V.\ Figueiredo}{R.M.V.~Figueiredo}.
\itemspace

\author{Nicholas Lacasse}						%LACASSE
\authorspace
\bsgitem{20xxa}{}  Minimal and disjoint negation sets in signed graphs.  Submitted.  \emph{}\ 
\arXiv{2010.02276}.  
\codes{\CSG:~\CFr,~\CSw}%
\itemspace

\bsgitem{20xxb}{}  The characteristic polynomial of the odd arrangement of hyperplanes.  Submitted.  \emph{}\ 
\codes{\CGG:~\CGeom:~\CInvar}%
\itemspace

\author{Richard Ladner}						%LADNER
\authorspace
\authref  See \hyperlink{Victor Klee, Richard Ladner, \& Rachel Manber}{V.~Klee}.
\itemspace

\authorx{George M.\ Lady}						%LADY
\author{George M.\ Lady, Thomas J.\ Lundy, \& John Maybee}
\authorspace
\bsgitem{1995a}{}  Nearly sign-nonsingular matrices.  \emph{Linear Algebra Appl.}\ 220 (1995), 229--248.
\MR{1334579} (96e:15007).  \Zbl{838.15013}.  
\annot{} The signed digraph $S(A)$ of square matrix $A$.  Thm.\ 1: $A$ is NSNS iff the rows can be permuted so that $S(A)$ has a negative loop at each vertex and no other negative cycles, and no vertex-disjoint positive cycles.
\dated{12 Jun, 24 Nov 2012} 
\codes{\CSD:~\CQM}%
\itemspace

\author{George M.\ Lady \& John S.\ Maybee}
\authorspace
\bsgitem{1983a}{}  Qualitatively invertible matrices.  \emph{Math.\ Social Sci.}\ 6 (1983), 397--407.  
\MR{0747746} (85f:15005).  \Zbl{547.15002}.   
\annot{} In terms of signed graphs, restates and completes the characterizations of sign-invertible matrices $A$ due to \xref{Bassett, Maybee, and Quirk}{1968a}{Lowell Bassett, John Maybee, \& James Quirk} and George M.\ Lady (The structure of qualitatively determinate relationships.  \emph{Econometrica} 51 (1983), 197--218.  \MR{0694457} (85c:90019).  \Zbl{517.15004}) and reveals the sign pattern of $A^{-1}$ in terms of path signs in the associated signed digraph.  
\codes{\CQM:~\CSol:~\CSD}%
\itemspace

\author{J.C.\ Lagarias}						%LAGARIAS
\authorspace
\bsgitem{1985a}{}  The computational complexity of simultaneous diophantine approximation problems.  \emph{SIAM J.\ Computing}  14 (1985), 196--209.  
\MR{0774939} (86m:11048).  \Zbl{563.10025}.					
\annot{} Theorem F:  Feasibility of integer linear programs with at most two variables per constraint is NP-complete.  
\codes{\CGN(\CInc):~\CDual:~\CAlgor}%
\itemspace

\author{Hong-Jian Lai}						%LAI HJ
\authorspace
\authref  See \hyperlink{Zhi-Hong Chen}{Z.H.~Chen}, \hyperlink{Shengjie He}{S.-J.~He}, \hyperlink{Yanting Liang}{Y.T.~Liang}, and \hyperlink{Jianbing Liu}{J.-B.~Liu}.
\itemspace

\author{Dimitri Lajou}							%LAJOU
\authorspace
\authref  See also \hyperlink{Florent Foucaud}{F.~Foucaud}.
\authrefspace

\bsgitem{2019a}{}  On the achromatic number of signed graphs.  \emph{Theor.\ Computer Sci.}\ 759 (2019), 50--60.
\MR{3913216}.  \Zbl{1405.05063}.  \arXiv{1902.04828}.  \HAL{hal-02014469}.		
\annot{} Introduces achromatic numbers $\chi_a(\Sigma)$ for sign-colored simple graphs and $\chi_s([\Sigma]) := \max\chi_a(\Sigma')$ over $\Sigma'\in[\Sigma]$ for switching classes, leading to two variants for switching classes and four for unsigned graphs: max and min of $\chi_a(\Sigma')$ for $\Sigma' \in [\Sigma]$, max and min of $\chi_a(\Sigma)$ over $|\Sigma|=\Gamma$, max and min of $\chi_s([\Sigma])$ over $|\Sigma|=\Gamma$.  Five of the eight, including $\chi_a$ and $\chi_s$, are shown NP-complete; the other three are open.  
A ``$k$-coloring'' of $\Sigma$ is $\alpha: V \to [k]$ such that $\forall\{i,j\}$ the set $\{uv\in E: \alpha(u)=i,\ \alpha(v)=j\}$ is homogeneously signed.  A $k$-coloring of $[\Sigma]$ is a $k$-coloring of any $\Sigma'\in[\Sigma]$.  
\annotctd{}  ]This coloring is unrelated to (signed) coloring defined in \xref{Zaslavsky}{1982b}{Thomas Zaslavsky}.  Is there an achromatic number for the latter?]
\annotctd{} Dictionary:  ``2-edge-colored graph'' = sign-colored graph $\Sigma$, ``re-signing'' = switching, ``signed graph'' = switching class $[\Sigma]$, ``2-edge-colored homomorphism'' = edge-sign-preserving homomorphism, ``signed homomorphism'' = switching (circle-sign-preserving) homomorphism, ``(un)bal\-anced path/cycle'' = (negative) positive path/circle.
\dated{28 Dec 2019} 
\codes{\CSGc,~\CSG:~\CCol(\CGen)}%
\itemspace

\bsgitem{2020a}{}  On Cartesian products of signed graphs.  In:  Manoj Changat and Sandip Das, eds., \emph{Algorithms and Discrete Applied Mathematics} (Proc.\ 6th Int.\ Conf., CALDAM 2020, Hyderabad, India, 2020), pp.\ 219--234.  \LNCS 12016. Springer, Cham, 2020.
\MR{4074911}.  \Zbl{1456.05133}.  \HAL{hal-02491883}.  
\annot{} Product defined by \xref{Germina, Hameed, and Zaslavsky}{2011a}{K.A.\ Germina, Shahul Hameed K, \& Thomas Zaslavsky}.  \Cf\ \xrefd{20xxa}{Dimitri Lajou}.
\codes{\CSG:~\CAlgor,~\CHom}%
\itemspace

\bsgitem{2021a}{}  \emph{On Various Graph Coloring Problems}.  [\emph{Sur divers probl\`emes de coloration de graphes}.]  Doctoral thesis, Univ.\ de Bordeaux, 2021. 
\HAL{tel-03522406}.  
\annot{} Part I, ``Signed graphs''.  \S2, ``Introduction to signed graphs''.  \S3, ``Complexity of edge-colored and signed graphs modification problems''.  \S4, ``Coloring signed graphs with small cyclomatic number''.  \S5, ``Cartesian product of signed graphs''.
\dated{24 Mar 2022} 
\par
\codes{\CSG:~\CCol,~\CHom,~\CAlgor}%
\itemspace

\bsgitem{2022a}{}  On Cartesian products of signed graphs.  \emph{Discrete Appl.\ Math.}\ 319 (2022), 533--555.
\MR{4457822}.  \Zbl{1494.05051}.  
\annot{} Algorithm for Cartesian product decomposition.  Homomorphic chromatic number defined via sign-preserving homomorphisms.
\dated{1 May 2021, 8 Mar 2023} 
\codes{\CSG:~\CAlgor,~\CHom}%
\itemspace

\author{Aparna Lakshmanan S}						%LAKSHMANAN
\authorspace
\authref  See \hyperlink{Jeepamol J.\ Palathingal}{J.J.~Palathingal}.
\itemspace

\author{R.\ Lakshmi}							%LAKSHMI
\authorspace
\authref  See \hyperlink{M.\ Venkata Anusha}{M.V.~Anusha}.
\itemspace

\author{P.\ Lallemand}						%LALLEMAND
\authorspace
\authref  See \hyperlink{Hung T.\ Diep}{H.T.~Diep}.
\itemspace

\author{[S.\ Ben Lamine]}						%LAMINE
\authorspace
\authref  See \hyperlink{Samia Ben Lamine}{S.~Ben~Lamine} (under ``B'').
\itemspace

\author{Yanhua Lan}							%LAN YH
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\author{Kelvin Lancaster} 						%LANCASTER*
\authorspace
\bsgitem{1981a}{}  Maybee's ``Sign solvability''.  
In:  Harvey J.\ Greenberg and John S.\ Maybee, eds., \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ Sympos., Boulder, Col., 1980), pp.\ 259--270.  
Academic Press, New York, 1981.  
\MR{0617930} (82g:00016) (book).  \Zbl{495.93001} (book).
\annot{} Comment on \xref{Maybee}{1981a}{John S.\ Maybee}.
\codes{\CQM:~\CSol:~\CSD}%
\itemspace

\authorx{J.W.\ Landry}						%LANDRY
\author{J.W.\ Landry \& S.N.\ Coppersmith}
\authorspace
\bsgitem{2002a}{}  Ground states of two-dimensional $\pm J$ Edwards-Anderson spin glasses.  \emph{Phys.\ Rev.\ B} 65 (2002), art.\ 134404, 15 pp.
\arXiv{cond-mat/0109136}.  
\annot{} Finds all ground states of small signed square lattice graphs, and their distribution, to investigate how physical properties vary with $x := \#E^+/\#E$.  
Dictionary:  ``ground state'' = switching with fewest negative edges.
\dated{10 Jan 2015} 
\codes{\CSG:~\CState(\Cfr):~\CAlgor,~\CPhys}%
\itemspace

\bsgitem{2004a}{}  Quantum properties of a strongly interacting frustrated disordered magnet.  \emph{Phys.\ Rev.\ B} 69 (2004), art.\ 184416, 6 pp.
\annot{} Similar to \xrefd{2002a}{J.W.\ Landry \& S.N.\ Coppersmith} with a ``quantum term'' added.  The effect is that of an extra vertex $v_0$ added to $\Sigma$, positively adjacent to all of $V$ with an arbitrary strength.  Quantum ground and low-energy states are linear combinations of ground states of $\Sigma$ in a single component of the ground=state graph.
Dictionary:  ``low-energy'' = relatively few negative edges, ``ground state graph'' has ground states $\zeta$ for vertices and an edge between ground states that differ by switching a vertex (necessarily having $d^+=d^-$).
\dated{10 Jan 2015} 
\codes{\CSG:~\CState(\Cfr):~\CAlgor,~\CPhys}%
\itemspace

\authorx{Carsten Lange}						%LANGE
\author{Carsten Lange, Shiping Liu, Norbert Peyerimhoff, \& Olaf Post}
\authorspace
\bsgitem{2015a}{}  Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians.  \emph{Calc.\ Variations Partial Diff.\ Eqns.}\ 54 (2015), no.\ 4, 4165--4196.
\MR{3426108}.  \Zbl{1330.05103}.  \arXiv{1502.06299}.				
\codes{\CPhys:~\Cgg,~\CBal}%
\itemspace

\author{Jan-Hendrik Lange}						%LANGE
\authorspace
\bsgitem{20xxa}{}  Flow-partitionable signed graphs.  Submitted.  \emph{}\ 
\arXiv{2005.01536}.  
\codes{\CSG:~\CFlows}%
\itemspace

\author{Steven Landy}						%LANDY
\authorspace
\bsgitem{1988a}{}  A generalization of Ceva's theorem to higher dimensions.  \emph{Amer.\ Math.\ Monthly} 95 (Dec., 1988), no.\ 10, 936--939.
\MR{0979140} (90c:51020).  \Zbl{663.51011}.					
\annot{} The theorem characterizes concurrence of lines drawn from each vertex of a rectilinear simplex to a point in the opposite side.  [\Problem.  Reformulate, maybe generalize, in terms of gain graphs.  \Cf\ \xref{Boldescu}{1970a}{Petre Boldescu}, \xref{Zaslavsky}{2003b}{Thomas Zaslavsky} \S2.6.]
\codes{\Cgg:~\CGeom}%
\itemspace

\authorx{Andrea S.\ LaPaugh}						%LAPAUGH
\author{Andrea S.\ LaPaugh \& Christos H.\ Papadimitriou}
\authorspace
\bsgitem{1984a}{}  The even-path problem for graphs and digraphs.  \emph{Networks} 14 (1984), 507--513.  
\MR{0767373} (86g:05057).  \Zbl{552.68059}.
\annot{} Fast algorithms for existence of even paths between two given vertices (or any two vertices) of a graph.  The corresponding digraph problem is NP-complete.  [Signed (di)graphs are similar, due to the standard reduction by negative subdivision.]  [Also see, e.g., works by \hyperlink{Carsten Thomassen}{Thomassen}.]
\par
\codes{\CPar:~\CPaths:~\CAlgor}\codesctd{\Csd:~\CPar:~\CPaths:~\CAlgor}%
\itemspace

\author{Michel Las Vergnas}						%LAS VERGNAS
\authorspace
\authref  See \hyperlink{Anders Bjorner}{A.~Bj\"orner}.
\itemspace

\authorx{Martin L\"atsch}						%LATSCH
\authorx{Martin Latsch}
\author{Martin L\"atsch \& Britta Peis}
\authorx{Martin Latsch \& Britta Peis}
\authorspace
\bsgitem{2008a}{}  On a relation between the domination number and a strongly connected bidirection of an undirected graph.  \emph{Discrete Appl.\ Math.}\ 156 (2008), 3194--3202.
\MR{2468789} (2010a:05139).  \Zbl{1176.05058}.					
\annot{} A bidirected graph $(\Gamma,\tau)$ (where $\tau$ assigns $+$ or $-$ to each incidence) is ``strongly connected'' if there is a coherent walk from any vertex to any other vertex.  The distance $\dist_{(\Gamma,\tau)}(u,v) :=$ the minimum length of a coherent $\vec{uv}$ walk.  The diameter $\diam (\Gamma,\tau) := \max_{(u,v) \in V^2} \dist_{(\Gamma,\tau)}(u,v)$.  In $\Gamma$ define $i :=$ number of isthmi, $\gamma :=$ domination number.
Thm.\ 5:  $\Gamma$ has a strongly connected bidirection iff $\#V=1$ or $\Gamma$ is connected and minimum degree $\geq 2$.  
Thm.\ 10:  If $\Gamma$ has strongly connected bidirections $\tau_j$ ($j=1,\ldots,k$), then $\min_i \diam(\Gamma,\tau_j) \leq 2i + 2\min(i,1) + 5\gamma - 1.$  When $i=0$, $\tau_j$ can be chosen so $\Sigma(\Gamma,\tau_j)$ is all positive.  \Conjecture.  Also true when $i>0$.
Thm.\ 11:  If $\Gamma$ has a strongly connected bidirection, then $\min_j \diam(\Gamma,\tau_j) \leq 6\gamma + 3.$  By Fig.\ 8 this bound must be at least $6\gamma + 1$ if isthmi are allowed.
The proofs are constructive, esp.\ by extending to $\Gamma$ a bidirection of a dominating subgraph.
Dictionary:  ``path'' = walk [not trail].
\dated{27 Apr 2007}
\codes{\Csg:~\COri:~\CInvar,~\Cdom}%
\itemspace

\author{Reinhard Laubenbacher}					%LAUBENBACHER
\authorspace
\authref  See \hyperlink{Eduardo D.\ Sontag}{E.~Sontag} and \hyperlink{Alan Veliz-Cuba}{A.~Veliz-Cuba}.
\itemspace

\author{Monique Laurent}						%LAURENT 
\authorspace 
\authref  See \hyperlink{Michel Marie Deza}{M.M.~Deza} and \hyperlink{A.M.H.\ Gerards \& M.\ Laurent}{A.M.H.~Gerards}.
\itemspace

\author{Eugene L.\ Lawler} 						%LAWLER
\authorspace
\bsgitem{1976a}{} \emph{Combinatorial Optimization:  Networks and Matroids}.  Holt, Rinehart and Winston, New York, 1976.  
\MR{0439106} (55 \#12005).  \Zbl{413.90040}.
Repr.:  Dover Publications, Mineola, N.Y., 2001.
\Zbl{1058.90057}.  
\annot{} Ch.\ 6:  ``Nonbipartite matching.'' \S3:  Bidirected flows. 
\codes{\Csg:~\COri}%
\annotctd{} Ch.\ 4:  ``Network flows.'' \S8:  ``Networks with losses and gains.''  \S12:  ``Integrality of flows and the unimodular property.''
\par
\codes{\CGN}\codesctd{\Csg:~\CInc,~\CBal}%
\itemspace

\author{Ben Lawson}							%LAWSON
\authorspace
\authref  See \hyperlink{Charalampos E.\ Tsourakakis}{C.E.~Tsourakakis}.
\itemspace

\author{Bac Hoai Le}							%LE BH
\authorspace
\authref  See \hyperlink{Tuyen-Thanh-Thi Ho}{T.T.T.~Ho}.
\itemspace

\author{Jason Leasure}						%LEASURE
\authorspace
\authref   See \hyperlink{Lori Fern}{L.~Fern}.
\itemspace

\author{Walter Lebrecht}						%LEBRECHT
\authorx{W.\ Lebrecht}
\authorspace
\authref  See also \hyperlink{J.F.\ Valdes}{J.F.~Vald\'es} and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\authrefspace

\author{W.\ Lebrecht \& J.F.\ Vald\'es}
\authorspace
\bsgitem{2013a}{}  $\pm J$ Ising model on mixed Archimedean lattices: $(3^3,4^2)$, $(3^2,4,3,4)$, $(3,12^2)$, $(4,6,12)$.  \emph{Physica A}\ 392 (2013), no.\ 19, 4549--4570.
\MR{3083111}.  \Zbl{1395.82049}.  
\codes{\CPhys:~\Csg:~\CFr}%
\itemspace

\author{W.\ Lebrecht, J.F.\ Vald\'es, \& E.E.\ Vogel}
\authorx{W.\ Lebrecht, J.F.\ Valdes, \& E.E.\ Vogel}
\authorspace
\bsgitem{2003a}{}  Frustration in mixed two-dimensional $\pm J$ Ising lattices.  \emph{Physica A} 323 (2003), 466--486.
\Zbl{1050.82009}.  
\annot{} Randomly signed Kagom\'e and five-point-star planar lattices with specified concentration $x$ of positive edges: frustration index (``frustration length'') \etal, with combinatorial and numerical results compared.  Also, compared with results for homogeneous lattices like square and triangular to analyze effects of degree (``coordination number''), plaquette shape (degree of polygonal faces), \etal
	
\dated{3 Jan 2015} 
\par
\codes{\CSG,~\CPhys:~\CFr}%
\itemspace

\bsgitem{2008a}{}  Local analysis of frustration based on Kagom\'e lattices.  \emph{Physica A} 387 (2008), 5147--5158.
\annot{} Ground state energy $l(\Sigma)$, \etal, as functions of $x := \#E^+/\#E$.  Analytical, probabilistic, and computational results are largely consistent.
	
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CFr,~\CSw}%
\itemspace

\author{W.\ Lebrecht \& E.E.\ Vogel}
\authorspace
\bsgitem{1996a}{}  Order parameters and percolation for ground-state of honeycomb lattices.  In:  F.\ Leccabue and V.\ Sagredo, eds., \emph{Magnetism, Magnetic Materials and Their Applications} (Proc., M\'erida, Venezuela, 1995), pp.\ 304--309.  World Scientific, Singapore, 1996.
\codes{\CSG:~\CFr:~\CState,~\CPhys}%
\itemspace

\author{W.\ Lebrecht, E.E.\ Vogel, J.\ Cartes, \& J.F.\ Vald\'es}
\authorx{W.\ Lebrecht, E.E.\ Vogel, J.\ Cartes, \& J.F.\ Valdes}
\authorspace
\bsgitem{2004a}{}  Plaquette distributions for $\pm J$ Ising lattices.  \emph{Physica A} 342 (2004), 90--96.
\annot{} In given $\Gamma$, $x := \#E^+/\#E$ implies an expected number of frustrated (negative) plaquettes.  $\Gamma$ is triangular, square, hexagonal, Kagom\'e, etc., with periodic boundary conditions (i.e., toroidal) or is the graph of a regular or semiregular polyhedron.  
	
Dictionary:  \cf\ \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}.
\dated{2 Jan 2015} 
\par
\codes{\CSG:~\CState,~\CPhys}%
\itemspace

\author{W.\ Lebrecht, E.E.\ Vogel, \& J.F.\ Vald\'es}
\authorx{W.\ Lebrecht, E.E.\ Vogel, \& J.F.\ Valdes}
\authorspace
\bsgitem{2002a}{}  Ising model on mixed two-dimensional lattices.  \emph{Physica B} 320 (2002), 343--347.
\annot{} Probabilistic and computational analysis of average states on signed toroidal Kagom\'e and five-point-star lattices.  Frustration, energy, \etal\ as functions of $x := \#E^+/\#E$.  Comparison to honeycomb, square, and triangular lattices (\cf\ other papers of the authors).
	
Dictionary:  \cf\ \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}.
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CState,~\CFr}%
\itemspace

\bsgitem{2004a}{}  Frustration in Archimedean $\pm J$ lattices.  \emph{J.\ Alloys Compounds} 369 (2004), 66--69.
\annot{} Toroidal (``periodic boundary conditions'') lattice $(3,4,6,4)$ (Gr\"unbaum--Shephard classification) with random signs having proportion $x$ of positive edges.  Distribution of frustrated (negative) plaquettes, proportion of satisfied edges, \etal, in ground states.  Comparison to other lattices (\cf\ other papers of the authors).
	
\dated{3 Jan 2015}
\par 
\codes{\CSG,~\CPhys:~\CState,~\CFr}%
\itemspace

\author{Bruno Leclerc} 						%LECLERC B
\authorspace
\bsgitem{1981a}{}  Description combinatoire des ultram\'etriques.  \emph{Math.\ Sci.\ Humaines} No.\ 73 (1981), 5--37.  
\MR{0623034} (82m:05083).  \Zbl{476.05079}.
\codes{\CSG:~\CBal}%
\itemspace

\authorx{Etienne Leclerc}						%LECLERC E
\author{Etienne Leclerc, Gary MacGillivray, \& Jacqueline M.\ Warren}
\authorspace
\bsgitem{20xxa}{}  Switching $(m, n)$-mixed graphs with respect to Abelian groups.  Submitted.  \emph{}\ 
\arXiv{2110.01576}.  
\codes{\Cgg:~\CSw}%
\itemspace

\authorx{J.\ Leclercq}						%LECLERCQ
\authorspace
\author{J.\ Leclercq \& R.\ Thomas}
\authorspace
\bsgitem{1981a}{}  Analyses bool\'eenne et continue de syst\`emes comportant des boucles de r\'etro\-action.  II.\  Syst\`eme \'a deux attracteurs form\'e d'une boucle positive et d'une boucle n\'egative conjugu\'ees.  (In French.)  \emph{Acad.\ Roy.\ Belg.\ Bull.\ Cl.\ Sci.}\ (5) 67 (1981), no.\ 3, 190--225 (1 foldout). 
\MR{0652794} (84h:92020b).  \Zbl{493.92004}.					
\annot{} Two signed digraph cycles of opposite signs, joined at a vertex, give a multiplicity of dynamics depending on the initial state and the delays, such as one or more stable states, an unstable cycle of states and related stable state cycles, etc.  
Part I: \xref{Richelle}{1980a}{Jean Richelle}.
\dated{10 Jul 2022} 
\codes{\Csd:~\CDyn:~\CBiol}%
\itemspace

\author{Euiwoong Lee}						%LEE EW
\authorspace
\authref  See \hyperlink{Vaggos Chatziafratis}{V.~Chatziafratis}.
\itemspace

\authorx{Gibaek Lee}						%LEE GB
\author{Gibaek Lee, Sang-Oak Song, \& En Sup Yoon}
\authorspace
\bsgitem{2003a}{}  Multiple-fault diagnosis based on system decomposition and dynamic PLS.  \emph{Indust.\ Engin.\ Chem.\ Res.}\ 42 (2003), 6145--6154.
\annot{} Combines signed digraphs and partial least squares for fault analysis in chemical engineering.
\codes{\CSD:~\CAppl}%
\itemspace

\author{Jaeun Lee} 							%LEE JU
\authorspace
\authref  See \hyperlink{Ian P.\ Goulden}{I.P.~Goulden}, \hyperlink{Dongseok Kim}{D.~Kim}, and \hyperlink{Jin Ho Kwak}{J.H.~Kwak}.
\itemspace

\author{Jon Lee} 							%LEE J
\authorspace
\bsgitem{1989a}{}  Subspaces with well-scaled frames.  \emph{Linear Algebra Appl.}\ 114/115 (1989), 21--56.  
\MR{0986864} (90k:90111).  \Zbl{675.90061}.						
\annot{} See \S9.
\codes{\Csg:~\COri:~\CInc,~\CFlows,~\CAlgor}%
\itemspace

\author{Jon Lee \& Matt Scobee}
\authorspace
\bsgitem{1999a}{}  A characterization of the orientations of ternary matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 77 (1999), no.\ 2, 263--291.
\MR{1719344} (2000k:05073).  \Zbl{1024.05016}.  
\annot{} The results imply that a ternary matroid, such as the frame matroid of a signed graph, has at most three orientation classes.  [Thanks to Stefan van Zwam.]
\dated{2 Apr 2013} 
\codes{\Csg:~\Cmatrd}%
\itemspace

\author{Orlando Lee}							%LEE O
\authorspace
\authref  See \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi}.
\itemspace

\authorx{Sang-Gu Lee}						%LEE SG
\author{Sang-Gu Lee \& Jin-Woo Park}
\authorspace
\bsgitem{2015a}{}  Sign idempotent sign pattern matrices that allow idempotence.  \emph{Linear Algebra Appl.}\ 487 (2015), 232--241.
\MR{3405558}.  \Zbl{1328.15042}.  
\annot{} See \S4, ``Nonnegativity of sign idempotent sign pattern matrices''.
\codes{\CQM:~\CSD,~\CSG:~\Csw,~\CBal}%
\itemspace

\author{Sang-Gu Lee, Se-Won Park, \& Han-Guk Seol}
\authorspace
\bsgitem{1998a}{}  Linear operators strongly preserving matrices whose sign patterns require the Perron property..  \emph{Linear Algebra Appl.}\ 269 (1998), 53--63.
\MR{1483520} (98i:\-15006).  \Zbl{887.15014}.					
\annot{} \Cf\ \xref{Eschenbach and Johnson}{1990a}{Carolyn A.\ Eschenbach \& Charles R.\ Johnson}.  
\dated{20 Oct 2020} 
\par
\codes{\CQM,~\Csd:~\Cbal(\CCycles),~\Csw}%
\itemspace

\author{Shyi-Long Lee}						%LEE SL
\authorspace
\authref   See also \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Prabhat K.\ Sahu}{P.K.~Sahu}.
\authrefspace

\bsgitem{1989a}{}  Comment on `Topological analysis of the eigenvalues of the adjacency matrices in graph theory: A difficulty with the concept of internal connectivity'.  \emph{J.\ Chinese Chem.\ Soc.}\ 36 (1989), no.\ 1, 63--65.  
\annot{} Response to \xref{Gutman}{1988a}{Ivan Gutman}.  Proposes weighted net sign: divide by number of nonzero vertex signs.  The goal is to have the ordering of net signs correlate more closely with that of eigenvalues.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\bsgitem{1989b}{}  Net sign analysis of eigenvectors and eigenvalues of the adjacency matrices in graph theory.  \emph{Bull.\ Inst.\ Chem., Academica Sinica} No.\ 36 (1989), 93--104.  
\annot{} Expounds principally \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu} and \xref{Lee and Gutman}{1989a}{Shyi-Long Lee \& Ivan Gutman}.  Examples include all connected, simple graphs of order $\leq 4$ and some aromatics.
\codes{\CVS,~\CSGw,~\CChem:~\CExp}%
\itemspace

\bsgitem{1992a}{}  Topological analysis of five-vertex clusters of group IVa elements.  \emph{Theoretica Chimica Acta} 81 (1992), 185--199.  
\annot{} See \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}.  More examples; again, eigenvalue and net-sign orderings are compared.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\author{Shyi-Long Lee \& Ivan Gutman}
\authorspace
\bsgitem{1989a}{}  Topological analysis of the eigenvectors of the adjacency matrices in graph theory:  Degenerate case.  \emph{Chem.\ Phys.\ Lett.}\ 157 (1989), 229--232.  
\annot{} Supplements \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu} to answer an objection by \xref{Gutman}{1988a}{Ivan Gutman}, by treating vertex signs corresponding to multidimensional eigenspaces.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\author{Shyi-Long Lee \& Chiuping Li}	
\authorspace
\bsgitem{1994a}{}  Chemical signed graph theory.  \emph{Int.\ J.\ Quantum Chem.}\ 49 (1994), 639--648.  
\annot{} Varies \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu} by taking net signs of all balanced signings, instead of only those obtained from eigenvectors, for small paths, circles, and circles with short tails.  The distribution of net sign, over all signings of each graph, is more or less binomial. 
\par
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\bsgitem{1994b}{}  On generating molecular orbital graphs:  the first step in signed graph theory.  \emph{Bull.\ Inst.\ Chem., Academica Sinica} No.\ 41 (1994), 69--75.
\annot{} Abbreviated presentation of \xrefd{1994a}{Shyi-Long Lee \& Chiuping Li}.  
\codes{\CVS,~\CSGw:~\CExp}%
\itemspace

\author{Shyi-Long Lee \& Feng-Yin Li}
\authorspace
\bsgitem{1990a}{}  Net sign approach in graph spectral theory.  \emph{J.\ Molecular Structure (Theochem)} 207 (1990), 301--317.  
\annot{} Similar topics to \xref{S.L.\ Lee}{1989a}{Shyi-Long Lee}, \xrefd{1989b}{Shyi-Long Lee}.  Several examples of order $6$.
\codes{\CVS,~\CSGw,~\CExp,~\CChem}%
\itemspace

\bsgitem{1990b}{}  Net sign analysis of five-vertex chemical graphs.  \emph{Bull.\ Inst.\ Chem., Academica Sinica} No.\ 37 (1990), 83--97.  
\annot{} See \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}.  Treats all connected, simple graphs of order $5$.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\author{Shyi-Long Lee, Feng-Yin Li, \& Friday Lin}
\authorspace
\bsgitem{1991a}{}  Topological analysis of eigenvalues of particle in one- and two-dimensional simple quantal systems:  Net sign approach.  \emph{Int.\ J.\ Quantum Chem.}\ 39 (1991), 59--70.  
\annot{} See \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}.  \S~II:  Net signs calculated for paths.  \S\S~III, IV:  Planar graphs with two different types of potential, yielding complicated results.
\codes{\CVS,~\CSG,~\CChem}%
\itemspace

\author{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}
\authorspace
\bsgitem{1987a}{}  Topological analysis of eigenvectors of the adjacency matrices in graph theory:  The concept of internal connectivity.  \emph{Chem.\ Phys.\ Lett.}\ 137 (1987), 279--284.  
\MR{0910752} (88i:05130).  
\annot{} Introduces the net sign of a (balanced) signed graph.  A graph has vertices signed according to the signs of an eigenvector $X_i$ of the adjacency matrix, $\mu(v_r) = \sgn(X_{ir})$, and $\sigma(v_rv_s) = \mu(v_r)\mu(v_s)$ [hence $\Sigma$ is balanced].  Note that a vertex can have `sign' $0$.  Net sign of a [hydrocarbon] chemical graph is applied to prediction of properties of molecular orbitals.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\author{Shyi-Long Lee, Yeung-Long Luo, \& Yeong-Nan Yeh}
\authorspace
\bsgitem{1991a}{}  Topological analysis of some special graphs.\ III.\ Regular polyhedra.  \emph{J.\ Cluster Sci.}\ 2 (1991), 105--116.  
\annot{} See \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}.  Net signs for the Platonic polyhedra (Table I).
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\author{Shyi-Long Lee \& Yeong-Nan Yeh}
\authorspace
\bsgitem{1990a}{}  Topological analysis of some special classes of graphs.\  Hypercubes.  \emph{Chem.\ Phys.\ Lett.}\ 171 (1990), 385--388.  
\annot{} Follows up \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu} and \xref{Lee and Gutman}{1989a}{Shyi-Long Lee \& Ivan Gutman}, calculating net signs of eigenspatially signed hypercube graphs of dimensions up to 6 by means of a general graph-product formula.
\par
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\bsgitem{1993a}{}  Topological analysis of some special classes of graphs.\ II.\ Steps, ladders, cylinders.  \emph{J.\ Math.\ Chem.}\ 14 (1993), 231--241.  
\MR{1262027} (95f:05079).  
\annot{} See \xref{Lee, Lucchese, and Chu}{1987a}{Shyi-Long Lee, Robert R.\ Lucchese, \& San Yan Chu}.  Net signs and eigenvalues are compared.
\codes{\CVS,~\CSGw,~\CChem}%
\itemspace

\authorx{G\'eraud Le Falher}						%LE FALHER
\authorx{Geraud Le Falher}
\author{G\'eraud Le Falher \& Fabio Vitale}
\authorx{Geraud Le Falher \& Fabio Vitale}
\authorspace
\bsgitem{2016a}{}  Even trolls are useful:  Efficient link classification in signed networks.  Manuscript, 2016.
\arXiv{1602.08986}.  
\codes{\CSD:~\CPsS:~\CAlgor}%
\itemspace

\author{Hanno Lefmann}						%LEFMANN
\authorspace
\bsgitem{1990a}{}  On families in finite lattices.  \emph{European J.\ Combin.}\ 11 (1990),165--179.
\MR{1044456} (91i:06009).  \Zbl{734.06007}.  
\annot{} Thm.\ 1.2 bounds the size of a family of lattice elements with prescribed meet ranks.  Dowling lattices are an example of this and related results.
\dated{9 Apr 2016} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Jen\H{o} Lehel}						%LEHEL
\authorx{Jeno Lehel}
\authorspace
\authref  See \hyperlink{Ralph J.\ Faudree}{R.J.~Faudree}.
\itemspace

\author{Ziyi Lei}							%LEI ZY
\authorspace
\authref  See \hyperlink{Zhiyun Cheng}{Z.-Y.~Cheng}.
\itemspace

\author{Frank Thomson Leighton}					%LEIGHTON
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{Samuel Leinhardt} 						%LEINHARDT
\authorspace
\authref  See also \hyperlink{James A.\ Davis}{J.A.~Davis} and \hyperlink{Paul W.\ Holland}{P.W.~Holland}.
\authrefspace

\author{Samuel Leinhardt, ed.}
\authorspace
\bsgitem{1977a}{}  \emph{Social Networks:  A Developing Paradigm}.  Academic Press, New York, 1977.
\annot{} An anthology reprinting some basic papers in structural balance theory, including some elementary signed-graph theory.
\codes{\CPsS,~\CSG:~\CBal,~\CClu}%
\itemspace

\authorx{P.W.H.\ Lemmens}		 				%LEMMENS
\author{P.W.H.\ Lemmens \& J.J.\ Seidel}
\authorspace
\bsgitem{1973a}{} Equiangular lines.  \emph{J.\ Algebra} 24 (1973), 494--512.
\MR{0307969} (46 \#7084).  \Zbl{255.50005}.					
Repr.\ in \xref{Seidel}{1991a}{J.J.\ Seidel}, pp.\ 127--145.
\annot{} Hints of graph switching; see \xref{van Lint and Seidel}{1966a}{J.H.\ van Lint \& J.J.\ Seidel}.
\codes{\CGeom,~\Csw}%
\itemspace

\authorx{Gloria de Leon-Calio}						%LEON-CALIO
\author{Gloria de Leon-Calio \& Yueh-Er Kuo}
\authorspace
\bsgitem{2003a}{}  Signed digraphs and their applications.  \emph{Soochow J.\ Math.}\ 29 (2003), no.\ 1, 69--81.
\MR{1975279} (2004c:05082).  \Zbl{1032.05062}.  
\annot{} Simple properties of pulse properties, as in \xref{Roberts}{1976a}{Fred S.\ Roberts}.  Seems largely expository.
\dated{5 Jun 2019} 
\codes{\CSDw:~\CExp}%
\itemspace

\authorx{Marianne Lepp} 						%LEPP
\author{Marianne Lepp [Marianne L.\ Gardner]}
\authorspace
\authref   See \hyperlink{Randy Shull}{R.~Shull}.
\itemspace

\author{J\"urgen Lerner}						%LERNER
\authorx{Jurgen Lerner}
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\authorx{Jure Leskovec}						%LESKOVEC
\author{Jure Leskovec, Daniel Huttenlocher, \& Jon Kleinberg}
\authorspace
\bsgitem{2010a}{}  Signed networks in social media.  
In:  \emph{CHI '10: Proceedings of the 28th ACM Conference on Human Factors in Computing Systems} (Atlanta, 2010), pp.\ 1361--1370.  Assoc.\ for Computing Machinery, New York, 2010.
\arXiv{1003.2424}.  
\par
\codes{\CSD,~\CSG:~\CBal,~\CClu}%
\itemspace

\bsgitem{2010b}{}  Predicting positive and negative links in online social networks.  
In:  \emph{WWW '10: Proceedings of the 19th International Conference on World Wide Web} (Raleigh, N.C., 2010).  Assoc.\ for Computing Machinery, New York, 2010.
\arXiv{1003.2429}.  
\codes{\CSD:~\CBal}%
\itemspace

\author{Adam N.\ Letchford}						%LETCHFORD
\authorspace
\authref  See \hyperlink{Konstantinos Kaparis}{K.~Kaparis}.
\itemspace

\authorx{Emily Leven}						%LEVEN
\author{Emily Leven, Brendon Rhoades, \& Andrew Timothy Wilson}
\authorspace
\bsgitem{2014a}{}  Bijections for the Shi and Ish arrangements.  \emph{ European J.\ Combin.}\ 39 (2014), 1--23.
\MR{3168512}.  \Zbl{1284.05331}.  \arXiv{1307.6523}.				
\codes{\Cgg:~\CGeom}%
\itemspace

\author{Richard Levins}						%LEVINS
\authorspace
\authref  See also \hyperlink{Jeffrey M.\ Dambacher}{J.M.~Dambacher} and \hyperlink{Charles J.\ Puccia}{C.J.~Puccia}.
\authrefspace

\bsgitem{1974a}{}  The qualitative analysis of partially specified systems.  \emph{Ann.\ N.Y.\ Acad.\ Sci.}\ 231 (1974), 123--138.
\Zbl{285.93028}.  
\codes{\CSD:~\CQM:~\CSta:~\CCycles}
\itemspace

\bsgitem{1975a}{}  Evolution in communities near equilibrium. 
In: M.\ Cody and J.M.\ Diamond, eds., \emph{Ecology and Evolution of Communities}, pp.\ 16--50.  Harvard Univ.\ Press, Cambridge, Mass., 1975. 
\codes{\CSD:~\CQM:~\CSta:~\CCycles}
\itemspace

\author{Vadim E.\ Levit}						%LEVIT
\authorspace
\authref  See \hyperlink{Yonah Cherniavsky}{Y.~Cherniavsky}.
\itemspace

\authorx{Mario Levorato}						%LEVORATO
\author{Mario Levorato, Rosa Figueiredo, Yuri Frota, \& L\'ucia Drummond}
\authorx{Mario Levorato, Rosa Figueiredo, Yuri Frota, \& Lucia Drummond}
\authorspace
\bsgitem{2017a}{}   Evaluating balancing on social networks through the efficient solution of correlation clustering problems.  \emph{EURO J.\ Computational Optim.}\ 5 (2017), no.\ 4, 467--498.
\MR{3736647}.  \Zbl{1386.90161}.  \HAL{hal-02178542}.  
Erratum.  \emph{ibid.}\ 5 (2017), no.\ 4, 499.
\MR{3736648} (no rev).  
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Thierry L\'evy}						%LEVY
\authorspace
\authref  See \hyperlink{Adrien Kassel}{A.~Kassel}.
\itemspace

\author{Mordechai Lewin}						%LEWIN M
\authorspace
\bsgitem{1977a}{}  On the extreme points of the polytope of symmetric matrices with given row sums.  \emph{J.\ Combin.\ Theory Ser.\ A} 23 (1977), no.\ 2, 223--231.
\MR{0444495} (56 \#2846).  \Zbl{362.05040}.						
\annot{} An equivalent of Thm.\ 8.2.1 in \xref{Brualdi}{2006a}{Richard A.\ Brualdi}.
\dated{13 Oct 2012} 
\par
\codes{\Csg:~\Cpar:~\CAdj}%
\itemspace

\author{Frank L.\ Lewis}						%LEWIS FL
\authorspace
\authref  See \hyperlink{Zhenyu Sun}{Z.-Y.~Sun}.
\itemspace

\authorx{Torina Lewis}						%LEWIS T
\author{Torina Lewis, Jenny McNulty, Nancy Ann Neudauer, Talmage James Reid \& Laura Sheppardson}
\authorspace
\bsgitem{2013a}{}  Bicircular matroid designs.  \emph{Ars Combin.}\ 110 (2013), 513--523.
\MR{3100270}.  
\annot{} The connected bicircular matroids in which all circuits have the same size, i.e., which are duals of matroid designs, are certain uniform subdivisions of uniform matroids.
\dated{9 Jun 2013} 
\codes{\CBic}%
\itemspace

\author{David W.\ Lewit}						%LEWIT
\authorspace
\authref   See \hyperlink{Elizabeth G.\ Shrader}{E.G.~Shrader}.
\itemspace

\author{Josef Leydold}						%LEYDOLD
\authorspace
\authref  See \hyperlink{Turker Biyikoglu}{T.~B{\i}y{\i}ko{\u g}lu}.
\itemspace

\author{Claire Lhuillier}						%LHUILLIER
\authorspace
\authref  See \hyperlink{Gregoire Misguich}{G.~Misguich}.
\itemspace

\author{Bao Feng Li}							%LI BF
\authorspace
\authref  See \hyperlink{Xiao Hui Hao}{X.H.~Hao}.
\itemspace


\author{Bo Li}							%LI B
\authorspace
\authref  See \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\authorx{Cai Heng Li}						%LI CH
\author{Cai Heng Li \& Jozef \v{S}ir\'a\v{n}}
\authorx{Cai Heng Li \& Jozef Siran}
\authorspace
\bsgitem{2007a}{}  M\"obius regular maps.  \emph{J.\ Combin.\ Theory Ser.\ B} 97 (2007), no.\ 1, 57--73.
\MR{2278124} (2007h:05043).  \Zbl{1106.05033}.						
\annot{} That is, signed expansion graphs $\pm\Gamma$, orientation embedded in a surface (M\"obius), whose map automorphisms act transitively on flags (regularity).  Properties of their automorphism groups.  [Follows \xref{Wilson}{1989a}{Steve Wilson}.]
\dated{rev.\ 31 Jul 2014}
\codes{\CSG:~\CTop:~\CAut}%
\itemspace

\author{Chang Li}							%LI C
\authorspace
\authref  See \hyperlink{Tero Harju}{T.~Harju}.
\itemspace

\author{Chiuping Li}							%LI CP
\authorspace
\authref   See \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\author{Chuandong Li}						%LI CD
\authorspace
\authref  See \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\authorx{Dan Li}							%LI D
\author{Dan Li, Huiqiu Lin, \& Jixiang Meng}
\authorspace
\bsgitem{2023a}{}  Extremal spectral results related to spanning trees of signed complete graphs.  \emph{Discrete Math.}\ 346 (2023), no.\ 2, art.\ 113250, 17 pp.
\MR{4510055}.  
\arXiv{2201.06729}.  
\annot{} First, max eigenvalue of $A(K_n,-T)$, $T=$ spanning tree.  Second, upper bounds on min eigenvalue of distance matrix of incomplete $\Sigma$.
\dated{23 Jan 2022} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}\codesctd{\CSG:~\CAdj(\CGen):~\CEig}%
\itemspace

\author{Deqiong Li}							%LI DQ
\authorspace
\authref  See \hyperlink{Qiao Guo}{Q.~Guo} and \hyperlink{Dijian Wang}{D.-J.~Wang}.
\itemspace

\authorx{Dong Li}							%LI D
\author{Dong Li, Cuihua Wang, Shengping Zhang, Guanglu Zhou, Dianhui Chu, \& Chong Wu}
\authorspace
\bsgitem{2017a}{}  Positive influence maximization in signed social networks based on simulated annealing.  \emph{Neurocomputing} 260 (2017), 69--78.
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Dong Li, Zhi-Ming Xu, Nilanjan Chakraborty, Anika Gupta, Katia Sycara, \& Sheng Li}
\authorspace
\bsgitem{2014a}{}  Polarity related influence maximization in signed social networks.  \emph{PLoS ONE} 9 (2014), no.\ 7, art.\ 102199, 12 pp.
\codes{\CSG,~\CPsS}%
\itemspace

\author{Feng-Hin Li}							%LI FH
\authorspace
\authref   See \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\author{Guangbin Li}							%LI GB
\authorspace
\bsgitem{2013a}{}  The signless Laplacian spectral radius of $C_4$-free graphs with even order.  \emph{Basic Sci.\ J.\ Textile Univ.\ / Fangzhi Gaoxia} 26 (2013), no.\ 2, 171--175.
\Zbl{1299.05224}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Guojun Li}							%LI GJ
\author{Guojun Li \& Aimei Yu}
\authorspace
\bsgitem{2015a}{}  A characterization of bicyclic signed graphs with nullity $n - 7$.  \emph{J.\ Math.\ Res.\ Appl.}\ 35 (2015), no.\ 1, 1--10.
\MR{3328496}.  \Zbl{1340.05114}.  
\annot{} That is, adjacency rank 7.
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Hao Li}							%LI H
\authorspace
\authref  See \hyperlink{Wenjie Ning}{W.J.~Ning}.
\itemspace

\author{Hiram W.\ Li}							%LI HW
\authorspace
\authref  See \hyperlink{Jeffrey M.\ Dambacher}{J.M.~Dambacher}.
\itemspace

\author{Hong-Hai Li}							%LI HH
\authorspace
\authref  See also \hyperlink{Li Su}{L.~Su}.
\authrefspace

\author{Hong-Hai Li \& Jiong-Sheng Li}
\authorspace
\bsgitem{2008a}{}  An upper bound on the Laplacian spectral radius of the signed graphs.  \emph{Discuss.\ Math.\ Graph Theory}  28 (2008), no.\ 2, 345--359.
\MR{2477235} (2010a:05115).  \Zbl{1156.05035}.					
\annot{} 
Dictionary:  See \xref{X.D.~Zhang and Li}{2002a}{Xiao-Dong Zhang \& Jiong-Sheng Li}.
\dated{23 Mar 2009}
\par
\codes{\CSG:~\Cinc,~\CEig}%
\itemspace

\bsgitem{2009a}{}  Note on the normalized Laplacian eigenvalues of signed graphs.  \emph{Australasian J.\ Combin.}\ 44 (2009), 153--162.
\MR{2527006} (2010i:05210).  \Zbl{1177.05050}.					
\par
\codes{\CSG:~\CEig}%
\itemspace

\author{Hong-Hai Li, Bit-Shun Tam, \& Li Su}
\authorspace
\bsgitem{2013a}{}  On the signless Laplacian coefficients of unicyclic graphs.  \emph{Linear Algebra Appl.}\ 439 (2013), no.\ 7, 2008--2028.
\MR{3090451}.  
\annot{} Minimum and maximum magnitudes and associated graphs are found (for $n\geq5$).  
Thms.\ 4.1, 5.1, 5.2, 5.3 on transforms of $\Gamma$ have two cases depending on whether (connected) $\Gamma$ is bipartite.  [\Conjecture.  The results generalize to signed graphs with two (connected) cases: balanced or not.]
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ji Li}							%LI J
\authorspace
\authref  See \hyperlink{Hongzhong Deng}{H.Z.~Deng}.
\itemspace

\author{Jiaao Li}							%LI JA
\authorspace
\authref  See also \hyperlink{Matt DeVos}{M.~DeVos}.
\authrefspace

\author{Jiaao Li, Yongtang Shi, Zhouningxin Wang, and Chunyan Wei}
\authorspace
\bsgitem{20xxa}{}  Homomorphisms to small negative even cycles.  In preparation.  \emph{}\ 
\codes{\CSG:~\CHom,~\CCol}%
\itemspace

\author{Jianxi Li}							%LI JX
\authorspace
\authref  See also \hyperlink{Ji Ming Guo}{J.M.~Guo}.
\authrefspace

\author{Jianxi Li \& Ji-Ming Guo}
\authorspace
\bsgitem{2013a}{}  The signless Laplacian spectral radii of modified graphs.  \emph{Math.\ Commun.}\ 18 (2013), 67--73.
\MR{3085789}.  \Zbl{1276.05070}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jing Li}							%LI J
\authorspace
\authref  See \hyperlink{Shiying Wang}{S.Y.~Wang}.
\itemspace

\author{Ke Li}							%LI K
\authorspace
\authref  See also \hyperlink{Ligong Wang}{L.G.~Wang}.
\authrefspace

\author{Ke Li, Ligong Wang, \& Guopeng Zhao}
\authorspace
\bsgitem{2011a}{}  The signless Laplacian spectral radius of tricyclic graphs and trees with $k$ pendant vertices.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 4, 811--822. 
\MR{2807235} (2012f:05179).  \Zbl{1220.05075}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011b}{}  The signless Laplacian spectral radius of unicyclic and bicyclic graphs with a given girth.  \emph{Electronic J.\ Combin.}\ 18 (2011), no.\ 1, Paper 183, 10 pp.
\MR{2836818} (2012g:05138).  \Zbl{1230.05200}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jiong-Sheng Li}						%LI JS
\authorspace
\authref  See \hyperlink{Yaoping Hou}{Y.P.~Hou}, \hyperlink{Hong-Hai Li}{H.H.~Li}, and \hyperlink{Xiao-Dong Zhang}{X.D.~Zhang}.
\itemspace

\author{Lulu Li}							%LI LL
\authorspace
\authref  See \hyperlink{Guohong Mu}{G.-H.~Mu}.
\itemspace

\author{Min Li}							%LI M
\authorspace
\authref  See \hyperlink{Sai Ji}{S.~Ji}.
\itemspace

\author{Nan Li}							%LI N
\authorspace
\authref  See \hyperlink{Atsushi Funato}{A.~Funato} and \hyperlink{Takayuki Hibi}{T.~Hibi}.
\itemspace

\authorx{Qian Li}							%LI QA
\author{Qian Li \& Bolian Liu}	
\authorspace
\bsgitem{2008a}{}  Bounds on the $k$th multi-$g$ base index of nearly reducible sign pattern matrices.  \emph{Discrete Math.}\ 308 (2008), 4846--4860.
\MR{2446095} (2010a:05037).  \Zbl{1167.15013}.					
\codes{\CQM:~\CSD}%
\itemspace

\author{Qian Li, Bolian Liu, \& Jeffrey Stuart}
\authorspace
\bsgitem{2010a}{}  Bounds on the $k$-th generalized base of a primitive sign pattern matrix.  \emph{Linear Multilinear Algebra} 58 (2010), no.\ 3, 355--366.
\MR{2663436} (2011c:15090).  \Zbl{1196.15030}.					
\codes{\CSD:~\CQM}%
\itemspace

\author{Qingdu Li}							%LI QD
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Rao Li}							%LI R
\authorspace
\bsgitem{2010a}{}  Inequalities on vertex degrees, eigenvalues and (signless) Laplacian eigenvalues of graphs.  \emph{Int.\ Math.\ Forum} 5 (2010), no.\ 37-40, 1855--1860.
\MR{2672449} (no rev).  \Zbl{1219.05088}.						
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Ruilin Li}							%LI RL
\author{Ruilin Li \& Jinsong Shi}
\authorspace
\bsgitem{2010a}{}  The minimum signless Laplacian spectral radius of graphs with given independence number.  
\emph{Linear Algebra Appl.}\ 433 (2010), no.\ 8-10, 1614--1622.
\MR{2718223} (2011m:05181).  \Zbl{1211.05075}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Rui-lin Li, Jin-song Shi, \& Bing-can Dong}
\authorspace
\bsgitem{2011a}{}  Maximal signless Laplacian spectral radius of bicyclic graphs with given independence number.  (In Chinese?)  \emph{J.\ East China Norm.\ Univ.\ Natur.\ Sci.\ Ed.}\ 2011 (2011), no.\ 3, 73--84, 99.
\MR{2867304} (no rev).  \Zbl{1240.05194}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Selena Li}							%LI S
\authorspace
\authref  See \hyperlink{Lucas J.\ Rusnak}{L.J.~Rusnak}.
\itemspace

\author{Sheng Li}							%LI S
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Shuang-Dong Li}						%LI SD
\authorspace
\authref  See \hyperlink{Yi Wang}{Y.~Wang}.
\itemspace

\author{Shuchao Li}							%LI SC
\authorspace
\authref  See also \hyperlink{Beifang Chen}{B.~Chen}, \hyperlink{Chen Chen}{C.~Chen}, \hyperlink{Xianya Geng}{X.Y.~Geng}, \hyperlink{Shushan He}{S.S.~He}, \hyperlink{Jing Huang}{J.~Huang}, \hyperlink{Guangfu Wang}{G.-F.~Wang}, \hyperlink{Wei Wei}{W.~Wei}, and \hyperlink{Minjie Zhang}{M.J.~Zhang}.
\authrefspace

\author{Shuchao Li, Wanting Sun, \& Baogen Xu}
\authorspace
\bsgitem{20xxa}{}  Relation between the adjacency rank of a complex unit gain graph and some classical parameters of its underlying graph.  Submitted. 
\codes{\CGG:~\CAdj}%
\itemspace

\author{Shuchao Li, Wanting Sun, \& Yuantian Yu}
\authorspace
\bsgitem{2022a}{}  Adjacency eigenvalues of graphs without short odd cycles.  \emph{Discrete Math.}\ 345 (2022), no.\ 1, art.\ 112633, 13 pp.
\MR{4318317}.  \Zbl{1480.05086}.  \arXiv{2109.04599}.  
\annot{} Upper bound on $\lambda_1^2+\lambda_2^2$ for odd girth $> 2k+1$.  Spectral radius can imply odd girth $\leq 2k+1$.  Also, proves theorem of \xref{Lin and Guo}{2021a}{Huiqiu Lin \& Hangtian Guo}.  [\Problem.  Generalize to negative girth of unbalanced signed graphs.]
\dated{28 Jun 2022} 
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Shuchao Li \& Yi Tian}
\authorspace
\bsgitem{2011a}{}  On the (Laplacian) spectral radius of weighted trees with fixed matching number $q$ and a positive weight set.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 6, 1202--1212.
\MR{2807144} (2012f:05180).  \Zbl{1222.05165}.  
\annot{} Weights $w: E \to \bbR_{>0}$.  Since $\Spec \Lap(\Gamma,w) = \Spec \Lap(-\Gamma,w)$, $\Lap(-\Gamma,w)$ is used to find $\lambda_1(\Lap(\Gamma,w))$.
\dated{21 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  Some bounds on the largest eigenvalues of graphs.  \emph{Appl.\ Math.\ Lett.}\ 25 (2012), 326--332.
\MR{2855981} (2012h:05195).  \Zbl{1243.05152}.  
\annot{} Laplacian matrix $\Lap(-\Gamma)$.
\dated{29 Jul 2019} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2015a}{}  Some results on the bounds of signless Laplacian eigenvalues.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.} 38 (2015), no.\ 1, 131--141.
\MR{3394043}.  \Zbl{1308.05074}.  
\annot{} Upper and lower bounds on the sum of largest eigenvalues of $\Lap(-\Gamma)$ and $\Lap(-\Gamma^c)$ for a simple graph.  
[Also see \xref{Oliveira and de Lima}{2016a}{Carla Oliveira \& Leonardo de Lima}.]
Eigenvalue and eigenvector bounds from $\Lap(-\Gamma)$ on the clique and stability numbers.  
\dated{7 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shuchao Li \& Shujing Wang}
\authorspace
\bsgitem{2012a}{}  The least eigenvalue of the signless Laplacian of the complements of trees.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 7, 2398--2405.
\MR{2890000}.  \Zbl{1238.05162}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2020a}{}  The energy of random signed graph.  \emph{Linear Algebra Appl.}\ 585 (2020), 227--240.
\MR{4019819}.  \Zbl{1426.05103}.  \arXiv{1812.11865}.				
\annot{} The model:  $\Prob(+e_{ij})=p$, $\Prob(-e_{ij})=q$, $\Prob(\text{no }e_{ij})=1-p-q$.  An integral expression for energy $\cE$ and Wigner matrices.  Thm. 2.7:  $\cE = n^{3/2} (\frac{8}{3\pi}\sqrt{p+q-(p-q)^2}+o(1))$.  
\S3, ``The energy of the random multipartite signed graph'': bounds.
\dated{12 Jul 2019} 
\codes{\CSG:~\CRand:~\CEig}%
\itemspace

\author{Shuchao Li \& Wei Wei}
\authorspace
\bsgitem{2020a}{}  The multiplicity of an $A_\alpha$-eigenvalue: A unified approach for mixed graphs and complex unit gain graphs.  \emph{Discrete Math.}\ 343 (2020), no.\ 8, art.\ 111916, 19 pp.
\MR{4082511}.  \Zbl{1441.05142}.  
\annot{} Via gain group $\bbT := \{z\in\bbC: |z|=1\}$, generalizes results on graphs, signed graphs, mixed graphs.
\dated{4 Mar 2022} 
\par
\codes{\CGG:~\CAdj(\CGen):~\CEig}%
\itemspace

\author{Shuchao Li \& Ting Yang}
\authorspace
\bsgitem{2022a}{}  On the relation between the adjacency rank of a complex unit gain graph and the matching number of its underlying graph.  \emph{Linear Multilinear Algebra} 70 (2022), no.\ 9, 1768--1787.
\MR{4429445}.  
\codes{\CGG:~\CAdj}%
\itemspace

\author{Shuchao Li \& Yuantian Yu}
\authorspace
\bsgitem{2022a}{}  Hermitian adjacency matrix of the second kind for mixed graphs.  \emph{Discrete Math.}\ 345 (2022), no.\ 5, art.\ 112798, 22 pp.
\MR{4368471}.  
\arXiv{2102.03760}.  
\annot{} The graphs are (and are treated as) gain graphs with gains 1 for undirected, $\omega$ for directed edges, where $\omega$ is a complex unit chosen [seemingly arbitrarily] to be $\sqrt[6]{1}$ as in \xref{Mohar}{2020a}{Bojan Mohar}.  [\Question.  How much is independent of the choice of $\omega$?]
\dated{25 Mar 2022} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Shuchao Li \& Li Zhang}
\authorspace
\bsgitem{2011a}{}  Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 2, 145--158.
\MR{2773647} (2012a:05194).  \Zbl{1239.05116}.					
\annot{} Sharp upper and lower bounds for $\per(\Lap(-\Gamma))$ when $\Gamma$ is unicyclic or bipartite, with or without girth, and characterization of extremal graphs.  
[Bipartite $\Gamma$ means they are doing $\Lap(\Gamma)$; the truly signed part is for unicyclic graphs only.]
\dated{19 Nov 2011} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter $d$.  \emph{Graphs Combin.}\ 28 (2012), no.\ 4, 531--546.
\MR{2944040}.  
\annot{} See \xref{Li and Zhang}{2011a}{Shuchao Li \& Li Zhang}.  Here, the second minimum of, and a lower bound for, $\per \Lap(-\Gamma)$. 
\dated{24 Jan 2012} 
\codes{\Cpar:~\CLap}%
\itemspace

\author{Shuchao Li \& Minjie Zhang}
\authorspace
\bsgitem{2012a}{}  On the signless Laplacian index of cacti with a given number of pendant vertices.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 12, 4400--4411.
\MR{2917417}.  \Zbl{1241.05082}.						
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shuchao Li, Siqi Zhang, \& Baogen Xu}
\authorspace
\bsgitem{2019a}{}  The relation between the $H$-rank of a mixed graph and the independence number of its underlying graph.  \emph{Linear Multilinear Algebra} 67 (2019), no.\ 11, 2230--2245.
\MR{4002346}.  \Zbl{1422.05064}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.
\itemspace

\author{Xiang Li}							%LI X
\authorspace
\authref  See \hyperlink{Rongrong Kang}{R.-R.~Kang}.
\itemspace

\author{Xiangwen Li}							%LI XW
\authorspace
\authref  See \hyperlink{Lili Hu}{L.L~Hu}.
\itemspace

\author{Xiaodi Li}							%LI XD
\authorspace
\authref  See \hyperlink{Guohong Mu}{G.-H.~Mu}.
\itemspace

\author{Xiao Ming Li}							%LI XM
\authorspace
\authref  See \hyperlink{F.T.\ Boesch}{F.T.~Boesch} and \hyperlink{Fenglei Tian}{F.L.~Tian}.
\itemspace

\author{Xiaowang Li}							%LI XW
\authorspace
\authref  See \hyperlink{Xiangyu Ren}{X.-Y.~Ren}.
\itemspace

\author{Xueliang Li}							%LI XL
\authorspace
\authref  See also \hyperlink{Xiaolin Chen}{X.-L.~Chen},  \hyperlink{Wenxue Du}{W.X.~Du}, \hyperlink{Dan Hu}{D.~Hu}, \hyperlink{Bofeng Huo}{B.F.~Huo}, and \hyperlink{Jianxi Liu}{J.-X.~Liu}.
\authrefspace

\author{Xueliang Li \& Wen Xia}	
\authorspace
\bsgitem{2019a}{}  Skew-rank of an oriented graph and independence number of its underlying graph.  \emph{J.\ Combin.\ Optim.}\ 38 (2019), 268--277.
\MR{3955753}.  \Zbl{1420.05133}.  
\annot{} \Cf\ \xref{Huang, Li, and Wang}{2018a}{Jing Huang, Shuchao Li, \& Hua Wang}.
\dated{15 Jul 2019} 
\codes{\Cgg:~\CAdj}%
\itemspace

\bsgitem{2020a}{}  Adjacency rank and independence number of a signed graph.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 43 (2020), no.\ 1, 993--1007.
\MR{4044914}.  \Zbl{1433.05141}.  
\codes{\CSG:~\CAdj}%
\itemspace

\author{Xueliang Li, Jianbin Zhang, \& Lusheng Wang}
\authorspace
\bsgitem{2009a}{}  On bipartite graphs with minimal energy.  \emph{Discrete Appl.\ Math.}\ 157 (2009), no.\ 4, 869--873. 
\MR{2499503} (2010f:05116).  \Zbl{1226.05161}.  
\annot{} [Bipartite energy is the energy of $A(-\Gamma)$ for bipartite $\Gamma$.  \Problem\ 1.  Generalize to antibalanced signed graphs.  \Problem\ 2.  Generalize to signed graphs.]
\dated{24 Jan 2012} 
\codes{\Csg:~\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\authorx{Yadong Li}							%LI YD
\author{Yadong Li, Jing Liu, \& Chenlong Liu}
\authorspace
\bsgitem{2014a}{}  A comparative analysis of evolutionary and memetic algorithms for community detection from signed social networks.  \emph{Soft Computing} 18 (2014), 329--348.
\codes{\CSG:~\CStr:~\CAlgor}%
\itemspace

\author{Yang Li}							%LI Y
\authorspace
\authref  See \hyperlink{Bo Yang}{B.~Yang}.
\itemspace

\authorx{Yanhua Li}							%LI YH
\author{Yanhua Li, Wei Chen, Yajun Wang, \& Zhi-Li Zhang}
\authorspace
\bsgitem{2013a}{}  Influence diffusion dynamics and influence maximization in social networks with friend and foe relationships.  In:  \emph{Proceedings of the Sixth ACM International Conference on Web Search and Data Mining} (WSDM '13, 2013), pp.\ 657--666.  ACM, New York, 2013.
\arXiv{1111.4729}.  
\codes{\CSG:~\CPsS}%
\itemspace

\bsgitem{2015a}{}  Voter model on signed social networks.  \emph{Internet Math.}\ 11 (2015), no.\ 2, 93--133.
\MR{3316858}.  \Zbl{1465.91082}.  
\codes{\CSD}%
\itemspace

\author{Yijia Li}							%LI YJ
\authorspace
\authref  See \hyperlink{Shasha Feng}{S.-S.~Feng}.
\itemspace

\author{Yiyang Li}							%LI YY
\authorspace
\authref  See \hyperlink{Wenxue Du}{W.X.~Du}.
\itemspace

\author{Yong Li}							%LI Y
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.-S.~Wu}.
\itemspace

\authorx{Yongtao Li}							%LI YT
\author{Yongtao Li \& Yuejian Peng}
\authorspace
\bsgitem{2022a}{}  The maximum spectral radius of non-bipartite graphs forbidding short odd cycles.  \emph{Electronic J.\ Combin.}\ 29 (2022), no.\ 4, art.\ 4.2.
\annot{} [\Problem.  Generalize to unbalanced signed graphs without short negative circles.]  
[For Laplacian analog \cf\ \xref{Liu, Miao, and Xue}{20xxa}{Ruifang Liu, Lu Miao, \& Jie Xue}.]
\dated{27 Dec 2022} 
\codes{\CPar:~\CAdj:~\CEig;~\CStr}%
\itemspace

\author{Yu Li}							%LI Y
\authorspace
\authref  See \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\author{Yuemeng Li}							%LI YM
\authorspace
\authref  See also \hyperlink{Leting Wu}{L.T.~Wu}.
\authrefspace

\author{Yuemeng Li, Xintao Wu, \& Aidong Lu}
\authorspace
\bsgitem{2017a}{}  On spectral analysis of directed signed graphs.  In:  \emph{2017 IEEE International Conference on Data Science and Advanced Analytics} (DSAA, Tokyo), pp.\ 539--548.  IEEE, 2017.
\annot{} Extended abstract of \xrefd{Li, Yuan, Wu, and Lu}{2018a}{Yuemeng Li, Shuhan Yuan, Xintao Wu, \& Aidong Lu}.
\codes{\CSD:~\CAdj:~\CEig:~\CClu,~\CAlgor}%
\itemspace

\author{Yuemeng Li, Shuhan Yuan, Xintao Wu, \& Aidong Lu}
\authorspace
\bsgitem{2018a}{}  On spectral analysis of directed signed graphs.  \emph{Int.\ J.\ Data Sci.\ Analytics} 6 (2018), 147--162.
\arXiv{1612.08102}.  
\codes{\CSD:~\CAdj:~\CEig:~\CClu,~\CAlgor}%
\itemspace

\author{Zhentao Li}							%LI ZT
\authorspace
\authref  See \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi}.
\itemspace

\author{Zhongshan Li}						%LI ZS
\authorspace
\authref  See also \hyperlink{Marina Arav, Frank J.\ Hall, Zhongshan Li, \& Hein van der Holst}{M.~Arav}, \hyperlink{Carolyn Eschenbach}{C.A.~Eschenbach}, \hyperlink{Frank J.\ Hall}{F.J.~Hall} and \hyperlink{Ling Zhang}{L.~Zhang}.
\authrefspace

\author{Zhongshan Li, Frank Hall, \& Carolyn Eschenbach}
\authorspace
\bsgitem{1994a}{}  On the period and base of a sign pattern matrix.  \emph{Linear Algebra Appl.}\ 212-213 (1994), 101--120.
\MR{1306974} (95m:15026).  \Zbl{821.15017}.					
\itemspace

\authorx{Chaohua Liang}						%LIANG CH
\author{Chaohua Liang, Bolian Liu, \& Yufei Huang}
\authorspace
\bsgitem{2010a}{}  The $k$th lower bases of primitive non-powerful signed digraphs.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 7, 1680--1690.
\MR{2592910} (2011b:15076).  \Zbl{1221.05190}.					
\codes{\CSD:~\Cqm}%
\itemspace

\authorx{Yanting Liang}						%LIANG YT
\author{Yanting Liang, Bolian Liu, \& Hong-Jian Lai}
\authorspace
\bsgitem{2009a}{}  Multi-$g$ base index of primitive anti-symmetric sign pattern matrices.  \emph{Linear Multilinear Algebra} 57 (2009), no.\ 6, 535--546.
\MR{2543715} (2010i:05151.  \Zbl{1221.15019}.					
\codes{\CQM:~\CSD}%
\itemspace

\author{Rui-Quan Liao}						%LIAO RQ
\authorspace
\authref  See \hyperlink{Bin Hu}{B.~Hu}.
\itemspace

\author{F.\ di Liberto}							%LIBERTO %DI LIBERTO
\authorspace
\authref  See \hyperlink{Antonio Coniglio}{A.~Coniglio}.
\itemspace

\author{Hans Liebeck} 						%LIEBECK H
\authorspace
\authref  See \hyperlink{David Harries}{D.~Harries}.
\itemspace

\author{Martin W.\ Liebeck} 						%LIEBECK M
\authorspace
\bsgitem{1980a}{}  Lie algebras, $2$-graphs and permutation groups.  \emph{Bull.\ London Math.\ Soc.}\ 33 (1982), 76--85.
\MR{0565479} (81f:05095).  \Zbl{499.05031}.					
\annot{} Examines the $F\,\Aut([\Sigma])$-module $F\,V(\Sigma)$, where $\Sigma$ is a signed complete graph and $F$ is a field of characteristic $2$.
\codes{\CTG:~\CAut}%
\itemspace

\bsgitem{1982a}{}  Groups fixing graphs in switching classes.  \emph{J.\ Austral.\ Math.\ Soc.\ (A)} 33 (1982), 76--85.  
\MR{0662362} (83h:05048).  \Zbl{499.05031}.
\annot{} Given an abstract group $\fA$, which of its permutation representations are exposable on every invariant switching class of signed complete graphs [see \xref{Harries and H.\ Liebeck}{1978a}{David Harries \& Hans Liebeck} for definitions]?
\codes{\Ckg:~\Csw,~\CTG:~\CAut}%
\itemspace

\author{Thomas M.\ Liebling} 						%LIEBLING
\authorspace
\authref  See \hyperlink{Heinz Groflin}{H.~Gr\"oflin}.
\itemspace

\author{Rainer Liebmann}						%LIEBMANN
\authorspace
\bsgitem{1986a}{\dag}  \emph{Statistical Mechanics of Periodic Frustrated Ising Systems}.  Lect.\ Notes in Phys., Vol.\ 251.  Springer-Verlag, Berlin, 1986.
\MR{0850837} (87k:82004).  
\annot{} Detailed and readable descriptions, often simplified and relatively combinatorial, of the state of knowledge about Ising systems in the form of signed graphs and weighted signed graphs.  [Relatively accessible to combinatorists.]  
Dictionary:  ''model'' = graph with signs and usually weights, ``ferromagnetic'' = positive edge, ``antiferromagnetic'' = negative edge, ``fully frustrated'' = all girth circles are negative, ``state'' = $s: V \to \signnos$, ``ground state'' = state with fewest frustrated edges, ``ground state degeneracy'' = number of ground states (1 being nondegenerate), ``excited state'' = non-ground state.
\S2.1.1, ``Ground state degeneracy of the ANNNI-chain'', on chains of triangles with two bond signs and strengths, $J_1$ and $J_2$.  (ANNI = Axial Next Nearest Neighbor Ising model.)  The number and description of ground states are treated in detail, as well as less combinatorial physical quantities.  
\S2.3.1, ``Periodic frustrated chains'':  All weights equal, so this is signed graphs.  Restates \xref{Doman and Williams}{1982a}{B.G.S.\ Doman \& J.K.\ Williams} in terms of a path with distance-2 edges, signed with period 4.  The path edges have constant sign (either $+$ or $-$ by switching) and weight $B$; the distance-2 edges are $+---$ with weight $J$.
\annotctd{} \S3.1.2b, ``Star-triangle transformation'':  Edge signs and weights transform.  The triangle-star transformation on a negative triangle gives imaginary signs.  [\Question.  Does this indicate a use for complex unit gains?]
\S3.2, ``Triangular lattice'':  Based on \xref{Houtappel}{1950a}{R.M.F.\ Houtappel}, \xrefd{1950b}{R.M.F.\ Houtappel} and \xref{Wannier}{1950a}{G.H.\ Wannier}.
\S3.3.1, ``Union Jack lattice'':  Square lattice, edges weighted $J_1$, with alternating diagonals in alternating squares weighted $J_2<0$.  All triangles are negative.  $|J_2|/J_1$ determines behavior.  For ratio $1$ (a signed graph), there are $\approx C^{\#V}$ ground states for a finite sublattice, where $C \geq \surd(17/8)$.
\S3.3.2, ``Villain's odd model'':  \Cf\ \xref{Villain}{1977a}{Jacques Villain}.
\S3.3.3, ``Hexagon lattice'': \Cf\ \xref{Wolff and Zittartz}{1982a}{W.F.\ Wolff \& J.\ Zittartz}, \xrefd{1983a}{W.F.\ Wolff \& J.\ Zittartz}.
\S3.3.4, ``Pentagon lattice'':  \Cf\ \xref{Waldor, Wolff, and Zittartz}{1985a}{M.H.\ Waldor, W.F.\ Wolff, \& J.\ Zittartz}.
\S3.3.5, ``Kagom\'e lattice'':  Various periodic sign patterns; references.
\S3.3.6, ``Connection between GS [ground state] degeneracy and existence of a phase transition at $T_c=0$'':  The conjecture of \xref{Hoever, Wolff, and Zittartz}{1981a}{P.\ Hoever, W.F.\ Wolff, \& J.\ Zittartz}.  Also, a conjecture of S\"uto on the exact conditions under which the ground states are connected in the state graph.
\S3.4, ``Frustrated Ising systems with crossing interactions'':  Several more complicated extensions of previous models, usually by adding distance-2 edges (``nnn interactions'').  See (2) below.
\annotctd{} \S4.1, ``fcc antiferromagnet'':  All-negative face-centered cubic lattice graph.  Interesting remarks on how ground state and near-ground state structure might influence physical properties.
\S4.2, ``Fully and partially frustrated simple cubic lattice'':  The fully frustrated planar square lattice can be stacked in various ways to produce differently frustrated cubic lattices.
\S4.3, ``AF pyrochlore model'':  All-negative tetrahedra joined at corners.
\S4.4, ``ANNNI-model'':  All-positive cubic lattice with negative distance-2 vertical edges.
\annotctd{} Two frequent remarks:  (1)  An external magnetic field reduces the number of ground states.  (2)  Slightly more complicated graphs give models that are not exactly solvable.  [Combinatorial explanations:  The magnetic field corresponds to an extra vertex, positively adjacent to all $V(\Sigma)$; see \xref{Barahona}{1982a}{Francisco Barahona}.  The more complicated graphs are nonplanar; \xref{Barahona}{1982a}{Francisco Barahona} and \xref{Istrail}{2000a}{Sorin Istrail} indicate that this is the obstacle to exact solution.]
\dated{28 Aug 2012} 
\par
\codes{\CPhys,~\CSG,~\CWG:~\CFr,~\CState(\Cfr):~\CExp,~\CRef}%
\itemspace

\authorx{Magnhild Lien}						%LIEN
\author{Magnhild Lien \& William Watkins}
\authorspace
\bsgitem{2000a}{}  Dual graphs and knot invariants.  \emph{Linear Algebra Appl.}\ 306 (2000) 123--130.
\MR{1740436} (2000k:05187).  \Zbl{946.05061}.
\annot{} The Laplacian matrices of a signed plane graph and its dual have the same invariant factors.  The proof is via the signed graphs of knot diagrams.
\codes{\CSGc:~\CDual,~\CLap,~\CEig,~\CKnot}%
\itemspace

\author{Frauke Liers}							%LIERS
\authorspace
\authref  See also \hyperlink{Matteo Palassini}{M.~Palassini} and \hyperlink{G.\ Pardella}{G.~Pardella}.
\authrefspace

\author{Frauke Liers, Michael J\"unger, Gerhard Reinelt, \& Giovanni Rinaldi}
\authorx{Frauke Liers, Michael Junger, Gerhard Reinelt, \& Giovanni Rinaldi}
\authorspace
\bsgitem{2004a}{}  Computing exact ground states of hard Ising spin glass problems by branch-and-cut.  In:  Alexander K.\ Hartmann \& Heiko Rieger, eds., \emph{New Optimization Algorithms in Physics}, pp.\ 47--69.  WILEY-VCH, Weinheim, 2004.
\Zbl{1059.90147}.  
\annot{} Reduces frustration index to max cut.  Algorithms for the latter.
\dated{6 Jul 2022} 
\codes{\Csg:~\CFr:~\CAlgor}%
\itemspace

\author{Ko-Wei Lih}							%LIH
\authorspace
\authref   See \hyperlink{Jing-Ho Yan}{J.H.~Yan}.
\itemspace

\author{Bart Litjens}							%LITJENS
\authorspace
\authref  See \hyperlink{Andrew Goodall}{A.~Goodall}.
\itemspace

\author{Chjan C.\ Lim}						%LIM CC
\authorspace
\bsgitem{1993a}{}  Nonsingular sign patterns and the orthogonal group.  \emph{Linear Algebra Appl.}\ 184 (1993), 1--12.
\MR{1209379} (94c:15036).  \Zbl{782.68098}.					
\annot{} A family of bipartite signed wheels that prevent $A = (A\inv)\transpose$.  A family of bipartite signed graphs which allow it.	
\dated{6 Mar 2011} 
\par
\codes{\CSG:~\CQM}%
\itemspace

\author{Ee-Peng Lim}						%LIM EP
\authorspace
\authref  See \hyperlink{David Lo}{D.~Lo}.
\itemspace

\author{Meng-Hiot Lim}						%LIM MH
\authorspace
\authref  See \hyperlink{Frank Harary, Meng-Hiot Lim, Amit Agarwal, \& Donald C.\ Wunsch}{Harary, Lim, \etal}
\itemspace

\authorx{Carlos V.G.C.\ Lima}						%LIMA
\author{Carlos V.G.C.\ Lima, Dieter Rautenbach, U\'everton S.\ Souza, \& Jayme L.\ Szwarcfiter}
\authorx{Carlos V.G.C.\ Lima, Dieter Rautenbach, Ueverton S.\ Souza, \& Jayme L.\ Szwarcfiter}
\authorspace
\bsgitem{2018a}{}  Bipartizing with a matching.  In:  Donghyun Kim, R.N.\ Uma, and Alexander Zelikovsky, eds., \emph{Combinatorial Optimization and Applications} (COCOA 2018, 12th Int.\ Conf., Atlanta), pp.\ 198--213.  \LNCS 11346.  Springer, Cham, 2018.
\MR{3893518}.  
\annot{} Slightly abbreviated version of \xrefd{2022a}{Carlos V.G.C.\ Lima, Dieter Rautenbach, Ueverton S.\ Souza, \& Jayme L.\ Szwarcfiter}.
\codes{\Cpar:~\CFr,~\CAlgor}%
\itemspace

\bsgitem{2022a}{}  On the computational complexity of the bipartizing matching problem.  \emph{Ann.\ Oper.\ Res.}\ 316 (2022), no.\ 2, 1235--1256.
\MR{4478058}.  \Zbl{1497.05093}.  \arXiv{1710.07741}.  
\annot{} It is NP-complete to decide which graphs become bipartite by deleting a matching, even for 3-colorable planar graphs with $\Delta(\Gamma)=4$, but linear-time solvable for subcubic graphs.  [\Question.  Which signed graphs have a balancing matching?]  
\dated{10 Sept 2022} 
\codes{\Cpar:~\CFr,~\CAlgor}%
\itemspace

\author{[Leonardo Silva de Lima]}					%LIMA
\authorspace
\authref  See \hyperlink{Leonardo Silva de Lima}{L.S.~de~Lima} (under `D').
\itemspace

\author{Enzo M.\ Li Marzi}						%LI MARZI
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo} and \hyperlink{Jianfeng Wang}{J.F.~Wang}.
\itemspace

\author{Friday Lin}							%LIN F
\authorspace
\authref   See \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\authorx{Huiqiu Lin}							%LIN HQ
\authorspace
\authref  See also \hyperlink{Dan Li}{D.~Li} and \hyperlink{Hangtian Guo}{H.T.~Guo}.
\authrefspace

\author{Huiqiu Lin \& Hangtian Guo}
\authorspace
\bsgitem{2021a}{}  A spectral condition for odd cycles in non-bipartite graphs.  \emph{Linear Algebra Appl.}\ 631 (2021), 83--93.
\MR{4308089}.  \Zbl{1483.05095}.  
\annot{} Max.\ spectral radius of $\Gamma$ with order $n$ and odd girth $\geq 2k+3$: only one graph.  [Also in \xref{Li, Sun, and Yu}{2022a}{Shuchao Li, Wanting Sun, \& Yuantian Yu}.]
[This is the case $\Sigma = -\Gamma$ of \Problem.  Generalize to unbalanced signed graphs with negative girth $k$.]
\dated{28 Jun 2022} 
\codes{\Cpar:~\CAdj:~\CEig:~\Cbal}%
\itemspace

\author{Lin Lin}							%LIN L
\authorspace
\authref  See \hyperlink{Jiu-Gang Dong}{J.-G.~Dong}.
\itemspace

\author{Shangwei Lin}						%LIN SW
\authorspace
\authref  See \hyperlink{Shiying Wang}{S.Y.~Wang}.
\itemspace

\authorx{Xue Lin}							%LIN X
\author{Xue Lin, Qiang Jiao, \& Long Wang}
\authorspace
\bsgitem{2019a}{}  Opinion propagation over signed networks: Models and convergence analysis.  \emph{IEEE Trans.\ Automatic Control} 64 (2019), no.\ 8, 3431--3438.
\MR{3992885}.  \Zbl{1482.91165}.  
\codes{\CSG:~\CPsS,~\CDyn}%
\itemspace

\author{Yen-Chi Roger Lin}						%LIN YC
\authorspace
\authref  See \hyperlink{Meng-Yue Cao}{M.-Y.~Cao}.
\itemspace

\author{Jonathan A.\ Lindquist}						%LINDQUIST
\authorspace
\authref  See \hyperlink{Steffen Klamt}{S.~Klamt}.
\itemspace

\author{Bernt Lindstr\"om} 						%LINDSTROM*
\authorx{Bernt Lindstrom}
\authorspace
\authref  See \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{Gabriele Lini}						%LINI
\authorspace
\authref  See \hyperlink{Claudio Altafini}{C.~Altafini}.
\itemspace

\author{Nathan Linial}						%LINIAL
\authorspace
\authref  See \hyperlink{Yonatan Bilu}{Y.~Bilu} and \hyperlink{Shlomo Hoory}{S.~Hoory}.
\itemspace

\author{S\'ostenes Lins} 						%LINS
\authorx{Sostenes Lins}
\authorspace
\bsgitem{1981a}{}  A minimax theorem on circuits in projective graphs.  \emph{J.\ Combin.\ Theory Ser.\ B}  30 (1981), 253--262.  
\MR{0624541} (82j:05074).  \Zbl{457.05057}.					
\annot{} For Eulerian $\Sigma$ in projective plane, max.\ number of edge-disjoint negative circles = min.\ number of edges cut by a noncontractible closed curve that avoids the vertices.  [Generalized by \xref{Schrijver}{1989a}{Alexander Schrijver}.]
\par
\codes{\CSG:~\CTop,~\Cfr,~\CCircles,~\CAlgor}%
\itemspace

\bsgitem{1982a}{}  Graph-encoded maps.  \emph{J.\ Combin.\ Theory Ser.\ B} \ 32 (1982), 171--181.  
\MR{0657686} (83e:05049).   \Zbl{465.05031}, (\Zbl{478.05040}).			
\annot{} See \S4.  	
[\Cf\ \xref{Zaslavsky}{1992a}{Thomas Zaslavsky} for related work.]
\codes{\Csg:~\CTop:~\Cbal}%
\itemspace

\bsgitem{1985a}{}  Combinatorics of orientation reversing circles.  \emph{Aequationes Math.}\ 29 (1985), 123--131.  
\MR{0819300} (87c:05051).  \Zbl{592.05019}.  
\codes{\Csg,~\Cpar:~\CTop,~\CBal,~\CFr} 
\itemspace

\authorx{J.H.\ van Lint}		 				%LINT
\author{J.H.\ van Lint \& J.J.\ Seidel}
\authorspace
\bsgitem{1966a}{} Equilateral point sets in elliptic geometry.  \emph{Proc.\ Koninkl.\ Ned.\ Akad.\ Wetenschap.\ Ser.\ A} 69 (= \emph{Indag.\ Math.}\ 28) (1966), 335--348.  
\MR{0200799} (34 \#685).  \Zbl{138.41702} (138, p.\ 417b).				
Repr.\ in \xref{Seidel}{1991a}{J.J.\ Seidel}, pp.\ 3--16.
\annot{} Introduces graph switching.
\codes{\Ctg,~\CGeom}%
\itemspace

\author{Svante Linusson}						%LINUSSON
\authorspace
\authref  See \hyperlink{Christos A.\ Athanasiadis}{C.A.~Athanasiadis}.
\itemspace

\authorx{Marc J.\ Lipman}			 			%LIPMAN
\author{Marc J.\ Lipman \& Richard D.\ Ringeisen}
\authorspace
\bsgitem{1978a}{} Switching connectivity in graphs.  
In:  F.\ Hoffman \etal, eds., \emph{Proc.\ of the Ninth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing} (Boca Raton, 1978), pp.\ 471--478.  
Congressus Numerantium, XXI.  Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1978.
\MR{0527972} (80k:05073).  \Zbl{446.05033}.
\codes{\CTG}%
\itemspace

\author{A.\ Lipshtat}							%LIPSHTAT
\authorspace
\authref  See \hyperlink{Avi Ma'ayan}{A.~Ma'ayan}.
\itemspace

\author{C.H.C.\ Little}						%LITTLE
\authorspace
\authref  See \hyperlink{Ilse Fischer}{I.~Fischer}.
\itemspace

\author{Simon Litsyn}						%LITSYN
\authorspace
\authref   See \hyperlink{I.\ Krasikov}{I.~Krasikov}.
\itemspace

\author{Charles H.C.\ Little}						%LITTLE
\authorspace
\authref   See \hyperlink{C.\ Paul Bonnington}{C.P.~Bonnington}.
\itemspace

\author{A-Ming Liu}							%LIU AM
\authorspace
\authref  See \hyperlink{Ming-Zhu Chen}{M.-Z.~Chen}.
\itemspace

\author{Bolian Liu}							%LIU BL
\authorspace
\authref  See also \hyperlink{Bo Cheng}{B.~Cheng}, \hyperlink{Yufei Huang}{Y.F.~Huang}, \hyperlink{Chang Li}{C.~Li}, \hyperlink{Qian Li}{Q.A.~Li}, \hyperlink{Yanting Liang}{Y.T.~Liang}, \hyperlink{Jianping Liu}{J.P.~Liu}, \hyperlink{Mu Huo Liu}{M.H.~Liu}, and \hyperlink{Lihua You}{Z.F.~You}.
\authrefspace

\bsgitem{2007a}{}  The period and base of a reducible sign pattern matrix.  \emph{Discrete Math.}\ 307 (2007), 3031--3039.
\MR{2371074} (2009i:15043).  \Zbl{1127.15018}.					
\codes{\CQM:~\CSD}%
\itemspace

\author{Bolian Liu, Muhuo Liu, \& Zhifu You}
\authorspace
\bsgitem{2013a}{}  The majorization theorem for signless Laplacian spectral radii of connected graphs.  \emph{Graphs Combin.}\ 29 (2013), no.\ 2, 281--287.
\MR{3027603}.  \Zbl{1263.05062}.  
\annot{} For a degree sequence $\pi$, define $\mu_c(\pi) := \max_\Gamma \lambda_1(\Lap(-\Gamma))$ over connected $\Gamma$ with degree sequence $\pi$ and $c$ circles.  Let $\pi \preccurlyeq \pi'$ in the majorization ordering.  
Thm.\ 2:  Under certain assumptions on $c$, $\pi$, $\pi'$, $\mu(\pi) \leq \mu(\pi')$.  
For the special cases of unicyclic and bicyclic graphs: \xref{X.D.~Zhang}{2009a}{Xiao-Dong Zhang} and \xref{Huang, Liu, and Liu}{2011a}{Rong Huang, Jianzhou Liu, \& Li Zhu}.  
For majorization also see \xref{Tam, Fan, and Zhou}{2008a}{Bit-Shun Tam, Yi-Zheng Fan, \& Jun Zhou}, \xref{M.H.~Liu and Liu}{2012a}{Muhuo Liu \& Bolian Liu}.
\dated{24 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Chenlong Liu}						%LIU CL
\authorspace
\authref  See \hyperlink{Yadong Li}{Y.D.~Li}.
\itemspace

\author{Daphne Der-Fen Liu}						%LIU DDF
\authorspace
\authref  See \hyperlink{Yiting Jiang}{Y.-T.~Jiang}.
\itemspace

\author{Fang Liu}							%LIU F
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\author{Feng Liu}							%LIU F
\authorspace
\authref  See \hyperlink{Xiao-Jun Tian}{X.-J.~Tian}.
\itemspace

\authorx{Gui Zhen Liu}						%LIU GZ
\author{Gui Zhen Liu \& Qiang Wu}
\authorspace
\bsgitem{1995a}{}  Applications of graph theory to social science.  (In Chinese.  English summary.)  \emph{Shandong Daxue Xuebao Ziran Kexue Ban} 30 (1995), no.\ 4, 361--366.
\MR{1387317} (97c:05149).  \Zbl{882.05116}.
\annot{} Describes some applications of and some results about balance in signed graphs.
\codes{\CSG:~\CBal,~\CPsS:~\CExp,~\CMatrd}
\itemspace

\authorx{Henry Liu}							%LIU H
\author{Henry Liu, Robert Morris, \& Noah Prince}
\authorspace
\bsgitem{2009a}{}  Highly connected monochromatic subgraphs of multicolored graphs.  \emph{J.\ Graph Theory} 61 (2009), no.\ 1, 22--44.
\MR{2514097} (2010d:05083).  \Zbl{1202.05079}.  
\annot{} See \xref{{\L}uczak}{2016a}{Tomasz Luczak}.
\dated{24 Jan 2016} 
\codes{\Csg:~\CStr}%
\itemspace

\author{Huan Liu}							%LIU H
\authorspace
\authref  See \hyperlink{Ghazaleh Beigi}{G.~Beigi} and \hyperlink{Jiliang Tang}{J.-L.~Tang}.
\itemspace

\author{Huiqing Liu}							%LIU HQ
\authorspace
\authref  See also \hyperlink{Qin Wen}{Q.~Wen}.
\authrefspace

\author{Huiqing Liu \& Mei Lu}
\authorspace
\bsgitem{2014a}{}  A conjecture on the diameter and signless Laplacian index of graphs.  \emph{Linear Algebra Appl.}\ 450 (2014), 158--174.
\MR{3192475}.  \Zbl{1286.05093}.  
\annot{} Corrects and proves a conjecture of \xref{Hansen and Lucas}{2010a}{Pierre Hansen \& Claire Lucas} on $\max \lambda_1(\Lap(-\Gamma)) D(\Gamma)$ for $\#V=n$, where $\lambda_1=$ the largest eigenvalue of $\Lap(-\Gamma)$ and $D=$ diameter.
\dated{23 Nov 2014} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ji Liu}							%LIU J
\authorspace
\authref  See \hyperlink{Wei Chen}{W.~Chen}.
\itemspace

\authorx{Jia-Bao Liu}							%LIU JB
\author{Jia-Bao Liu \& Shaohui Wang}
\authorspace
\bsgitem{2017a}{}  A note on ``Extremal graphs with bounded vertex bipartiteness number''.  Manuscript, 2017.
\arXiv{1704.02867}.  
\annot{} Counterexamples and correction to \xref{Robbiano, Morales, and San Martin}{2016a}{Maria Robbiano, Katherine Tapia Morales, \& Bernardo San Martin}.  [\Problem.  Generalize to frustration number of $\Sigma$.]
\dated{19 May 2018} 
\codes{\Csg:~\CPar:~\CFr,~\CEig}%
\itemspace

\authorx{Jianbing Liu}							%LIU JB
\author{Jianbing Liu, Miaomiao Han, \& Hong-Jian Lai}
\authorspace
\bsgitem{2022a}{}  Weighted modulo orientations of graphs and signed graphs.  \emph{Electronic J.\ Combin.}\ 29 (2022), no.\ 4, art.\ P4.47, 20 pp.
\annot{} Despite the title this is about (existence of) modular flows with specified boundaries, not weights or orientations.	
\dated{17 Dec 2022} 
\par
\codes{\CSG:~\CFlows}%
\itemspace

\authorx{Jianping Liu}						%LIU JP
\author{Jianping Liu \& Bolian Liu}
\authorspace
\bsgitem{2008a}{}  The maximum clique and the signless Laplacian eigenvalues.  \emph{Czechoslovak Math.\ J.}\ 58(133) (2008), no.\ 4, 1233--1240.
\MR{2471179} (2010a:05116).  \Zbl{1174.05079}.					
\annot{} Bounds on the clique number $\omega(\Gamma)$ based on the least and greatest eigenvalues of $\Lap(-\Gamma)$.  A similar lower bound on the stability number $\alpha(\Gamma)$.
\dated{23 Nov 2014} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Jianxi Liu}							%LIU JX
\author{Jianxi Liu \& Xueliang Li}
\authorspace
\bsgitem{2015a}{}  Hermitian-adjacency matrices and Hermitian energies of mixed graphs.  \emph{Linear Algebra Appl.}\ 466 (2015), 182--207.
\MR{3278246}.  \Zbl{1302.05106}.  
\annot{} ``Hermitian adjacency matrix'' = $A(\Phi)$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.  Thm.\ 2.7:  Sachs formula for $\det A(\Phi)$.  Thm.\ 2.8:  Coefficients of characteristic polynomial.  Imaginary-gain circles contribute nothing to the formulas.  Thm.\ 2.16:  If $|\Phi_1| = |\Phi_2|$ and gains differ on some edge in a cut, then $\Spec A(\Phi_1) = \Spec A(\Phi_2)$ [obviously false as stated].  
[\Cf\ \xref{Mohar}{2016a}{Bojan Mohar}.]
\S3, ``Bounds of Hermitian energy''.  
\S4, ``Mixed graphs that share the same spectrum with their underlying graphs''.  Thm.\ 4.1:  $\Spec A(\Phi) = \Spec A(\|\Phi\|)$ iff $\Phi$ is balanced.  Thm.\ 4.2:  Cartesian product [as in] preserves balance.  
Dictionary:  ``value'' = gain; ``rank, corank'' = those of matroid $\bfF(\|\Phi\|)$; ``positive'' = balanced, ``generalized orientation'' = partial orientation.  
\S5, ``Oriented graphs''.  \S6, ``Integral representation for Hermitian energy''.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CAdj,~\CSw:~\CEig}%
\itemspace

\author{Jianzhou Liu}						%LIU JZ
\authorspace
\authref  See \hyperlink{Rong Huang}{R.~Huang}.
\itemspace

\author{Jiming Liu}							%LIU JM
\authorspace
\authref  See \hyperlink{Bo Yang}{B.~Yang}.
\itemspace

\author{Jing Liu}							%LIU J
\authorspace
\authref  See \hyperlink{Yadong Li}{Y.D.~Li}.
\itemspace

\author{Lele Liu}							%LIU LL
\authorspace
\authref  See \hyperlink{Xiying Yuan}{X.Y.~Yuan}.
\itemspace

\authorx{Lily L.\ Liu}							%LIU LL
\author{Lily L.\ Liu \& Yi Wang}
\authorspace
\bsgitem{2007a}{}  A unified approach to polynomial sequences with only real zeros.  \emph{Adv.\ Appl.\ Math.}\ 38 (2007), 542--560.
\MR{2311051} (2008a:05013).  \Zbl{1123.05009}.  
\annot{} \S3.5., ``Compositions of multisets and Dowling lattices''.
	
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Lu Liu}							%LIU L
\authorspace
\authref  See \hyperlink{Yanzhi Wu}{Y.-Z.~Wu}.
\itemspace

\author{Mu Huo Liu}							%LIU MH
\authorspace
\authref  See also \hyperlink{Bolian Liu}{B.L.~Liu}. 
\authrefspace

\author{Muhuo Liu \& Bolian Liu}
\authorspace
\bsgitem{2010a}{}  The signless Laplacian spread.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 2-3, 505--514.
\MR{2577696} (2011d:05226).  \Zbl{1206.05064}.					
\annot{} Spread $S =$ difference of largest and smallest eigenvalues, studied for $\Lap(-\Gamma)$.  Let $m(v) :=$ average degree in $N(v)$.  Thm.\ 2.1:  $\Delta-\delta+1 \leq S \leq \max_v \{d(v)+m(v)\}$.  Other lower bounds in terms of $\sum_v d(v)^2$, average degree of independent set of vertices.  Thm.\ 2.5:  Min spread of unicyclic graphs.
[\Cf\ \xrefd{2011a}{Muhuo Liu \& Bolian Liu}; \xref{Oliveira, de Lima, de Abreu, and Kirkland}{2010a}{Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, \& Steve Kirkland}; \xref{Fan and Fallat}{2012a}{Yi-Zheng Fan \& Shaun Fallat}.]
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011a}{}  On the spectral radii and the signless Laplacian spectral radii of $c$-cyclic graphs with fixed maximum degree.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 12, 3045--3055. 
\MR{2831596} (2012h:05197).  \Zbl{1226.05138}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  New method and new results on the order of spectral radius.  \emph{Computers Math.\ Appl.}\ 63 (2012), no.\ 3, 679--686.
\MR{2871667} (2012i:05171).  \Zbl{1238.05164}.  
\annot{} 
Also see \xref{Liu, Liu, and You}{2013a}{Bolian Liu, Muhuo Liu, \& Zhifu You}.
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2015a}{}  On the signless Laplacian spectra of bicyclic and tricyclic graphs.  \emph{Ars Combin.}\ 120 (2015), 169--180.
\MR{3363272}.  \Zbl{1349.05213}.  
\annot{} The 2 or 4 largest eigenvalues and spreads of $\Lap(-\Gamma)$.  
[\Problem.  Generalize to signed graphs, or complex unit gain graphs.  \Cf\ \xref{Reff}{2012a}{Nathan Reff}.]
\dated{19 May 2018} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Muhuo Liu, Bolian Liu, \& Fuyi Wei}
\authorspace
\bsgitem{2011a}{}  Graphs determined by their (signless) Laplacian spectra.  \emph{Electronic J.\ Linear Algebra} 22 (2011), 112--124.
\MR{2781040} (2012g:05141).  \Zbl{1227.05185}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Muhuo Liu, Xuezhong Tan, \& Bolian Liu}
\authorspace
\bsgitem{2010a}{}  The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with $n$ vertices and $k$ pendant vertices.  \emph{Czechoslovak Math.\ J.}\ 60 (2010), no.\ 3, 849--867.
\MR{2672419} (2011f:05185).  \Zbl{1224.05311}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011a}{}  The largest signless Laplacian spectral radius of connected bicyclic and tricyclic graphs with $n$ vertices and $k$ pendant vertices.  (In Chinese.)  \emph{Appl.\ Math.\ J.\ Chinese Univ.\ Ser.\ A} 26 (2011), no.\ 2, 215--222.
\MR{2838952} (2012e:05238).  \Zbl{1240.05197}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace


\authorx{Ning Liu}							%LIU N
\author{Ning Liu \& William J.\ Stewart}
\authorspace
\bsgitem{2011a}{}  Markov chains and spectral clustering.  In:  \emph{Performance Evaluation of Computer and Communication Systems:  Milestones and Future Challenges}, pp.\ 87--98.  \LNCS 6821.  Springer-Verlag, Berlin, 2011.
\par
\codes{\CPar:~\CEig:~\CAppl}%
\itemspace
� a method which was more broadly applicable to nonsymmetric systems. Enlightened by these, somewhat orthogonal, results and combining them together, we show that spectral partitioning can be viewed in the framework of state clustering on Markov chains. Our research results to date are two-fold. First, we prove that the second eigenvector of the **signless Laplacian** provides a heuristic solution to the NP-complete state clustering problem which is the dual of the minimum cut problem. Second, we propose two clustering techniques for Markov chains based on two different clustering measures.

\author{Ruifang Liu}							%LIU RF
\authorspace
\authref  See also \hyperlink{Mingqing Zhai}{M.Q.~Zhai}.
\authrefspace

\author{Ruifang Liu, Lu Miao, \& Jie Xue}
\authorspace
\bsgitem{20xxa}{}  Maxima of the $Q$-index of non-bipartite $C_3$-free graphs.  Submitted.  \emph{}\ 
\arXiv{2209.00801}.  
\annot{} [\Problem.  Generalize to unbalanced signed graphs without negative triangles.]  
[For a stronger adjacency analog \cf\ \xref{Yongtao Li \& Yuejian Peng}{2022a}{Yongtao Li \& Yuejian Peng}.]
\dated{27 Dec 2022} 
\codes{\CPar:~\CLap:~\CEig;~\CStr}%
\itemspace

\author{Shiping Liu}							%LIU SP
\authorspace
\authref  See also \hyperlink{Fatihcan M.\ Atay}{F.M.~Atay}, \hyperlink{Chuanyuan Ge}{C.-Y.~Ge}, \hyperlink{Chunyang Hu}{C.-Y.~Hu}, and \hyperlink{Carsten Lange}{C.~Lange}.
\authrefspace

\author{Shiping Liu, Norbert Peyerimhoff, \& Alina Vdovina}
\authorspace
\bsgitem{2020a}{}  Signatures, lifts, and eigenvalues of graphs.  In:  Fatihcan M.\ Atay \etal, eds., \emph{Discrete and Continuous Models in the Theory of Networks}, pp.\ 255--269.  Operator Theory Adv.\ Appl., Vol.\ 281.  Birkh\"auser/Springer, Cham, 2020.
\MR{4181350}.  \Zbl{1473.05176}.  \arXiv{1412.6841}.				
\annot{} Gains in the group $\bbT_3$ of cube roots of unity.   Eigenvalues of the $\bbT_3$-covering graph in terms of those of the gain graph $\Phi$ and underlying graph $\|\Phi\|$.	
\dated{30 Oct 2017} 
\codes{\CGG:~\CAdj,~\CCov,~\CEig}%
\itemspace

\author{Vivian Liu}							%LIU V
\authorspace
\authref  See \hyperlink{Gina Chen}{G.~Chen}.
\itemspace

\author{Weichan Liu}							%LIU WC
\author{Weichan Liu, Chen Gong, Lifang Wu, \& Xin Zhang}
\authorspace
\bsgitem{2018a}{}  A note on the vertex arboricity of signed graphs.  \emph{Utilitas Math.}\ 106 (2018), 251--258.
\MR{3753793}.  \Zbl{1393.05139}.  \arXiv{1708.03077}.  
\newcommand\va{\operatorname{va}}
\annot{} Introduces vertex arboricity $\va(\Sigma) := \min m$ such that $\Sigma$ has an $m$-coloring $c$ whose improper edge set $I(c)$ satisfies: $I(c){:}c\inv(\pm i)$ is a forest $\forall i\geq0$.  $c$ is a ``signed tree coloring''.  Thm.\ 2:  $va$ is switching invariant.  Lemma 3:  Signed tree colorings combine under $K_2$- and $K_3$-unions.  Thm.\ 4:  A signed near-triangulation (of the plane) is list-tree-colorable under assumptions on the lists.  Cor.\ 5:  A balanced signed triangulation is tree 3-colorable [but easily proved by Thm.\ 2 and known theorem that $\va(\Gamma) \leq 3$ for planar $\Gamma$].  Thm.\ 11: $\va(\Gamma) \leq 3$ if $\Gamma$ has no $K_5$ minor (expressed in terms of balanced signatures).
\dated{28 Jun 2022} 
\par
\codes{\CSG:~\CCol(\CGen),~\CSw}%
\itemspace

\authorx{Wenzhong Liu}						%LIU WZ
\author{Wenzhong Liu, Qing Cui, \& Yunzhuo Wang}
\authorspace
\bsgitem{2022a}{}  On 4-connected 4-regular graphs without even cycle decompositions.  \emph{Discrete Math.}\ 345 (2022), no.\ 10, art.\ 113012.
\MR{4432158}.  
\annot{} On decomposing $E(-\Gamma)$ into positive circles.  \Cf\ \xref{Rizzi}{2001a}{Romeo Rizzi}.
\dated{11 Jul 2022} 
\codes{\CPar:~\CStr:~\CCircles}%
\itemspace

\author{Wenzhong Liu, Huazheng You, \& Qing Cui}
\authorspace
\bsgitem{2020a}{}  On even cycle decompositions of line graphs of cubic graphs.  \emph{Discrete Math.}\ 343 (2020), no.\ 7, art.\ 111904, 15 pp.
\MR{4078918}.  \Zbl{1440.05169}.  
\annot{} Question of \xref{Markstr\"om}{2012a}{Klas Markstrom} solved for oddness $\leq4$.
\dated{11 Nov 2020} 
\codes{\Cpar:~\CLG:~\CStr}%
\itemspace

\authorx{Xianzhu Liu}						%LIU XZ
\author{Xianzhu Liu, Zhijian Ji, \& Ting Hou}
\authorspace
\bsgitem{2019a}{}  Graph partitions and the controllability of directed signed networks.  \emph{Sci.\ China Inform.\ Sci.}\ 62 (2019), art.\ 042202, 11 pp.
\MR{3919757}.  
\par
\codes{\CSD,~\CSG,~\CWG:~\CEig,~\CBal:~\CAlgor}%
\itemspace

\author{Xiaogang Liu}						%LIU XG
\authorspace
\authref  See also \hyperlink{Dan Hu}{D.~Hu} and \hyperlink{Yuanping Zhang}{Y.P.~Zhang}.
\authrefspace

\author{Xiaogang Liu, Suijie Wang, Yuanping Zhang, \& Xuerong Yong}
\authorspace
\bsgitem{2011a}{}  On the spectral characterization of some unicyclic graphs.  \emph{Discrete Math.}\ 311 (2011), 2317--2336.
\MR{2832132}.  \Zbl{1242.05165}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xiaoyu Liu}							%LIU XY
\authorspace
\authref  See \hyperlink{Yu Qing Chen}{Y.Q.~Chen}.
\itemspace

\author{Xin Liu}							%LIU X
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.-H.~Yu}.
\itemspace

\author{Xueyan Liu}							%LIU XY
\authorspace
\authref  See \hyperlink{Bo Yang}{B.~Yang}.
\itemspace

\author{Yan Pei Liu}							%LIU YP
\authorspace
\authref  See \hyperlink{Rong Xia Hao}{R.X.~Hao}.
\itemspace

\author{Yingluan Liu}							%LIU YL
\authorspace
\authref  See \hyperlink{Yufei Huang}{Y.F.~Huang}.
\itemspace

\authorx{Yu Liu}							%LIU Y
\author{Yu Liu \& Lihua You}
\authorspace
\bsgitem{2014a}{}  Further results on the nullity of signed graphs.  \emph{J.\ Appl.\ Math.}\ 2014 (2014), art.\ 483735, 8 pp.
\MR{3173331}.  \Zbl{1406.05064}.  \arXiv{1309.0174}.				
\annot{} Thm.\ 1:  Sachs formula for coefficients of the characteristic polynomial of $A(\Sigma, w)$ where $w: E \to \bbR_{>0}$.  [\Cf\ \xref{Trinajsti\'c}{1977a}{Nenad Trinajstic}, \xrefd{1983a}{Nenad Trinajstic} (no weights).]
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Yue Liu}							%LIU Y
\authorspace
\authref  See also \hyperlink{Xiying Yuan}{X.Y.~Yuan}.
\authrefspace

\author{Yue Liu, Jia-Yu Shao, \& Ling-Zhi Ren}
\authorspace
\bsgitem{2011a}{}  Characterization of ray pattern matrix whose determinantal region has two components after deleting the origin.  \emph{Linear Algebra Appl.}\ 435 (2011), 3139--3150.
\MR{2831602} (2012f:15054).  \Zbl{1226.15021}.  
\annot{} 
Dictionary:  ``arc-weighted digraph of $A$'' = complex gain digraph whose adjacency matrix is $A$.
\codes{\CQM:~\Cgg:~\CAdj}%
\itemspace

\author{Etera R.\ Livine}						%LIVINE
\authorspace
\authref  See \hyperlink{Remi C.\ Avohou}{R.C.~Avohou}.
\itemspace

\author{Paulette Lloyd}						%LLOYD
\authorspace
\authref  See \hyperlink{Phillip Bonacich}{P.~Bonacich} and \hyperlink{Patrick Doreian}{P.~Doreian}.
\itemspace

\authorx{David Lo}							%LO D
\author{David Lo, Didi Surian, Philips Kokoh Prasetyo, Kuan Zhang, \& Ee-Peng Lim}
\authorspace
\bsgitem{2013a}{}  Mining direct antagonistic communities in signed social networks.  \emph{Inform.\ Process.\ Management} 49 (2013), 773--791.
\codes{\CSD:~\Cclu:~\CAlgor}%
\itemspace

\author{Martin Loebl}							%LOEBL
\authorspace
\authref  See also \hyperlink{Yves Crama}{Y.~Crama} and \hyperlink{Anna Galluccio}{A.~Galluccio}.  
\authrefspace

\author{Martin Loebl \& Iain Moffatt}
\authorspace
\bsgitem{2008a}{}  The chromatic polynomial of fatgraphs and its categorification.  \emph{Adv.\ Math.}\ 217 (2008), no.\ 4, 1558--1587.
\MR{2382735} (2008j:05114).  \Zbl{1131.05036}.  \arXiv{math/0511557}.			
\codes{\CSGc:~\CTop,~\CInvar}
\itemspace

\author{[Shobana Loganathan]}						%LOGANATHAN
\authorspace
\authref  See \hyperlink{L.\ Shobana}{L.~Shobana}.
\itemspace

\authorx{D.O.\ Logofet}						%LOGOFET
\author{D.O.\ Logofet \& N.B.\ Ul'yanov}
\authorspace
\bsgitem{1982a}{}  Necessary and sufficient conditions for the sign stability of matrices.  (In Russian.)  \emph{Dokl.\ Akad.\ Nauk SSSR} 264 (1982), 542--546.  
\MR{0659759} (84j:15018).  \Zbl{509.15008}.
\annot{} Necessity of \xref{Jeffries'}{1974a}{Clark Jeffries} sufficient conditions.
\codes{\CSta}%
\itemspace

\bsgitem{1982b}{}  (as D.O.\ Logofet and N.B.\ Ul'janov)  Necessary and sufficient conditions for the sign stability of matrices.  \emph{Soviet Math.\ Dokl.}\ 25 (1982), 676--680.  
\MR{0659759} (84j:15018).  \Zbl{509.15008}.
\annot{} English trans.\ of \xrefd{1982a}{D.O.\ Logofet \& N.B.\ Ul'yanov}.
\codes{\CSta}%
\itemspace

\author{Michael Lohman}						%LOHMAN
\authorspace
\authref  See \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}.
\itemspace

\author{V.\ Lokesha}							%LOKESHA
\authorspace
\authref  See also \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\authrefspace

\author{V.\ Lokesha, P.S.\ Hemavathi, \& S.\ Vijay}
\authorspace
\bsgitem{2018a}{}  Note on super radial signed graphs.  \emph{Ultra Sci.\ Phys.\ Sci., Sect.\ A} 30 (2018), no.\ 1, 40--44.
\Zbl{1416.05132}.  
\codes{\CSG:~\CSw}%
\itemspace

\author{V.\ Lokesha, P.\ Siva Kota Reddy, \& S.\ Vijay}
\authorspace
\bsgitem{2009a}{}  The triangular line $n$-sigraph of a symmetric $n$-sigraph.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 19 (2009), no.\ 1, 123--129.
\MR{2542128} (2010k:05121).  \Zbl{1213.05120}.					
\annot{} Definitions and notation as in \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Generalization of \xref{Subramanya and Reddy}{2009a}{M.S.\ Subramanya \& P.\ Siva Kota Reddy} to symmetric $n$-signed graphs, with similar definitions and results.  
[The results remain true without assuming symmetry.]
\dated{10 Apr 2009}
\par
\codes{\CSG(\CGen),~\Cgg:~\CBal,~\CLG(\CGen),~\CSw}%
\itemspace

\author{Andreas Lommatzsch}						%LOMMATZSCH
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Bo Long}							%LONG B
\authorspace
\authref  See \hyperlink{Shuang Hong Yang}{S.H.~Yang}.
\itemspace

\author{M.\ Lor\'ea}							%LOREA
\authorx{M.\ Lorea}
\authorspace
\bsgitem{1979a}{}  On matroidal families.   \emph{Discrete Math.}\ 28 (1979), 103--106.
\MR{0542941} (81a:\-05029).  \Zbl{409.05050}.
\annot{} Discovers the ``linearly bounded'' (or ``count'') matroids of graphs.  [See \xref{White and Whiteley}{1983a}{Neil White \& Walter Whiteley}, \xref{Whiteley}{1996a}{Walter Whiteley}, \xref{Schmidt}{1979a}{Rudiger Schmidt}.]
\par
\codes{\CMtrdF:~\CBic,~\CGen}%
\itemspace

\authorx{Martin Lotz}							%LOTZ
\author{Martin Lotz \& Johann A. Makowsky}
\authorspace
\bsgitem{2004a}{}  On the algebraic complexity of some families of coloured Tutte polynomials.  \emph{Adv.\ Appl.\ Math.}\ 32 (2004), 327--349.
\MR{2037634} (2004k:05116).  \Zbl{1041.05042}.  \arXiv none.				
\codes{\CSGw:~\CInvar:~\CAlgor}%
\itemspace

\author{E.\ Loukakis}							%LOUKAKIS
\authorspace
\bsgitem{2003a}{}   A dynamic programming algorithm to test a signed graph for balance.  \emph{Int.\ J.\ Computer Math.}\ 80 (2003), no.\ 4, 499--507.
\MR{1983308}.  \Zbl{1024.05034}.						
\annot{} Another algorithm for detecting balance [\cf\ \xref{Hansen}{1978a}{Pierre Hansen}, \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}].  Also, once again proves that all-negative frustration index [obviously equivalent to Max Cut] is NP-complete.
\par
\codes{\CSG:~\CBal,~\CFr:~\CAlgor}%
\itemspace

\author{Janice R.\ Lourie} 						%LOURIE
\authorspace
\bsgitem{1964a}{}  Topology and computation of the generalized transportation problem.  \emph{Management Sci.}\  11 (1965) (Sept., 1964), no.\ 1, 177--187.  
\codes{\CGN:~\CMatrd(\Cbases)}%
\itemspace

\author{L\'aszl\'o Lov\'asz}						%LOVASZ
\authorx{Laszlo Lovasz}
\authorspace
\authref   See also \hyperlink{J.A.\ Bondy}{J.A.~Bondy}, \xref{Gerards, Lov\'asz, \etal{}}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}, and \hyperlink{Martin Grotschel}{M.~Gr\"otschel}. 
\authrefspace

\bsgitem{1965a}{}  On graphs not containing independent circuits.  (In Hungarian.)  \emph{Mat.\ Lapok} 16 (1965), 289--299.
\MR{0211902} (35 \#2777).  \Zbl{151.33403} (151, p.\ 334c).
\annot{} Characterization of the graphs having no two vertex-disjoint circles.  See \xref{Bollob\'as}{1978a}{Bela Bollobas} for exposition in English.  [\emph{Major} \Problem.  Characterize the biased graphs having no two vertex-disjoint unbalanced circles.  This theorem is the contrabalanced case.  For the sign-biased case see \xref{Slilaty}{2007a}{Daniel C.\ Slilaty}.  \xref{McCuaig}{1993a}{William McCuaig} might be relevant to the general problem.]
\codes{\CGG:~\CCircles:~\CStr}%
\itemspace

\bsgitem{1979a}{}  \emph{Combinatorial Problems and Exercises}.  North-Holland, Amsterdam, and Aka\-d\'emiai Kiad\'o, Budapest, 1979.
\MR{0537284} (80m:05001).  \Zbl{439.05001}.					
\annot{} Prob.\ 7.21 finds $\rk \incid(-\Gamma)$ [\cf\ \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}].  
Prob.\ 10.18:  The vertex frustration number of a contrabalanced graph vs.\ the circle edge-packing number.  
\dated{16 Jun 2012} 
\codes{\Csg:~\Cpar:~\CInc}\codesctd{\Cgg:~\Cfr}%
\itemspace

\bsgitem{1983a}{} Ear-decompositions of matching-covered graphs.  \emph{Combinatorica} 3 (1983), 105--117.  
\MR{0716426} (85b:05143).  \Zbl{516.05047}.
\annot{} It is hard to escape the feeling that we are dealing with all-negative signed graphs and their $-K_4$ and $-K_2^{\circ}$ minors.  [And indeed, see \xref{Gerards and Schrijver}{1986a}{A.M.H.\ Gerards \& A.\ Schrijver} and \xref{Gerards, Lov\'asz, \etal{}}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper} and the notes on \xref{Seymour}{1995a}{P.D.\ Seymour}.]
\codes{\CPar:~\CStr}%
\itemspace

\bsgitem{1993a}{}  \emph{Combinatorial Problems and Exercises},.  Second ed.  Elsevier, Amsterdam, and Akad\'emiai Kiad\'o, Budapest, 1993.
\MR{1265492} (94m:05001).  \Zbl{785.05001}.					
\annot{} See \xrefd{1979a}{Laszlo Lovasz}.
\dated{16 Jun 2012} 
\codes{\Csg:~\Cpar:~\CInc}\codesctd{\Cgg:~\Cfr}%
\itemspace

\bsgitem{2007a}{}  \emph{Combinatorial Problems and Exercises}.  Second ed., corr.\ reprint.  AMS Chelsea Publ., American Mathematical Soc., Providence, R.I., 2007.
\MR{2321240} (no rev).  \Zbl{439.05001}.						
\annot{} See \xrefd{1979a}{Laszlo Lovasz}.
\dated{16 Jun 2012} 
\codes{\Csg:~\Cpar:~\CInc}\codesctd{\Cgg:~\Cfr}%
\itemspace

\bsgitem{2011a}{}  Subgraph densities in signed graphons and the local Simonovits--Sidorenko conjecture.  \emph{Electronic J.\ Combin.}\ 18 (2011), \#P127.
\MR{2811096} (2012f:05158).  \Zbl{1219.05084}.  \arXiv{1004.3026}.			
\codes{\CSG}%
\itemspace

\author{L.\ Lov\'asz \& M.D.\ Plummer}
\authorx{L.\ Lovasz \& M.D.\ Plummer}
\authorspace
\bsgitem{1986a}{}  \emph{Matching Theory}.  North-Holland Math.\ Stud., Vol.\ 121.  Ann.\ Discrete Math., Vol.\ 29.  Akad\'emiai Kiad\'o, Budapest, and North-Holland, Amsterdam, 1986.  
\MR{0859549} (88b:90087).  \Zbl{618.05001}.						
\annot{} Pp.~247--248:  Shortest odd/even $uv$-path problem in $\Gamma$.  Lemma 6.6.9 reduces min length of odd path to a min-weight perfect matching problem in a modified graph.  Exerc.\ 6.6.10--11 are similar for even paths and odd/even circles.  [\Problem.  Generalize to negative/positive paths and circles in signed graphs.]
\S6.6, p.\ 252:  $l(-\Gamma)$ [i.e., max cut in $\Gamma$], $l(\Sigma)$ for signed planar graphs.  Cor.\ 6.19:  For planar $\Gamma$, $l(-\Gamma) = \frac12($max number of circles in a 2-packing of negative circles).  [\Question:  How does this generalize to signed planar graphs?]
Pp.\ 252--253:  Odd-circle packing and 2-packing.  
\dated{10 Nov 2010} 
\par
\codes{\Csg,~\Cpar:~\Cfr,~\CPaths,~\CCircles:~\CExp}%
\annotctd{} \S8.7, pp.\ 353--354:  Weighted non-ferromagnetic Ising model.
\dated{10 Nov 2010} 
\codes{\CSG,~\CWG:~\CPhys,~\Cfr:~\CExp}%
\itemspace

\bsgitem{2009a}{}  \emph{Matching Theory}.  AMS Chelsea Publ.\ (Amer.\ Math.\ Soc.), Providence, R.I., 2009.
\MR{2536865}.  \Zbl{618.05001}.
\annot{} Reprint of \xrefd{1986a}{L.\ Lovasz \& M.D.\ Plummer} with errata and an appendix of updates.
\dated{10 Nov 2010}
\par
\codes{\Csg:~\Cpar:~\CCircles,~\CPaths,~\Cfr:~\CExp}\codesctd{\Csg,~\CWG:~\CPhys,~\Cfr:~\CExp}%
\itemspace

\author{L.\ Lov\'asz, L.\ Pyber, D.J.A.\ Welsh, \& G.M.\ Ziegler}
\authorx{L.\ Lovasz, L.\ Pyber, D.J.A.\ Welsh, \& G.M.\ Ziegler}
\authorspace
\bsgitem{1995a}{}  Combinatorics in pure mathematics.  
In:  R.L.\ Graham, M.\ Gr\"otschel, and L.\ Lov\'asz, eds., \emph{Handbook of Combinatorics}, Vol.\ II, Ch.\ 41, pp.\ 2039--2082.  
North-Holland (Elsevier), Amsterdam, and MIT Press, Cambridge, Mass., 1995.  
\MR{1373697} (97f:00003).  \Zbl{851.52017}.
\annot{} \S7:  ``Knots and the Tutte polynomial'', considers the signed graph of a knot diagram (pp.\ 2076--77).
\codes{\CSGc:~\CKnot}%
\itemspace

\author{Aidong Lu}							%LU AD
\authorspace
\authref  See \hyperlink{Yuemeng Li}{Y.-M.~Li} and \hyperlink{Leting Wu}{L.T.~Wu}.
\itemspace

\author{Lingfei Lu}							%LU LF
\authorspace
\authref  See \hyperlink{Min Zhu}{M.~Zhu}.
\itemspace

\authorx{Lu Lu}							%LU L
\author{Lu Lu, Jianfeng Wang, \& Qiongxiang Huang}
\authorspace
\bsgitem{2021a}{}  Complex unit gain graphs with exactly one positive eigenvalue.  \emph{Linear Algebra Appl.}\ 608 (2021), 270--281.
\MR{4152791}.  \Zbl{1458.05148}.  
\codes{\CGG:~\CAdj:~\CEig,~\CSw}%
\itemspace

\author{Mei Lu}							%LU M
\authorspace
\authref  See \hyperlink{Huiqing Liu}{H.Q.~Liu} and \hyperlink{Wenjie Ning}{W.J.~Ning}.
\itemspace

\author{Yong Lu}							%LU Y
\authorspace
\authref  See also \hyperlink{Qi Wu}{Q.~Wu} and \hyperlink{Li Zhang}{L.~Zhang}.
\authrefspace

\author{Yong Lu, Ligong Wang, \& Peng Xiao}
\authorspace
\bsgitem{2017a}{}  Complex unit gain bicyclic graphs with rank 2, 3 or 4.  \emph{Linear Algebra Appl.}\ 523 (2017) 169--186.
\MR{3624673}.  \Zbl{1370.05139}.  \arXiv{1511.07589}.				
\annot{} Rank of $A(\Phi)$.
\codes{\CGG:~\CAdj}%
\itemspace

\author{Yong Lu, Ligong Wang, \& Qiannan Zhou}
\authorspace
\bsgitem{2017a}{}  Hermitian-Randi\'c matrix and Hermitian-Randi\'c energy of mixed graphs.  \emph{J.\ Inequalities Appl.}\  2017 (2017), paper no.\ 54, 14 pp.
\MR{3619588}.  \Zbl{1358.05177}.  \arXiv{610.09783}.				
\annot{} Normalized adjacency matrix with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.
\codes{\Cgg:~\CAdj}%
\itemspace

\bsgitem{2018a}{}  The rank of a signed graph in terms of the rank of its underlying graph.  \emph{Linear Algebra Appl.}\ 538 (2018), 166--186.
\MR{3722834}.  \Zbl{1374.05133}.  
\annot{} Rank of $A(\Sigma)$ vs.\ $A(|\Sigma|)$.
\codes{\CSG:~\CAdj}%
\itemspace

\bsgitem{2018b}{}  Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph.  \emph{Filomat} 32 (2018), no.\ 4, 1303--1312.
\MR{3848106}.  
\codes{\Cgg:~\CAdj}%
\itemspace

\bsgitem{2019a}{}  The rank of a complex unit gain graph in terms of the rank of its underlying graph.  \emph{J.\ Combin.\ Optim.}\ 38 (2019), 570--588.
\MR{3969335}.  \Zbl{1420.05087}.  \arXiv{1711.11448}.				
\annot{} Rank of $A(\Phi)$ vs.\ $A(\|\Phi\|)$.
\codes{\CGG:~\CAdj}%
\itemspace

\author{Yong Lu \& Jingwen Wu}
\authorspace
\bsgitem{2021a}{}  No signed graph with the nullity $\eta(G,\sigma) = |V(G)| - 2m(G) + 2c(G) - 1$.  \emph{Linear Algebra Appl.}\ 615 (2021), 175--193.
\MR{4200821}.  \Zbl{1459.05112}.  \arXiv{2006.00002}.  
\annot{} $\rk A(\Sigma) + 2\xi  - 2\mu = s$ is impossible for $s=1$, where $\mu :=$ matching number, $\xi :=$ cyclomatic number.  Fixing $\xi$, infinitely many $\Sigma$ exist for which $s \in \{0\} \cup [2,3\xi]$.
\dated{6 Feb 2021} 
\codes{\CSG:~\CAdj}%
\itemspace

\bsgitem{2021b}{}  Bounds for the rank of a complex unit gain graph in terms of its maximum degree.  \emph{Linear Algebra Appl.}\ 610 (2021), 73--85.
\MR{4159284}.  \Zbl{1458.05149}.  
\annot{} $\rk A(\Phi) \geq n/\Delta(G)$.  Equality is characterized.
\dated{4 Jun 2022} 
\codes{\CGG:~\CAdj}%
\itemspace

\author{Yong Lu \& Qi Wu}
\authorspace
\bsgitem{20xxa}{}  Inertia indices of a complex unit gain graph in terms of matching number.  \emph{Linear Multilinear Algebra} (to appear).
\arXiv{2108.01443}.  
\annot{} \Cf\ \xref{He, Hao, and Dong}{2020a}{Shengjie He, Rong-Xia Hao, \& Fengming Dong}.  Four inequalities and characterizations of equality.
\dated{4 Jun 2022} 
\codes{\CGG:~\CAdj}%
\itemspace

\author{Yong Lu \& Wei-Ru Xu}
\authorspace
\bsgitem{2021a}{}  An lower bound of the rank of a signed graph in terms of order and maximum degree.  \emph{ScienceAsia}\ 47 (2021), no.\ 6, 779--784.
\annot{} Thms.:  $\rk A(K_{a,b},\sigma) = 2$ iff $(K_{a,b},\sigma)$ is balanced.  $\rk A(\Sigma) \geq n/\Delta$, $=$ iff $\Sigma \sim k$ copies of $+K_{\Delta,\Delta}$.  $\rk A(\Sigma) = (n+1)/\Delta$ iff $\Sigma$ is the preceding with one missing vertex.
\dated{4 Jun 2022} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{You Lu}							%LU Y
\authorspace
\authref  See also \hyperlink{Jian Cheng}{J.-A.~Cheng}, \hyperlink{Matt DeVos}{M.~DeVos}, and \hyperlink{Xiao Wang}{X.~Wang}.
\authrefspace

\author{You Lu, Rong Luo, Michael Schubert, Eckhard Steffen, \& Cun-Quan Zhang}
\authorspace
\bsgitem{2020a}{}  Flows on signed graphs without long barbells.  \emph{SIAM J.\ Discrete Math.}\ 34 (2020), no.\ 4, 2166--2182. 
\MR{4164490}.  \Zbl{1450.05037}.  \arXiv{1908.11004}.  
\annot{} In many ways they behave like flows on unsigned graphs.	
\codes{\CSG:~\CFlows}%
\itemspace

\author{You Lu, Rong Luo, \& Cun-Quan Zhang}
\authorspace
\bsgitem{2018a}{}  Multiple weak 2-linkage and its applications on integer flows of signed graphs.  \emph{European J.\ Combin.}\ 69 (2018), 36--48.
\MR{3738139}.  \Zbl{1376.05066}.  
\annot{} $\Sigma$ without edge-disjoint negative circles has a nowhere-zero 6-flow, satisfying \xref{Bouchet's}{1983a}{Andre Bouchet} conjecture.
\dated{26 Nov 2020} 
\codes{\CSG:~\CFlows}%
\itemspace

\author{Alexander Lubotzky}						%LUBOTZKY
\authorspace
\authref  See \hyperlink{Tali Kaufman}{T.~Kaufman}.
\itemspace

\author{Claire Lucas}							%LUCAS
\authorspace
\authref  See \hyperlink{Mustapha Aouchiche}{M.~Aouchiche} and \hyperlink{Pierre Hansen \& Claire Lucas}{P.~Hansen}.
\itemspace

\author{Robert R.\ Lucchese}						%LUCCHESE
\authorspace
\authref   See \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\author{Henri Luchian}						%LUCHIAN
\authorspace
\authref  See \hyperlink{Andrei Bautu, Elena Bautu, \& Henri Luchian}{A.~B\u{a}utu}.
\itemspace

\author{Tomasz {\L}uczak}						%LUCZAK
\authorx{Tomasz Luczak}
\authorspace
\authref  See also \hyperlink{Ervin Gyori}{E.~Gy\"ori}.
\authrefspace

\bsgitem{2016a}{}  Highly connected monochromatic subgraphs of two-colored complete graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 117 (2016), 88--92.
\MR{3437613}.  \Zbl{1329.05172}.  
\annot{} Thm.\ If $2\leq k \leq (n+3)/4$, then $\Sigma=(K_n,\sigma)$ contains a $k$-connected homogeneously signed subgraph of order $> n-2(k-1)$ or $k$-connected all-positive and all-negative subgraphs of order $n-2(k-1)$.  Completes work of \xref{Bollob\'as and Gy\'arf\'as}{2008a}{Bela Bollobas \& Andras Gyarfas} (who conjectured most of this), \xref{Liu, Morris, and Prince}{2009a}{Henry Liu, Robert Morris, \& Noah Prince}, and \xref{Fujita and Magnant}{2011a}{Shinya Fujita \& Colton Magnant}.
\dated{24 Jan 2016} 
\codes{\Csg:~\CStr}%
\itemspace

\author{Mark Ludwig}						%LUDWIG
\authorspace
\authref  See also \hyperlink{Peter Abell}{P.~Abell} and \hyperlink{Bernard Kujawski}{B.~Kujawski}.
\authrefspace

\author{M.\ Ludwig \& P.\ Abell}
\authorspace
\bsgitem{2007a}{}  An evolutionary model of social networks.  \emph{European Phys.\ J.\ B} 58 (2007), 97--105.
\annot{} Signed edges are added to and deleted from a fixed set of nodes under a balancing rule.  Imbalance measured by frustrated triangles impels evolution, which converges under some conditions.
	
\dated{20 Jun 2011} 
\codes{\CSG:~\CBal,~\CFr:~\CDyn}%
\itemspace

\authorx{J.\ Lukic}							%LUKIC
\author{J.\ Lukic, A.\ Galluccio, E.\ Marinari, O.C.\ Martin, \& G.\ Rinaldi}
\authorspace
\bsgitem{2004a}{}  Critical thermodynamics of the two-dimensional $\pm J$ Ising spin glass.  \emph{Phys.\ Rev.\ Lett.}\ 92 (2004), no.\ 11, \#117202.
\arXiv{cond-mat/0309238}.  
\annot{} Physical properties of a signed toroidal square lattice graph, from computation of the exact partition function (energy distribution) via \xref{Galluccio, Loebl, and Vondr\'ak}{2000a}{Anna Galluccio, Martin Loebl, \& Jan Vondrak}, \xrefd{2001a}{Anna Galluccio, Martin Loebl, \& Jan Vondrak}.  E.g., the approximate proportion of negative edges is important.
\dated{18 Aug 2012} 
\par
\codes{\CSG:~\CPhys,~\CFr}%
\itemspace

\author{Robert Lukot'ka}						%LUKOTKA
\authorspace
\authref  See \hyperlink{Tomas Kaiser}{T.~Kaiser}.
\itemspace

\author{J.\ Richard Lundgren} 						%LUNDGREN
\authorspace
\authref  See \hyperlink{Harvey J.\ Greenberg}{H.J.~Greenberg} and \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{Thomas J.\ Lundy}						%LUNDY
\authorspace
\authref  See also \hyperlink{George M.\ Lady}{G.M.~Lady}.
\authrefspace

\author{Thomas J.\ Lundy, John Maybee, \& James Van Buskirk}
\authorspace
\bsgitem{1996a}{}  On maximal sign-nonsingular matrices.  \emph{Linear Algebra Appl.}\ 247 (1996), 55--81.
\MR{1412740} (97k:15020) (\qv).  \Zbl{862.15019}.					
\annot{} Constructions of such matrices.  A matrix definition of $C_4$-cockades.
\dated{6 Mar 2011} 
\codes{\CSG:~\CSol}%
\itemspace

\author{Feng Luo}							%LUO F
\authorspace
\authref  See \hyperlink{Haotian Wang}{H.-T.~Wang}.
\itemspace

\author{Guoqiang Luo}						%LUO GQ
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Rong Luo}							%LUO R
\authorspace
\authref  See \hyperlink{Jian Cheng}{J.-A.~Cheng}, \hyperlink{Matt DeVos}{M.~DeVos}, \hyperlink{You Lu}{Y.~Lu}, \hyperlink{Xingqin Qi}{X.Q.~Qi}, \hyperlink{Li Zhang}{L.~Zhang}, and \hyperlink{Xiao-Dong Zhang}{X.D.~Zhang}.
\itemspace

\author{Yeung-Long Luo}						%LUO YL
\authorspace
\authref   See \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Shyi-Long Lee}{S.L.~Lee}.
\itemspace

\author{Bob Lutz}							%LUTZ B
\authorspace
\bsgitem{2019a}{}  \emph{Electrical Networks, Hyperplane Arrangements and Matroids.}  Ph.D. thesis, University of Michigan, 2019.
\MR{4071609} (no rev).  
\codes{\Cgg:~\Cmatrd,~\CGeom}%
\itemspace

\bsgitem{2019b}{}  Electrical networks and hyperplane arrangements.  \emph{Adv.\ Appl.\ Math.}\ 110 (2019), 375--402.
\MR{3990769}.  \Zbl{1427.52017}.  \arXiv{1709.01227}.				
\annot{} ``Dirichlet hyperplane arrangements'' are a kind of gain-graphic arrangement; \cf\ \xrefd{2022a}{Bob Lutz}.
	
\dated{16 Nov 2018} 
\codes{\Cgg:~\Cmatrd,~\CGeom}%
\itemspace

\bsgitem{2019c}{}  Koszulness and supersolvability for Dirichlet arrangements.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 147 (2019), no.\ 11, 4937--4947. 
\MR{4011525}.  \Zbl{1427.52016}.  \arXiv{1810.03518}.				
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{2022a}{}  Matroids arising from electrical networks.  \emph{Adv.\ Appl.\ Math.}\ 137 (2022), art.\ 102331, 32 pp. 
\MR{4388465}.  \Zbl{1486.05038}.  \arXiv{1809.10100}.				
\annot{} ``Dirichlet matroids'' are complete lift matroids of a kind of biased graph.  Dirichlet arrangements \xrefd{2019b}{Bob Lutz} are hyperplane representations of them.
	
\dated{16 Nov 2018, 25 Oct 2019} 
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\author{Shengxiang Lv [Shengxiang Lyu]}					%LV SX
\authorx{Shengxiang Lv}
\authorspace
\bsgitem{2015a}{}  The largest demigenus over all signatures on $K_{3,n}$.  \emph{Graphs Combin.}\ 31 (2015), 169--181.
\MR{3293474}.  \Zbl{306.05086}.  
\annot{} For $n\geq3$, $D(K_{3,n})):=\max_\sigma d(K_{3,n},\sigma)=2\lfloor\frac14(n-2)\rfloor+0$ if $n\equiv3\ (\mod 4)$, $+1$ otherwise ($d=$ demigenus, ``Euler genus'').  Most interesting feature:  The maximum is attained (not uniquely) with only a single negative edge.  
\dated{7 Nov 2017}.
\codes{\CSG:~\CTop}%
\itemspace

\author{Shengxiang Lv \& Zihan Yuan}
\bsgitem{2018a}{}  The smallest surface that contains all signed graphs on $K_{4,n}$.  \emph{Discrete Math.}\ 341 (2018), 732--747.
\MR{3754386}.  \Zbl{1378.05084}.  
\annot{} Thm.: $D(K_{4,n}):=\max_\sigma d(K_{4,n}, \sigma)=d(K_{4,n})+1$ for $n>4$, $D(K_{4,4})=4$.  As with $K_{2,n}$ and $K_{3,n}$ (\cf\ \xref{Lv}{2015a}{Shengxiang Lv}), the maximum is attained with only one negative edge, except for $K_{4,4}$, where the maximum requires a negative perfect matching.
\Question.  Does this pattern continue for $K_{m,n}$, $m>4$?
\dated{7 Nov 2017}
\codes{\CSG:~\CTop}%
\itemspace

\authorx{Shengxiang Lyu}						%LYU SX
\author{Shengxiang Lyu [Shengxiang Lv]}
\authorspace
\authref  See \hyperlink{Shengxiang Lv}{S.-X.~Lv}.
\itemspace

\author{Yu.I.\ Lyubich}						%LYUBICH
\authorspace
\authref  See \hyperlink{G.R.\ Belitskii}{G.R.~Belitskii}.
\itemspace


\label{LETTER M} \hypertarget{LETTER M}{}	% LETTER  M


\author{Baoli Ma}							%MA BL
\authorspace
\authref  See \hyperlink{Mingjun Du}{M.J.~Du}.
\itemspace

\author{Hongping Ma}						%MA HP
\authorspace
\authref  See also \hyperlink{Longqin Wang}{L.Q.~Wang} and \hyperlink{Wei Wei}{W.~Wei}.
\authrefspace

\bsgitem{2009a}{}  Bounds on the local bases of primitive, non-powerful, minimally strong signed digraphs.  \emph{Linear Algebra Appl.}\ 430 (2009),  no.\ 2-3, 718--731.
\MR{2473178} (2009i:\-05100).  \Zbl{1151.05020}.					
\codes{\CSD:~\CAdj,~\Cqm}%
\itemspace

\author{Jun Ma}							%MA J
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Lijia Ma}							%MA LJ
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai}.
\itemspace

\author{Hongping Ma \& Zhengke Miao}
\authorspace
\bsgitem{2011a}{}  Imprimitive non-powerful sign pattern matrices with maximum base.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 4, 371--390.
\MR{2802520} (2012k:05170).  \Zbl{1221.15042}.  
\codes{\CQM:~\CSD:~\CAdj}%
\itemspace

\author{M.\ Ma}							%MA M
\authorspace
\authref  See \hyperlink{Daniel Blankschtein}{D.~Blankschtein}.
\itemspace

\authorx{Xiaobin Ma}							%MA XB
\author{Xiaobin Ma, Genhong Ding, \& Long Wang}
\authorspace
\bsgitem{20xxa}{}  On the nullity and the matching number of unicyclic signed graphs.  Submitted.  
\annot{} Further develops \xref{Y.Z.~Fan, Wang, and Wang}{2013a}{Yi-Zheng Fan, Yue Wang, \& Yi Wang}.  Employs the matching number to express the adjacency nullity and to characterize adjacency rank $6$ and $7$ of a signed unicyclic graph.
\dated{17 Dec 2011} 
\codes{\CSG:~\CEig}%
\itemspace

\author{Xiaobin Ma, Dein Wong, \& Fenglei Tian}
\authorspace
\bsgitem{2016a}{}  Skew-rank of an oriented graph in terms of matching number.  \emph{Linear Algebra Appl.}\ 495 (2019), 242--255.
\MR{3462998}.  \Zbl{1331.05181}.  
\annot{} $\Phi :=$ gain graph with gains $\pm\mathrm{i}$.  Problem: $\rk A(\Phi)$ [special case of adjacency rank for complex unit gain graphs; \cf\ \xref{Reff}{2012a}{Nathan Reff}].  Thm.\ 3.1(i):  $2\mu -2\xi \leq \rk A(\Phi) \leq 2\mu$, where $\mu :=$ matching number of $\|\Phi\|$, $\xi :=$ cyclomatic number.  Thm.\ 4.1 characterizes $2\mu -2\xi = \rk A(\Phi)$.  [Generalized in \xref{Fenglei Tian, Li Chen, \& Rui Chu}{2018a}{Fenglei Tian, Li Chen, \& Rui Chu}.]
Dictionary:  $\mathrm{i}=\sqrt{-1}$, $G := \|\Phi\|$, $\vec G :=$ orientation making all gains $\mathrm{i}$, skew adjacency matrix of $\vec G :=  \mathrm{i}\inv A(\Phi)$, skew rank of $\vec G = \rk A(\Phi)$, $\sgn(C) = \phi(C)/\mathrm{i}^{|C|}$.
\dated{10 May 2019} 
\codes{\Cgg:~\CAdj,~\CSG}%
\itemspace

\authorx{Avi Ma'ayan}						%MAAYAN
\author{A.\ Ma'ayan, A.\ Lipshtat, R.\ Iyengar, \& E.D.\ Sontag}
\authorspace
\bsgitem{2008a}{}  Proximity of intracellular regulatory networks to monotone systems.  \emph{ET Systems Biol.}\ 2 (2008), no.\ 3, 103--112.
\codes{\CSD,~\CBiol:~\CDyn:~\CFr:~\CAlgor}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a}						%MACAJOVA
\authorx{Edita Macajova}
\authorspace
\authref  See also \hyperlink{Tomas Kaiser}{T.~Kaiser} and \hyperlink{A.\ Kompisova}{A.~Kompi\v{s}ov\'a}.
\authrefspace

\author{Edita M\'a\v{c}ajov\'a \& J\'an Maz\'ak}
\authorx{Edita Macajova \& Jan Mazak}
\authorspace
\bsgitem{2013a}{}  On even cycle decompositions of 4-regular line graphs.  \emph{Discrete Math.}\ 313 (2013), 1697--1699.
\MR{3061005}.  \Zbl{1277.05067}.  
\annot{} Includes positive-circle decomposition (called ``even cycle decomposition'') of a signed graph; \cf\ \xref{Markstr\"om}{2012a}{Klas Markstrom}.  Thm.\ 2:  An infinite class of 4-regular, 4-connected $\Sigma$ without such a decomposition.  \Question\ 1:  Does every signed line graph of a cubic graph without isthmus, with even $\#E^-$, have such a decomposition?
\dated{4 Jun 2017} 
\codes{\CSG:~\CCircles:~\CStr}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a, Andr\'e Raspaud, Edita Rollov\'a, \& Martin \v{S}koviera}
\authorx{Edita Macajova, Andre Raspaud, Edita Rollova, \& Martin Skoviera}
\authorspace
\bsgitem{2016a}{}  Circuit covers of signed graphs.  \emph{J.\ Graph Theory}\ 81 (2016), no.\ 2, 120--133.
\MR{3433634}.  \Zbl{1332.05066}.  
\annot{} 	
Dictionary:  ``signed circuit'' (not a circuit with signs) = sign circuit = frame circuit.  
Improved in \xref{Kaiser, Lukot'ka, \etal\!}{2019a}{Tomas Kaiser, Robert Lukotka, Edita Macajova, \& Edita Rollova}.
\codes{\CSG:~\CMatrd}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a, Andr\'e Raspaud, \& Martin \v{S}koviera}
\authorx{Edita Macajova, Andre Raspaud, \& Martin Skoviera}
\authorspace
\bsgitem{2014a}{}  The chromatic number of a signed graph.  In:  \emph{Bordeaux Graph Workshop 2014}, pp.\ 29--30.  LaBRI, Bordeaux, 2014.  
URL \url{http://bgw.labri.fr/2014/bgw2014-booklet.pdf}
\annot{} Extended abstract of \xrefd{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}.
\dated{19 Mar 2017} 
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{2016a}{\dag}  The chromatic number of a signed graph.  \emph{Electronic J.\ Combin.}\ 23 (2016), no.\ 1, art.\ P1.14, 10 pp.
\MR{3484719}.  \Zbl{1329.05116}.  \arXiv{1412.6349}.				
\annot{$\chi(\Sigma)$} 
Coloring is exactly as in \xref{Zaslavsky}{1982b}{Thomas Zaslavsky} [despite authors' statement to the contrary].  Chromatic number $\chi(\Sigma) :=$ size of smallest set of signed colors [a better definition than Zaslavsky's].  
\annotctd{} Main results:  Thm.\ 6 (Brooks' Theorem for signed simple graphs): $\chi(\Sigma) \leq \Delta(\Sigma)$ except for balanced $(K_n,\sigma)$ and $(C_{\text{odd}},\sigma)$ and unbalanced $(C_\text{even},\sigma)$.  
[\xref{Fleiner and Wiener}{2016a}{T.\ Fleiner \& G.\ Wiener} have a short list-coloring proof.  \xref{Zaj\c{a}c}{20xxa}{Mariusz Zajac} has a short, more general proof.]
Prop.\ 4(i):  $\Sigma\not\geq K_4 \implies \chi(\Sigma)\leq3$.  
Thm.\ 10:  If $|\Sigma|$ is planar, $\chi(\Sigma)\leq5$, $\leq 4$ if $C_3$-free, $\leq3$ if girth $\geq 5$.  
\Conjecture:  Every planar signed graph is 4-colorable.
\dated{14 Sept 2015, 7 May 2018, rev 9 Feb, 2 Jul 2020, 5 Jan 2021}
\par
\codes{\CSG:~\CCol:~\CInvar}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a \& Edita Rollov\'a}
\authorx{Edita Macajova \& Edita Rollova}
\authorspace
\bsgitem{2011a}{}  On the flow numbers of signed complete and complete bipartite graphs.  \emph{Electronic Notes Discrete Math.}\ 38 (2011), 591--596.
\Zbl{1274.05207}.  
\codes{\CSG:~\CFlows}%
\itemspace

\bsgitem{2015a}{}  Nowhere-zero flows on signed complete and complete bipartite graphs.  \emph{J.\ Graph Theory} 78 (2015), no.\ 2, 108--130.
\MR{3293079}.  \Zbl{1307.05096}.  
\codes{\CSG:~\CFlows}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a \& Martin \v{S}koviera}
\authorx{Edita Macajova \& Martin Skoviera}
\authorspace
\bsgitem{2011a}{}  Determining the flow numbers of signed eulerian graphs.  \emph{Electronic Notes Discrete Math.}\ 38 (2011), 585--590.
\Zbl{1274.05283}.  \arXiv{1408.1703}.  
\codes{\CSG:~\CFlows}%
\itemspace

\bsgitem{2015a}{}  Remarks on nowhere-zero flows in signed cubic graphs.  \emph{Discrete Math.}\ 338 (2015), no.\ 5, 809--815.
\MR{3303859}.  \Zbl{1306.05087}.  
\codes{\CSG:~\CFlows}%
\itemspace

\bsgitem{2016a}{}  Characteristic flows on signed graphs and short circuit covers.  \emph{Electronic J.\ Combin.}\ 223 (2016), no.\ 3, art.\ P3.30, 10 pp.
\MR{3558067}.  \Zbl{1344.05066}.  \arXiv{1407.5268}.				
\codes{\CSG:~\CFlows,~\Cmatrd}%
\itemspace

\bsgitem{2017a}{}  Odd decompositions of Eulerian graphs.  \emph{SIAM J.\ Discrete Math.}\ 31 (2017), no.\ 3, 1923--1936.
\MR{3691723}.  \Zbl{1370.05126}.  \arXiv{1607.00053}.				
\codes{\CSG:~\CFlows}%
\itemspace

\bsgitem{2017b}{}  Nowhere-zero flows on signed Eulerian graphs.  \emph{SIAM J.\ Discrete Math.}\ 31 (2017), no.\ 3, 1937--1952.
\MR{3691724}.  \Zbl{1370.05085}.  \arXiv{1408.1703}.				
\codes{\CSG:~\CFlows}%
\itemspace

\author{Edita M\'a\v{c}ajov\'a \& Eckhard Steffen}
\authorx{Edita Macajova \& Eckhard Steffen}
\authorspace
\bsgitem{2015a}{}  The difference between the circular and the integer flow number of bidirected graphs.  \emph{Discrete Math.}\ 338 (2015), no.\ 6, 866--867.
\MR{3318624}.  \Zbl{1371.05112}.  
\annot{} The difference can be $\geq 3$. It is at most $3$ if \xref{Bouchet's}{1983a}{Andre Bouchet} conjecture is true.
\dated{20 Jun 2022} 
\codes{\CSG:~\CFlows}%
\itemspace

\author{Ben D.\ MacArthur}						%MACARTHUR
\authorspace
\authref  See \hyperlink{Raffaella Mulas}{R.~Mulas}.
\itemspace

\author{Enzo Maccioni} 						%MACCIONI
\authorspace
\authref  See \hyperlink{Francisco Barahona}{F.~Barahona}.
\itemspace

\author{Gary MacGillivray}						%MACGILLIVRAY
\authorspace
\authref  See also \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}, \hyperlink{Etienne Leclerc}{E.~Leclerc}, and \hyperlink{Keivan Hassani Monfared}{K.~Hassani Monfared}.
\authrefspace

\author{Gary MacGillivray, Ben Tremblay, \& Jacqueline M.\ Warren}
\authorspace
\bsgitem{20xxa}{}  Colourings of $m$-edge-coloured graphs and switching.  Submitted.  
\annot{} Great generalization of \xref{Brewster and Graves}{2009a}{Richard C.\ Brewster \& Timothy Graves}.
\codes{\Cgg(\CGen),~\CCov}%
\itemspace

\author{Amila P.\ Macodi-Ringia}					%MACODI-RINGIA
\authorspace
\authref  See \hyperlink{Mahid M.\ Mangontarum}{M.M.~Mangontarum}.
\itemspace

\author{Jayakrishnan Madathil}						%MADATHIL
\authorspace
\authref  See \hyperlink{Avinandan Das}{A.~Das}.
\itemspace

\author{Bolette Ammitzb{\o}ll Madsen}					%MADSEN
\authorspace
\authref  See \hyperlink{Jesper Makholm Byskov}{J.M.~Byskov}.
\itemspace

\author{K.V.\ Madhusudhan}						%MADHUSUDHAN
\authorspace
\authref  See also \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\authrefspace

\author{K.V.\ Madhusudhan \& S.\ Vijay}
\authorspace
\bsgitem{2018a}{}  Distance divisor signed graphs.  \emph{IOSR J.\ Engineering} 8 (2018), no.\ 10, 25--27.
\annot{} $DD(|\Sigma|)$ is signed using the canonical vertex signature, by $\sigma_{DD}(uv) = \mu_\sigma(u)\mu_\sigma(v)$.
\dated{26 May 2021} 
\codes{\CSG,~\CVS,~\CSw}%
\itemspace

\bsgitem{2019a}{}  A note on detour radial signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2 (2019), 112--116.
\annot{} $DR(|\Sigma|)$ is signed by $\sigma_{DR}(uv)=\mu_\sigma(u)\mu_\sigma(v)$.
\dated{12 Jul 2019} 
\par
\codes{\CSG,~\CVS}%
\itemspace

\author{[A.\ El Maftouhi, Abdelakim El Maftouhi]}				%MAFTOUHI
\authorspace
\authref  See \hyperlink{Abdelhakim El Maftouhi}{A.~El Maftouhi} (under `E').
\itemspace

\author{Mohammad Maghasedi}					%MAGHASEDI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}, \hyperlink{Soudabeh Dalvandi}{S.~Dalvandi}, \hyperlink{N.\ Kafai}{N.~Kafai}, and \hyperlink{Mona Souri}{M.~Souri}.
\itemspace

\author{Colton Magnant}						%MAGNANT
\authorspace
\authref  See \hyperlink{Shinya Fujita}{S.~Fujita}.
\itemspace

\author{Thomas L.\ Magnanti}						%MAGNANTI
\authorspace
\authref   See \hyperlink{Ravindra K.\ Ahuja}{R.K.~Ahuja}.
\itemspace

\author{N.V.R.\ Mahadev} 						%MAHADEV
\authorspace
\authref  See also \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}.
\authrefspace

\author{N.V.R.\ Mahadev \& U.N.\ Peled}
\authorspace
\bsgitem{1995a}{}  \emph{Threshold Graphs and Related Topics}.  Ann.\ Discrete Math., Vol.\ 56.  North-Holland, Amsterdam, 1995.  
\MR{1417258} (97h:05001).  \Zbl{950.36502}.
\annot{} \S8.3: ``Bithreshold graphs'' (from \xref{Hammer and Mahadev}{1985a}{P.L.\ Hammer \& N.V.R.\ Mahadev}), and \S8.4: ``Strict 2-threshold graphs''  (from \xref{Hammer, Mahadev, and Peled}{1989a}{Peter L.\ Hammer, N.V.R.\ Mahadev, \& Uri N.\ Peled}), characterize two types of threshold-like graph.  In each, a different signed graph $H$ is defined on $E(\Gamma)$ so that $\Gamma$ is of the specified type iff $H$ is balanced.  (The negative part of $H$ is the ``conflict graph'', $\Gamma^*$.)  The reason is that one wants $\Gamma$ to decompose into two subgraphs, and the subgraphs, if they exist, must be the two parts of the Harary bipartition of $H$.  [Thus one also gets a fast recognition algorithm, though not the fastest possible, for the desired type from the fast recognition of balance.]
\codes{\CSG:~\CBal:~\CAppl}%
\annotctd{} \S8.5: ``Recognizing threshold dimension $2$.''  Based on \xref{Raschle and Simon}{1995a}{Thomas Raschle \& Klaus Simon}.  Given: $\Gamma \subseteq K_n$ such that $\Gamma^*$ is bipartite.  Orient $-K_n$ so that $\Gamma$-edges are introverted and the other edges are extroverted.  Their ``alternating cycle'' is a coherent closed walk in this orientation.  Let us call it ``black'' (in a given black-white proper coloring of $\Gamma^*$) if its $\Gamma$-edges are all black.  
Thm.\ 8.5.2 (\xref{Hammer, Ibaraki, and Peled}{1981a}{P.L.\ Hammer, T.\ Ibaraki, \& U.\ Peled}):  If there is a black coherent closed walk in $E_0$, then there is a coherent tour (closed trail) of length $6$ (which is a pair of joined triangles or a hexagon---their $AP_5$ and $AP_6$).  Thm.\ 8.5.4:  Given that there is no black coherent hexagon, one can recolor quickly so there is no black coherent $6$-tour.  Thm.\ 8.5.9:  Given that there is no `double' coherent hexagon (the book's ``double $AP_6$''), one can recolor quickly so there is no black coherent hexagon.  Thm.\ 8.5.28:  Any $2$-coloring of $\Gamma^*$ can be quickly transformed into one with no `double' coherent hexagon.  
[\Question.  Can any of this, especially Thm.\ 8.5.2, be generalized to arbitrary oriented all-negative graphs $\Beta$?  Presumably, this would require first defining a conflict graph on the introverted edges of $\Beta$.  More remotely, consider generalizing to bidirected complete or arbitrary graphs.]
\codes{\Cpar:~\Cori,~\CAlgor}%
\annotctd{} \S9.2.1: ``Threshold signed graphs.''  See \xref{Benzaken, Hammer, and de Werra}{1981a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}, \xrefd{1985a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}.  In this version it's not clear where the signs are! (and their role is trivial).  Real weights are assigned to the vertices and an edge receives the sign of the weight product of its endpoints.
\par
\codes{\Csg:~\Cbal}%
\itemspace

\bsgitem{1988a}{}  Strict 2-threshold graphs.  \emph{Discrete Appl.\ Math.}\ 21 (1988), 113--131.
\MR{0959424} (89i:05234).  \Zbl{0658.05063}.  
\annot{} Uses the auxiliary signed graph of \xref{Hammer and Mahadev}{1985a}{P.L.\ Hammer \& N.V.R.\ Mahadev}.
\dated{22 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal}%
\itemspace

\authorx{John Maharry}						%MAHARRY
\author{John Maharry, Neil Robertson, Vaidy Sivaraman, \& Daniel Slilaty}
\authorspace
\bsgitem{2017a}{}  Flexibility of projective-planar embeddings.  \emph{J.\ Combin.\ Theory Ser.\ B} 122 (2017), 241--300.
\MR{3575205}.  \Zbl{1350.05020}.  
\annot{} \S1:  The flexibility is connected to duality of signed-graphic frame matroids by \xref{Slilaty}{2005a}{Daniel C.\ Slilaty}.
\dated{20 Dec 2011} 
\codes{\CSG:~\CTop}%
\itemspace

\author{Ali Ridha Mahjoub} 						%MAHJOUB
\authorspace
\authref  See \hyperlink{Francisco Barahona}{F.~Barahona} and D.~Cornaz.
\itemspace

\author{J.M.\ Maillard} 						%MAILLARD
\authorspace
\authref  See \hyperlink{T.\ Garel}{T.~Garel} and \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\itemspace

\author{Hamid Reza Maimani}						%MAIMANI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari} and \hyperlink{Ebrahim Ghorbani}{E.~Ghorbani}.
\itemspace

\author{[Leila Parsaei Majd]}						%MAJD
\authorspace
\authref  See \hyperlink{Leila Parsaei-Majd}{L.~Parsaei-Majd} (under `P').
\itemspace

\author{Arezoo Majidi}						%MAJIDI
\authorspace
\authref  See \hyperlink{Ebrahim Ghorbani}{E.~Ghorbani}.
\itemspace

\author{Konstantin Makarychev}					%MAKARYCHEV K
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{Yury Makarychev}						%MAKARYCHEV Y
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{J.A.\ Makowsky}						%MAKOWSKY
\authorspace
\authref  See also \hyperlink{E.\ Fischer}{E.~Fischer} and \hyperlink{Martin Lotz}{M.~Lotz}.
\authrefspace

\bsgitem{2001a}{}  Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width.  In:  \emph{Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms} (Washington, D.C., 2001), pp.\ 487--495.  Soc.\ for Industrial and Appl.\ Math., Philadelphia, Pa., 2001.
\MR{1958441} (no rev).  \Zbl{988.05087}.  
\annot{} See \xrefd{2005a}{J.A.\ Makowsky}.
\codes{\CSGc:~\CInvar:~\CAlgor}%
\itemspace

\bsgitem{2005a}{}  Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width.  \emph{Discrete Appl.\ Math.}\ 145 (2005), no.\ 2, 276--290.
\MR{2113147} (2005m:05214).  \Zbl{1084.05505}.  
\annot{} Polynomial-time computability for edge-colored graphs of bounded tree width.  [Also see \xref{Traldi}{2006a}{Lorenzo Traldi}.]
\codes{\CSGc:~\CGen:~\CInvar:~\CAlgor,~\CKnot}%
\itemspace

\author{A.Z.\ Maksymowicz}						%MAKSYMOWICZ
\authorspace
\authref  See \hyperlink{M.J.\ Krawczyk}{M.J.~Krawczyk}.
\itemspace

\author{Krzysztof Malarz}						%MALARZ
\authorspace
\authref  See \hyperlink{M.J.\ Krawczyk}{M.J.~Krawczyk} and \hyperlink{Bosiljka Tadic}{B.~Tadi\'c}.
\itemspace

\authorx{H.A.\ Malathi}						%MALATHI
\author{H.A.\ Malathi \& H.C.\ Savithri}
\authorspace
\bsgitem{2010a}{}  A note on jump symmetric $n$-sigraph.  \emph{Int.\ J.\ Math.\ Combin.}\ 2010 (2010), vol.\ 2, 65--67.
\Zbl{1216.05051}.  
\codes{\CSG(\CGen):~\CLG}%
\itemspace

\authorx{M.\ Malek-Zavarei}		 				%MALEK-ZAVAREI
\author{M.\ Malek-Zavarei \& J.K.\ Aggarwal}
\authorspace
\bsgitem{1971a}{}  Optimal flow in networks with gains and costs.  \emph{Networks} 1 (1971), 355--365.  
\MR{0295831} (45 \#4896).  \Zbl{236.90026}.
\codes{\CGN:~\Cbal}%
\itemspace

\author{Piotr Malicki}							%MALICKI
\authorspace
\authref  See \hyperlink{Mariusz Kaniecki}{M.~Kaniecki}.
\itemspace

\author{G\"unther Malle}						%MALLE
\authorx{Gunther Malle}
\authorspace
\bsgitem{1982a}{}  On maximum bipartite subgraphs.  \emph{J.\ Graph Theory} 6 (1982), 105--113.
\MR{655195} (83i:05047).  \Zbl{493.05035}.					
\annot{} Frustration index $l(-\Gamma)$.  Criterion for unique minimum deletion set in $-\Gamma$.  All-negative Petersen and dodecahedron characterized via minimum deletion sets.  [\Problem.  Extend to other signed graphs.]
\dated{28 Jun 2022} 
\codes{\Csg:~\Cpar:~\CFr}%
\itemspace

\authorx{Fragkiskos D.\ Malliaros}					%MALLIAROS
\author{Fragkiskos D.\ Malliaros, Christos Giatsidis, Apostolos N.\ Papadopoulos, \& Michalis Vazirgiannis}
\authorspace
\bsgitem{2020a}{}  The core decomposition of networks: theory, algorithms and applications.  \linebreak \emph{VLDB J.}\ 29 (2020), 61--92.
\HAL{hal-01986309}.						
\annot{} \S2.2.3, ``Signed graphs'', defines ``core'' := maximum subgraph with $d^+_\textrm{in,out}$, $d^-_\textrm{in,out} \geq$ given bounds.
\dated{2 Feb 2020} 
\codes{\CSD}%
\itemspace

\author{Sudipta Mallik}						%MALLIK
\authorspace
\authref  See also \hyperlink{Ryan Hessert}{R.~Hessert} and \hyperlink{Jerad Ipsen}{J.~Ipsen}.
\authrefspace

\bsgitem{20xxa}{}  Matrix tree theorem for the net Laplacian matrix of a signed graph.  \emph{Linear Multilinear Algebra} (to appear).
\arXiv{2210.03711}.  
\annot{} The theorem arises by introducing $\sqrt{-1}$ into the incidence matrix of an orientation of $|\Sigma|$, not $\Sigma$.  Unusually for oriented $|\Sigma|$, it is not about digraph properties.
\dated{28 Dec 2022} 
\codes{\CSG:~\CIncid,~\CLap(\CGen)}%
\itemspace

\author{R.B.\ Mallion}						%MALLION
\authorspace
\authref   See \hyperlink{A.C.\ Day, R.B.\ Mallion, \& M.J.\ Rigby}{A.C.~Day}.
\itemspace

\author{Devlin Mallory}						%MALLORY
\authorspace
\authref  See also \hyperlink{John Brown}{J.~Brown}.
\authrefspace

\author{Devlin Mallory, Abigail Raz, Christino Tamon, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2013a}{}  Which exterior powers are balanced?  \emph{Electronic J.\ Combin.}\ 20 (2013), no.\ 2, art.\ P43, 14 pp.
\MR{3084585}.  \Zbl{1266.05047}.  \arXiv{1301.0973}.				
\codes{\CSG:~\CBal}%
\itemspace

\authorx{C.L.\ Mallows} 						%MALLOWS
\author{C.L.\ Mallows \& N.J.A.\ Sloane}
\authorspace
\bsgitem{1975a}{}  Two-graphs, switching classes and Euler graphs are equal in number.  \emph{SIAM J.\ Appl.\ Math.}\ 28 (1975), 876--880. 
\MR{0427128} (55 \#164).  \Zbl{275.05125}, (\Zbl{297.05129}).				
\annot{} Thm.\ 1: For all $n$, the number of unlabelled two-graphs of order $n$ [i.e., switching isomorphism classes of signed $K_n$'s] equals the number of unlabelled even-degree simple graphs on $n$ vertices.  The key to the proof is that a permutation fixing a switching class fixes a signing in the class.  (\xref{Seidel}{1974a}{J.J.\ Seidel} proved the odd case, where the fixing property is simple.)  Thm.\ 2: The same for the labelled case.  [More in \xref{Cameron}{1977b}{Peter J.\ Cameron}, \xref{Cameron and Wells}{1986a}{P.J.\ Cameron \& Albert L.\ Wells, Jr.}, \xref{Cheng and Wells}{1984a}{Ying Cheng \& Albert L.\ Wells, Jr.}, \xrefd{1986a}{Ying Cheng \& Albert L.\ Wells, Jr.}]
\annotctd{} To prove the fixing property they find the conditions under which a given permutation $\pi$ of $V(K_n)$ and switching set $C$ fix some signed $K_n$.  [More in \xref{Harries and Liebeck}{1978a}{David Harries \& Hans Liebeck}, \xref{Liebeck}{1982a}{Martin W.\ Liebeck}, and \xref{Cameron}{1977b}{Peter J.\ Cameron}.]
\codes{\CTG:~\CAut,~\CEnum}%
\itemspace

\author{Aleksander Malni\v{c}}						%MALNIC
\authorx{Aleksander Malnic}
\authorspace
\authref  See also \hyperlink{Istvan Kovacs}{I.~Kov\'acs}.
\authrefspace

\bsgitem{1998}{}  Group actions, coverings and lifts of automorphisms.  \emph{Discrete Math.}\ 182 (1998), 203--218.
\MR{1603687} (98k:57006).  \Zbl{893.57001}.  
\annot{} Gain graphs (``voltage graphs'') are the tool.
\codes{\CGG:~\CCov,~\CAut}%
\itemspace

\bsgitem{2002a}{}  Action graphs and coverings.  \emph{Discrete Math.}\ 244 (2002), 299--322.
\MR{1844040} (2003b:05081).  \Zbl{996.05067}.  
\annot{} Gain graphs (``voltage graphs'') and lifting automorphisms of their underlying graphs are a main example.
\dated{11 Jun 2012} 
\par
\codes{\CGG:~\CAut,~\CCov}%
\itemspace

\author{Aleksander Malni\v{c}, Roman Nedela, \& Martin \v{S}koviera}
\authorx{Aleksander Malnic, Roman Nedela, \& Martin Skoviera}
\authorspace
\bsgitem{2000a}{}  Lifting graph automorphisms by voltage assignments.  \emph{European J.\ Combin.}\ 21 (2000), 927--947.
\MR{1787907} (2001i:05086).  \Zbl{966.05042}.  
\annot{} Automorphisms of gain graphs that lift to the covering graph.	
\dated{18 Apr 2012} 
\codes{\CGG:~\CCov,~\CAut}%
\itemspace

\bsgitem{2002a}{}  Regular homomorphisms and regular maps.  \emph{European J.\ Combin.}\ 23 (2002), 449--461.
\MR{1914482} (2003g:05045).  \Zbl{1007.05062}.  
\annot{} \S6, ``Invariance of voltage assignments'', concerns automorphisms of a gain graph that preserve the gains, in connection with lifting automorphisms to the regular covering graph.  The treatment is via maps as gain graphs with rotation systems.
\dated{18 Apr 2012} 
\codes{\CGG:~\CCov,~\CAut}%
\itemspace

\author{John W.\ Mamer}						%MAMER
\authorspace
\authref  See \hyperlink{Richard D.\ McBride}{R.D.~McBride}.
\itemspace

\author{Rachel Manber} 						%MANBER
\authorspace
\authref  See also \hyperlink{Ron Aharoni}{R.~Aharoni} and \hyperlink{Victor Klee, Richard Ladner, \& Rachel Manber}{V.~Klee}.
\authrefspace

\bsgitem{1982a}{}  Graph-theoretical approach to qualitative solvability of linear systems.  \emph{Linear Algebra Appl.}\ 48 (1982), 457--470.  
\MR{0683238} (84g:68054).  \Zbl{511.15008}.					
\par
\codes{\CSD,~\CQM:~\CQSol}%
\itemspace

\author{Rachel Manber \& Jia-Yu Shao} 
\authorspace
\bsgitem{1986a}{}  On digraphs with the odd cycle property.  \emph{J.\ Graph Theory} 10 (1986), 155--165.  
\MR{0890220} (88i:05090).  \Zbl{593.05032}.  
\codes{\CSD,~\CSG:~\CPar}%
\itemspace

\author{Federico Mancini}						%MANCINI
\authorspace
\authref  See \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender}.
\itemspace

\authorx{Santanu Mandal}						%MANDAL
\author{Santanu Mandal \& Ranjit Mehatari}
\authorspace
\bsgitem{20xxa}{}  On the Seidel spectrum of threshold graphs.  Submitted.  \emph{}\ 
\arXiv{2101.03364}.  
\annot{} 
[\Cf\ \xref{Milica An{\dj}eli\'c, Tamara Koledin, \& Zoran Stani\'c}{2019a}{Milica Andelic, Tamara Koledin, \& Zoran Stanic}.]
\par
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Santanu Mandal, Ranjit Mehatari, \& Kinkar Chandra Das}
\authorspace
\bsgitem{20xxa}{}  On the spectrum and energy of Seidel matrix for chain graphs.  Submitted.  \emph{}\ 
\arXiv{2205.00310}.  
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Tabitha Agnes Mangam}					%MANGAM
\authorspace
\authref  See \hyperlink{Divya Antoney}{D.~Antoney}.
\itemspace

\authorx{Mahid M.\ Mangontarum}					%MANGONTARUM
\author{Mahid M.\ Mangontarum, Amila P.\ Macodi-Ringia, \& Normalah S.\ Abdulcarim}
\authorspace
\bsgitem{2014a}{}  The translated Dowling polynomials and numbers.  \emph{Int.\ Scholarly Res.\ Notices Discrete Math.}\ 2014 (2014), art.\ 678408, 8 pp.
\Zbl{1460.05014}.  
\annot{} Introduces $0$-Dowling polynomials $D_{n,m,0}(x) := \sum_k W_{m,0}(n,k) x^k$; \cf\ \xref{Belbachir and Bousbaa}{2013a}{Hacene Belbachir \& Imad Eddine Bousbaa} and $0$-Dowling numbers $D_{n,m,0}(1)$.  Formulas, identities, integrals, etc., for $0$-Whitney and $0$-Dowling.
Dictionary:  ``translated Dowling'' = $0$-Dowling [so named and coordinated with $r$-Whitney and $r$-Dowling numbers in \xref{Gyimesi and Nyul}{2018a}{Eszter Gyimesi \& Gabor Nyul}]; polynomials and numbers $\widetilde{D}_{(m)}(x)$,  $\widetilde{D}_{(m)}(n)$ = $D_{n,m,0}(x)$, $D_{n,m,0}(1)$ [notation of Gyimesi--Nyul].
\dated{28 May 2018} 
\codes{\Cgg:~\Cmatrd~\CInvar}%
\itemspace

\author{Silviu Maniu}							%MANIU
\authorspace
\authref  See \hyperlink{Christos Giatsidis}{C.~Giatsidis}.
\itemspace

\author{Anna Ma\'nka}						%MANKA
\authorx{Anna Manka}
\authorspace
\authref  See \hyperlink{Anna Manka-Krason}{A.~Ma\'nka-Kraso\'n}.
\itemspace

\authorx{Anna Ma\'nka-Kraso\'n}					%MANKA-KRASON
\authorx{Anna Manka-Krason}
\author{Anna Ma\'nka-Kraso\'n \& Krzysztof Ku{\l}akowski}
\authorx{Anna Manka-Krason \& Krzysztof Kulakowski}
\authorspace
\bsgitem{2009a}{}  Magnetism of frustrated regular networks.  \emph{Acta Phys.\ Polonica B} 40 (2009), no.\ 5, 1455--1461.
\itemspace

\bsgitem{2010a}{}  Frustration and collectivity in spatial networks.  In:  Roman Wyrzykowski, \etal, eds.,  \emph{Parallel Processing and Applied Mathematics} (8th Int.\ Conf., PPAM 2009, Wroclaw, 2009), Revised Selected Papers, Part II, pp.\ 539--546.  \LNCS 6068.  Springer, Berlin, 2010.
\arXiv{0904.4002}.  
\codes{\Csg:~\Cpar:~\CFr}%
\itemspace

\author{Anna Ma\'nka, Krzysztof Malarz, \& Krzysztof Kul{\l}akowski}
\authorx{Anna Manka, Krzysztof Malarz, \& Krzysztof Kullakowski}
\authorspace
\bsgitem{2007a}{}  Clusterization, frustration and collectivity in random networks.  \emph{Int.\ J.\ Modern Phys.\ C} 18 (2007), no.\ 11, 1765--1773.
\Zbl{1170.82371}.  
\annot{} Computer experiments on physics aspects of all-negative signed graphs.
\dated{14 Feb 2011} 
\codes{\CPar:~\CFr}%
\itemspace

\author{R.\ Lawrence Joseph Manoharan}					%MANOHARAN
\authorspace
\authref  See \hyperlink{P.\ Lawrence Rozario Raj}{P.L.~Rozario Raj}.
\itemspace

\author{Y.\ Manoussakis}						%MANOUSSAKIS
\authorspace
\authref  See \hyperlink{Abdelhakim El Maftouhi}{A.~El Maftouhi}.
\itemspace

\authorx{Rafael Esteves Mansano}					%MANSANO
\author{Rafael Esteves Mansano, Luiz Emilio Allem, Renata Raposo Del-Vecchio, \& Carlos Hoppen}
\authorspace
\bsgitem{20xxa}{}  Balanced portfolio via signed graphs and spectral clustering in the Brazilian stock market.  \emph{Quality \& Quantity}\ (to appear).
\annot{} \Cf.\ \xref{Harary, Lim, and Wunsch}{2002a}{Frank Harary, Meng-Hiot Lim, \& Donald C.\ Wunsch}.  Applies \xref{B.D.\ Acharya}{1980a}{B.\ Devadas Acharya}.
\dated{7 Feb 2022} 
\codes{\CAppl:~\CSG,~\CWG:~\CAdj:~\CEig,~\CClu}%
\itemspace

\author{Toufik Mansour}						%MANSOUR
\authorspace
\authref  See \hyperlink{Jonathan L.\ Gross}{J.L.~Gross}.
\itemspace

\author{Vassily Olegovich Manturov}					%MANTUROV
\authorspace
\bsgitem{2010a}{}  \emph{Virtual Knots:  The State of the Art}.  (In Russian.)  NIC, 2010.
\annot{} Original Russian ed.\ of \xref{Manturov and Ilyutko}{2013a}{Vassily Olegovich Manturov \& Denis Petrovich Ilyutko}.
\par
\codes{\CSGc,~\CVSc:~\CExp}%
\itemspace

\author{Vassily Olegovich Manturov \& Denis Petrovich Ilyutko}
\authorspace
\bsgitem{2013a}{}  \emph{Virtual Knots:  The State of the Art}.  Ser.\ Knots Everything, Vol.\ 51.  World Scientific, New Jersey, 2013.
\MR{2986036}.  \Zbl{1270.57003}.  
\annot{} English trans.\ of \xref{Manturov}{2010a}{Vassily Olegovich Manturov}.  
Ch.\ 9, ``Theory of graph-links'':  0/1-labelled and sign-colored chords in chord diagrams.  0/1-labelled and sign-colored vertices.  0/1-labelled and sign-colored graphs.
\dated{13 Sept 2013} 
\codes{\CSGc,~\CVSc:~\CExp}%
\itemspace

\author{Haik Manukian}						%MANUKIAN
\authorspace
\authref  See also \hyperlink{Yan Ru Pei}{Y.R.~Pei}.
\authrefspace

\author{Haik Manukian, Yan Ru Pei, Sean R.B.\ Bearden, \& Massimiliano Di Ventra}	
\authorspace
\bsgitem{2020a}{}  Mode-assisted unsupervised learning of restricted Boltzmann machines.  \emph{Commun.\ Phys.}\ 3 (2020), art.\ 105, 8 pp. 
\arXiv{2001.05559}.  
\par
\codes{\CSG:~\CFr:~\CPhys,~\CState}%
\itemspace

\author{Yanqi Mao}							%MAO YQ
\authorspace
\authref  See \hyperlink{Xiying Yuan}{X.Y.~Yuan}.
\itemspace

\authorx{Zeev Maoz}							%MAOZ
\author{Zeev Maoz, Lesley G.\ Terris, Ranan D.\ Kuperman, \& Ilan Talmud}
\authorspace
\bsgitem{2007a}{}  What is the enemy of my enemy?  Causes and consequences of imbalanced international relations, 1816--2001.  
\emph{J.\ Politics} 69 (2007), no.\ 1, 100--115. 
\par
\codes{\CPsS,~\CSG:~\CBal}%
\itemspace

\author{D\u{a}nu\c{t} Marcu}						%MARCU
\authorx{Danut Marcu}
\authorspace
\authref [I cannot vouch for the authenticity of these articles.  See \MR1324075 (97a:\-05095) and \Zbl{701.51004}.  Also see \MR{1038400} (92a:51002), \MR{1094344} (92b:51026), \MR{1107637} (92h:11026), \MR{1427830} (97k:05050); and \xref{Marcu}{1981b}{Danut Marcu}.]
\authrefspace

\bsgitem{1980a}{}  On the gradable digraphs.  \emph{An.\ \c{S}tiin\c{t}.\ Univ.\ ``Al.\ I.\ Cuza'' Ia\c{s}i Sec\c{t}.\ I a Mat.\ (N.S.)} 26 (1980), 185--187.  
\MR{0582484} (82k:05056) (\qv).  \Zbl{438.05032}.
\annot{} See \xref{Harary, Norman, and Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} for the definition.
\par
\codes{\CGD:~\Cbal}%
\itemspace

\bsgitem{1981a}{}  No tournament is gradable.  \emph{An.\ Univ.\ Bucure\c{s}ti Mat.}\ 30 (1981), 27--28.  
\MR{0643669} (83c:05069).  \Zbl{468.05028}.
\annot{} See \xref{Harary, Norman, and Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} for the definition.  The tournaments of order $3$ are [trivially] not gradable, whence the titular theorem.
\codes{\CGD:~\Cbal}%
\itemspace

\bsgitem{1981b}{}  Some results concerning the even cycles of a connected digraph.  \emph{Studia Univ.\ Babe\c{s}-Bolyai Math.}\ 26 (1981), 24--28.
\MR{0654119} (83e:05058).  \Zbl{479.05032}.						
\annot{} \S1, ``Preliminary considerations'', appears to be an edited, unacknowledged transcription of parts of \xref{Harary, Norman, and Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright} (or possibly \xrefd{1968a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright}), pp.\ 341--345.  Wording and notation have been modified, a trivial corollary has been added, and some errors have been introduced; but the mathematics is otherwise the same down to details of proofs.  \S2, ``Results'', is largely a list of the corollaries resulting from setting all signs negative.  The exception is Thm.\ 2.5, for which I am not aware of a source; however, it is simple and well known.
\codes{\Csg(\CSD):~\CBal}%
\itemspace

\bsgitem{1987a}{}  Note on the matroidal families.  \emph{Riv.\ Math.\ Univ.\ Parma} (4) 13 (1987), 407--412.
\MR{0977694} (89k:05025).  
\annot{} Matroidal families of (multi)graphs (see \xref{Sim\~oes-Pereira}{1973a}{J.M.S.\ Simoes-Pereira}) correspond to functions on all isomorphism types of graphs that are similar to matroid rank functions, e.g., submodular.  This provides insight into matroidal families, e.g., it immediately shows there are infinitely many.  
\par
\codes{\CMtrdF:~\CBic,~\CECyc:~\CGen}%
\itemspace

\authorx{Adam W.\ Marcus}						%MARCUS AW
\author{Adam W.\ Marcus, Daniel A.\ Spielman, \& Nikhil Srivastava}
\authorspace
\bsgitem{2013a}{}  Interlacing families I:  Bipartite Ramanujan graphs of all degrees.  In:  \emph{2013 IEEE 54th Annual Symposium on Foundations of Computer Science} (Proc., FOCS 2013, Berkeley, Calif.), pp.\ 529--537.  IEEE Computer Soc., Los Alamitos, Calif., 2013.
\MR{3246256}.  \Zbl{1334.68008} (book).  
\annot{} Preliminary version of \xrefd{2015a}{Adam W.\ Marcus, Daniel A.\ Spielman, \& Nikhil Srivastava}; the latter has slight (and occasionally important) additions, deletions, and corrections.
\dated{18 Oct 2015} 
\codes{\CSG:~\CCov,~\CAdj:~\CEig}%
\itemspace

\bsgitem{2015a}{}  Interlacing families I:  Bipartite Ramanujan graphs of all degrees.  \emph{Ann.\ Math.}\ (2) 182 (2015), no.\ 1, 307--325.
\MR{3374962}.  \Zbl{1316.05066}.  
\annot{} 	
Dictionary:  ``2-lift'' of $\Sigma$ = signed covering graph $\tilde\Sigma$.  ``Double-cover'' of $\Gamma$ = that of ${-\Gamma}$.
\codes{\CSG:~\CCov,~\CAdj:~\CEig}%
\itemspace

\author{Grzegorz Marczak}						%MARCZAK
\authorspace
\authref  See also \hyperlink{Mariusz Kaniecki}{M.~Kaniecki}.
\authrefspace

\author{Grzegorz Marczak, Daniel Simson \& Katarzyna Zaj\c{a}c}
\authorx{Grzegorz Marczak, Daniel Simson \& Katarzyna Zajac}
\authorspace
\bsgitem{2013a}{}  On computing non-negative loop-free edge-bipartite graphs.  In:  Nikolaj Bj\"orner \etal, eds., \emph{15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2013, Timisoara, Romania, 2013), pp.\ 68--75.  IEEE, 2013.
\codes{\CSG}%
\itemspace

\authorx{Stuart Margolis}						%MARGOLIS
\author{Stuart Margolis, John Rhodes, \& Pedro V.\ Silva}
\authorspace
\bsgitem{2021a}{\dag}  On the Dowling and Rhodes lattices and wreath products.  \emph{J.\ Algebra} 578 (2021), 55--91.
\MR{4234797}.  \Zbl{1461.05026}.  \arXiv{1710.05314}.  
\annot{} The \xref{Dowling lattice}{1973b}{Thomas A.\ Dowling} $Q_n(\fG)$ of a group $\fG$ and the Rhodes lattice in semigroup theory are constructed from the same complete $\fG$-gain graph $\fG K_n\full$ through somwhat opposite orderings.  The frame matroid $\bfF(\fG K_n\full)$ and lift matroid $\bfL(\fG K_n)$ are recovered from the Dowling and Rhodes lattices via ``boolean representable simplicial complexes'' (\cf\ Rhodes and Silva, \emph{Boolean Representations of Simplicial Complexes and Matroids}, 2015).
\dated{7 Apr 2021} 
\codes{\CGG,~\CMatrd}%
\itemspace

\author{Enzo Marinari}						%MARINARI
\authorspace
\authref  See also \hyperlink{S.\ Cabasino}{S.~Cabasino}, \hyperlink{Barbara Coluzzi}{B.~Coluzzi}, \hyperlink{M.\ Falcioni}{M.~Falcioni}, and \hyperlink{J.\ Lukic}{J.~Lukic}.
\authrefspace

\author{Enzo Marinari, Giorgio Parisi, \& Felix Ritort}
\authorspace
\bsgitem{1995a}{}  The fully frustrated hypercubic model is glassy and aging at large $D$.  \emph{J.\ Phys.\ A} 28 (1995), 327--334.
\annot{} Ising (spins, i.e., vertex values, $\in \bbS^0 = \signnos$) and $XY$ (spins $\in \bbS^1$, i.e., complex units) models behave differently on a fully frustrated signed hypercube graph $Q_D$ (all squares are negative).  
Numerical study of Ising spins of two such signatures:  $\sigma_1(x,x+e_\mu) = (-1)^{x_1+\cdots+x_{\mu-1}}$, while $\sigma_2$ [``simplex'' in the construction must mean hypercube] is from \xref{Derrida, Pomeau, Toulouse, and Vannimenus}{1979a}{B.\ Derrida, Y.\ Pomeau, G.\ Toulouse, \& J.\ Vannimenus}; ``with identical results''.  [Reason:  $\Sigma_1 \cong \Sigma_2$ under the coordinate transformation $i \leftrightarrow D+1-i$.]  
Based on simulations with $D \leq 47$, Ising ground states seem to be few and hard to find.  Near-ground states are easier to find but, apparently, tend to be far from ground states.  
\annotctd{} For positive temperature $T$, as $A(\Sigma)$ (``interaction matrix $J_{x,y}$'') is orthogonal [up to scaling], one can approximate by averaging over orthogonal adjacency matrices. 
\annotctd{} In simulations with $XY$ spins the ground state is highly accessible.
\annotctd{} Dictionary:  ``ground state'' = switching with minimum $\#E^-$.
\dated{19 Jun 2012} 
\codes{\CPhys,~\CSG}%
\itemspace

\bsgitem{1995b}{}  Replica theory and large-$D$ Josephson junction hypercubic models.  \emph{J.\ Phys.\ A} 28 (1995), 4481--4503.
\MR{1352169} (96g:82032).  \Zbl{925.82088}.  \arXiv{cond-mat/9502067}.				
\annot{} Physics on hypercube $Q_D$  with complex unit gains and three types of spin, after \xref{Parisi}{1994a}{Giorgio Parisi}, via simulations for $3 \leq D \leq 16$.
\dated{19 Jun 2012} 
\codes{\CPhys,~\Cgg}%
\itemspace

\bsgitem{2000a}{}  On the {\sc{3D}} Ising spin glass.  \emph{J.\ Phys.\ A} 27 (1994), no.\ 8, 2687--2708.
\MR{1280826} (no rev).  \arXiv{cond-mat/9310041}.  
\codes{\CPhys:~\CSG}%
\itemspace

\authorx{Fabrizio Marinelli}						%MARINELLI
\author{Fabrizio Marinelli \& Angelo Parente}
\authorspace
\bsgitem{2016a}{}  A heuristic based on negative chordless cycles for the maximum balanced induced subgraph problem.  \emph{Computers Oper.\ Res.}\ 69 (2016), 68--78.  
\MR{3458032}.  
\par
\codes{\CSG:~\CFr:~\CVS,~\CAlgor}%
\itemspace

\author{A.V.\ Markovski{\u\i}}						%MARKOVSKII
\authorx{A.V.\ Markovskii}
\authorspace
\bsgitem{1997a}{}  Analysis of the structure of signed directed graphs.  (In Russian.)  \emph{Izv.\ Akad.\ Nauk Teor.\ Sist.\ Upr.}\ 1997 (1997), no.\ 5, 144--149.  Eng.\ trans., \emph{J.\ Comput.\ Systems Sci.\ Int.}\ 36 (1997), no.\ 5, 788--793.
\MR{1679025} (2000a:05099).  \Zbl{898.05078}.
\codes{\CSD:~\CWG}%
\itemspace

\author{Harry Markowitz} 						%MARKOWITZ
\authorspace
\bsgitem{1955a}{}  Concepts and computing procedures for certain $X_{ij}$ programming problems.  
In:  H.A.\ Antosiewicz, ed., \emph{Proceedings of the Second Symposium in Linear Programming} (Washington, D.C., 1955), Vol.\ II, pp.\ 509--565.  Nat.\ Bur.\ Standards of U.S.\ Dept.\ of Commerce, and Directorate of Management Analysis, DCS Comptroller, HQ, U.S.\ Air Force, 1955.  Sponsored by Office of Scientific Res., Air Res.\ and Develop.\ Command.  
\MR{0075673} (17, 789).  
\annot{} Also see RAND Corporation Paper P-602, 1954. 
\codes{\CGN:~\Cmatrd(\Cbases)}%
\itemspace

\author{Klas Markstr\"om}						%MARKSTROM
\authorx{Klas Markstrom}
\authorspace
\bsgitem{2012a}{}  Even cycle decompositions of 4-regular graphs and line graphs.  \emph{Discrete Math.}\ 312 (2012), no.\ 17, 2676--2681.
\MR{2935419}.  \Zbl{1246.05087}.  
\annot{} Can $-\Gamma$ be decomposed into positive circles?  Studied for 4-regular graphs and line graphs of cubic graphs.  
\Cf\ \xref{C.Q.~Zhang}{1994a}{Cun-Quan Zhang}, \xref{M\'a\v{c}ajov\'a and Maz\'ak}{2013a}{Edita Macajova \& Jan Mazak} (especially), and \xref{Liu--You--Cui}{2020a}{Wenzhong Liu, Huazheng You, \& Qing Cui}.  
Dictionary:  ``oddness'' of $\Gamma$ = min \# negative circles in a 2-factor of $-\Gamma$.
\dated{12 Jan 2012, rev 11 Nov 2020} 
\codes{\Cpar:~\CStr}%
\itemspace

\author{Clifford W.\ Marshall} 						%MARSHALL
\authorspace
\bsgitem{1971a}{}  \emph{Applied Graph Theory}.  Wiley-Interscience, New York, 1971.  
\MR{0323595} (48 \#1951).  \Zbl{226.05101}.
\annot{} ``Consistency of choice'' discusses signed graphs, pp.\ 262--266.
\par
\codes{\CSG:~\CBal,~\CAdj:~\CExp}%
\itemspace

\author{T.H.\ Marshall}						%MARSHALL
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{Matteo Marsili}						%MARSILI
\authorspace
\authref  See \hyperlink{George C.M.A.\ Ehrhardt}{G.C.M.A.~Ehrhardt}.
\itemspace

\author{Florian Martin}						%MARTIN F
\authorspace
\bsgitem{2017a}{}  Frustration and isoperimetric inequalities for signed graphs.  \emph{Discrete Appl.\ Math.}\ 217, no.\ 2, 276--285.
\MR{3581352}.  \Zbl{1358.05133}.  
\codes{\CSG:~\CFr}%
\itemspace

\author{O.C.\ Martin}							%MARTIN OC
\authorspace
\authref  See \hyperlink{J.-P.\ Bouchaud}{J.-P.~Bouchaud} and \hyperlink{J.\ Lukic}{J.~Lukic}.
\itemspace

\author{Jeremy L.\ Martin}						%MARTIN JL
\authorspace
\authref  See \hyperlink{Joshua Hallam}{J.~Hallam}.
\itemspace

\author{Samuel Martin}						%MARTIN S
\authorspace
\authref  See \hyperlink {Saeed Ahmadizadeh} {S.~Ahmadizadeh}.
\itemspace

\author{V.\ Martin-Mayor}						%MARTIN-MAYOR
\authorspace
\authref  See \hyperlink{L.A.\ Fernandez}{L.A.~Fern\'andez}.
\itemspace

\authorx{Jos\'e Mart\'{i}nez-Bernal}					%MARTINEZ-BERNAL
\authorx{Jose Martinez-Bernal}
\author{Jos\'e Mart\'{i}nez-Bernal, Miguel A.\ Valencia-Bucio, \& Rafael H.\ Villarreal}
\authorx{Jose Martinez-Bernal, Miguel A.\ Valencia-Bucio, \& Rafael H.\ Villarreal}
\authorspace
\bsgitem{2020a}{}  Linear codes over signed graphs.  \emph{Designs Codes Cryptogr.}\ 88 (2020), no.\ 2, 273--296.
\MR{4055232}.  \Zbl{1452.94117}.  \arXiv{1904.09487}.				
\codes{\CSG:~\CInc,~\CMatrd,~\CFr}%
\itemspace

\authorx{Xavier MartInez-Rivera}					%MARTINEZ-RIVERA
\author{Xavier Mart\'{\i}nez-Rivera}
\authorspace
\authref  See \hyperlink{Steve Butler}{S.~Butler}.
\itemspace

\author{Enide Andrade Martins}						%MARTINS
\authorspace
\authref  See \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~de~Abreu} and I.~Gutman.
\itemspace

\author{Dragan Maru\v{s}i\v{c}}						%MARUSIC
\authorx{Dragan Marusic}
\authorspace
\authref  See \hyperlink{Istvan Kovacs}{I.~Kov\'acs}.
\itemspace

\author{Tatsuya Maruta}						%MARUTA
\authorspace
\authref  See \hyperlink{Yoshitaka Koga}{Y.~Koga}.
\itemspace

\authorx{Seth A.\ Marvel}						%MARVEL
\author{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}
\authorspace
\bsgitem{2011a}{}  Continuous-time model of structural balance.  \emph{Proc.\ Nat.\ Acad.\ Sci.\ (U.S.A.)} 108 (2011), no.\ 5, 1771--1776.
\arXiv{1010.1814}.  
\annot{} A differential equation model of balancing processes, based on \xref{Ku{\l}a\-kow\-ski, Gawro\'nski, \& Gronek}{2005a}{Krzysztof Kulakowski, Premiyslaw Gawronski, \& Piotr Gronek}.  Conclusion:  Final state is balance.
\Cf\ \xrefd{2011b}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}.  
[See commentary, \xref{Srinivasan}{2011a}{Aravind Srinivasan}.]
\dated{6 Feb 2011} 
\codes{\CSG:~\CKG:~\CFr,~\CDyn}%
\itemspace

\bsgitem{2011b}{}  Supporting information.  \emph{Proc.\ Nat.\ Acad.\ Sci.\ (U.S.A.)} 108 (2011), URL \url{http://www.pnas.org/cgi/doi/10.1073/pnas.1013213108}.
\annot{} The mathematical support for \xrefd{2011a}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}.
\dated{6 Feb 2011} 
\par
\codes{\CSG:~\CKG:~\CFr,~\CDyn}%
\itemspace

\author{Seth A.\ Marvel, Steven H.\ Strogatz, \& Jon M.\ Kleinberg}
\authorspace
\bsgitem{2009a}{}  Energy landscape of social balance.  \emph{Phys.\ Rev.\ Lett.}\ 103 (2009), art.\ 198701, 4 pp.
\annot{} Signed complete graphs under \xref{Antal, Krapivsky, and Redner's}{2005a}{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner} ``constrained triad dynamics'':  Imbalance measured by triangles; an edge is negated if it is in more negative than positive triangles.  Paley graphs $P$ give $K_P$ with equally many positive and negative triangles on each edge (normalized ``energy'' = 0).  Other such states exist.  
[\xref{Zyga}{2009a}{Lisa Zyga} gives a popular exposition.]  
[\Questions.  Do unbalanced locally minimal regions with more than one point (graph) exist?  How does the landscape look for switching classes?]
\dated{5 May 2010, 26 Jan 2011} 
\par
\codes{\CSG:~\CKG:~\CFr,~\CState:~\CDyn}%
\itemspace

\author{[Enzo M.\ Li Marzi]}						%MARZI
\authorspace
\authref  See \hyperlink{Enzo M.\ Li Marzi}{E.M.~Li~Marzi} (under `L').
\itemspace

\author{Andrew J.\ Mason}						%MASON AJ
\authorspace
\authref  See \hyperlink{Samin Aref}{S.~Aref}.
\itemspace

\author{J.H.\ Mason}							%MASON JH
\authorspace
\bsgitem{1977a}{}  Matroids as the study of geometrical configurations.  
In:  \emph{Higher Combinatorics} (Proc.\ NATO Adv.\ Study Inst., Berlin, 1976), pp.\ 133--176.  NATO Adv.\ Study Inst.\ Ser., Ser.\ C: Math.\ Phys.\ Sci., Vol.\ 31.  Reidel, Dordrecht, 1977.
\MR{0519783} (80k:05037).  \Zbl{358.05017}.					
\annot{} \S\S2.5--2.6: ``The lattice approach'' and ``Generalized coordinates'', pp.\ 172--174, propose a purely matroidal and more general formulation of \xref{Dowling's}{1973b}{Thomas A.\ Dowling} construction of his lattices.
\codes{\Cgg(\CGen):~\CMatrd}%
\itemspace

\bsgitem{1981a}{}  Glueing matroids together:  A study of Dilworth truncations and matroid analogues of exterior and symmetric powers.  
In:  \emph{Algebraic Methods in Graph Theory} (Proc., Szeged, 1978), Vol.~II, pp.\ 519--561.  Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 25.  North-Holland, Amsterdam, 1981.
\MR{0642060} (84i:05041).  \Zbl{477.05022}.
\annot{} Dowling matroids are an example in \S1.  
\codes{\Cgg:~\CMatrd}%
\itemspace

\authorx{R.\ Masoumi}						%MASOUMI
\author{R.\ Masoumi, F.\ Oloomi, S.\ Sajjadi, A.H.\ Shirazi, \& G.R.\ Jafari}
\authorspace
\bsgitem{2022a}{}  Modified Heider balance on Erd\"os-R\'enyi networks.  \emph{Phys.\ Rev.\ E}\ 106 (2022), art.\ 034309.
\arXiv{2204.04469}.  
\annot{} 
arXiv title: ``Modified Heider balance on sparse random networks''.
\codes{\CSG:~\CFr:~\CPhys}%
\itemspace

\authorx{Nancy Matar}						%MATAR
\author{Nancy Matar, Lon H.\ Mitchell, \& Sivaram K.\ Narayan}
\authorspace
\bsgitem{2022a}{}  On the minimum semidefinite rank of signed graphs.  \emph{Linear Algebra Appl.}\ 642 (2022), 73--85.
\MR{4381684}.  
\annot{} 
\Cf\ \xref{Al-Aqtash}{2014a}{Ansam I.\ Al-Aqtash}.
\codes{\CSG:~\CAdj:~\CQM}%
\itemspace

\authorx{A.M.\ Mathai}						%MATHAI
\author{A.M.\ Mathai \& Thomas Zaslavsky}
\authorspace
\bsgitem{2012a}{}  On adjacency matrices and descriptors of signed cycle graphs.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 359--372.
\Zbl{1301.05157}.  \arXiv{1303.3082}.  
\annot{} Eigenvalues of $A(C_n,\sigma)$ (previously stated by \xref{Fowler}{2002a}{Patrick W.\ Fowler}; equivalent to \xref{Fan}{2007a}{Yi-Zheng Fan}'s Laplacian eigenvalues) by an elegant matrix method.  [\Cf\ \xref{Germina and Hameed}{2010a}{K.A.\ Germina \& Shahul Hameed K (as K.\ Shahul Hameed)}.]  Some ways to partially or wholly distinguish different signatures of $C_n$ are compared.
\dated{6 Sept 2010, 7 Nov 2013, 13 Jan 2015} 
\codes{\CSG:~\CEig}%
\itemspace

\author{Albin Mathew}						%MATHEW
\authorspace
\authref  See also \hyperlink{Shahul Hameed K}{S.~Hameed}.
\authrefspace

\author{Albin Mathew \& Germina K A}
\authorspace
\bsgitem{20xxa}{}  Mycielskian of signed graphs.  Submitted.  
\arXiv{2302.00946}.  
\annot{} Does not respect balance or switching.  Thm.\ 2.9:  Chromatic number compared with that of $\Sigma$.  \S2.3:  Matrices in terms of those of $\Sigma$.
\dated{14 Feb 2023} 
\codes{\CSG:~\CBal,~\CSw,~\CCol,~\CAdj,~\CIncid,~\CLap}%
\itemspace

\bsgitem{20xxb}{}  Vector valued switching in the products of signed graphs.  Submitted.  
\annot{} \Cf\ \xref{Hameed, Mathew, Germina, and Zaslavsky}{20xxa}{Shahul Hameed K, Albin Mathew, Germina K A, \& Thomas Zaslavsky}.
Inequalities for balancing dimension of Cartesian, tensor, strong, and lexicographic products.
\dated{16 Jun 2023} 
\codes{\CSG:~\CSw(\CGen)}%
\itemspace

\author{Albin Mathew, T.V.\ Shijin, Roshni T Roy, P.\ Soorya, Shahul K.\ Hameed, \& K.A.\ Germina}
\authorspace
\bsgitem{20xxa}{}  Social balance - a signed detour distance analysis.  \emph{J.\ Math.\ Sociology} (in press).
\annot{} Similar to \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky} but for detour distance matrices, meaning longest $vw$-paths instead of shortest.
\dated{31 Dec 2021} 
\codes{\CSG:~\CWG,~\CAdj,~\CBal,~\CEig,~\CSw}%
\itemspace

\authorx{Anisha Jean Mathias}						%MATHIAS
\author{Anisha Jean Mathias, V.\ Sangeetha, \& Mukti Acharya}	
\authorspace
\bsgitem{2020a}{}  Restrained domination in signed graphs.  \emph{Acta Univ.\ Sapientiae, Math.}\ 12 (2020), no.\ 1, 155--163.
\MR{4126665}.  \Zbl{1452.05078}.  
\annot{} Restrained dominating set $D$:  $E \setm E:D^c$ is balanced.
\dated{3 Sept 2020} 
\codes{\CSG:~\CDom}%
\itemspace

\author{A.J.\ Mathias, V.\ Sangeetha, \& M.\ Acharya}
\authorspace
\bsgitem{2023a}{}  Restrained critical and abundant signed graphs.  \emph{Adv.\ Appl.\ Discrete Math.}\ 28 (2023), no.\ 1, 49--68.
\codes{\CSG}%
\itemspace

\bsgitem{20xxa}{}  Critical concepts of restrained domination in signed graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ (in press).
\annot{} \Cf\ \xrefd{2020a}{Anisha Jean Mathias, V.\ Sangeetha, \& Mukti Acharya}.  Effect of edge deletion or addition, esp.\ in examples.
\dated{1 Oct 2021} 
\codes{\CSG:~\CDom}%
\itemspace

\author{R.A.\ Mathon} 						%MATHON
\authorspace
\authref  See \hyperlink{F.C.\ Bussemaker}{F.C.~Bussemaker} and \xref{Seidel}{1991a}{J.J.\ Seidel}.
\itemspace

\authorx{Tetsushi Matsui}						%MATSUI
\author{Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, \& Takayuki Hibi}
\authorspace
\bsgitem{2011a}{}  Roots of Ehrhart polynomials arising from graphs.  \emph{J.\ Algebraic Combin.}\ 34 (2011), 721--749.
\MR{2842918} (2012i:52024).  \Zbl{1229.05122}.  \arXiv{1003.5444}.			
\annot{} Edge polytope (\cf\ \xref{Ohsugi and Hibi}{1998a}{Hidefumi Ohsugi \& Takayuki Hibi}), and introducing symmetric edge polytope of graph = convex hull of vectors for both directions of each edge.
	
\dated{4 Jun 2022} 
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\authorx{Tatsuya Matsuoka}						%MATSUOKA
\author{Tatsuya Matsuoka \& Shun Sato}
\authorspace
\bsgitem{2020a}{}  Making bidirected graphs strongly connected.  \emph{Algorithmica} 82 (2020), no.\ 4, 787--807.
\MR{406879}.  \Zbl{1441.05127}.  \arXiv{1709.00824}.				
\codes{\Csg:~\COri:~\CStr}%
\itemspace

\author{Hisayoshi Matsuyama} 						%MATSUYAMA
\authorspace
\authref  See \hyperlink{Masao Iri}{M.~Iri} and \hyperlink{J.\ Shiozaki}{J.~Shiozaki}.
\itemspace

\author{Amelia R.W.\ Mattern}						%MATTERN
\authorspace
\bsgitem{2020a}{}  \emph{Deficiency in Signed Graphs.}  Doctoral dissertation, Binghamton University, 2020.
\MR{4144528} (no rev).							
\annot{} Contains \xrefd{2021a}{Amelia R.W.\ Mattern} and \xrefd{20xxb}{Amelia R.W.\ Mattern}.
\dated{28 May 2020} 
\par
\codes{\CSG:~\CCol:~\CAlgor,~\CSw}%
\itemspace

\bsgitem{2021a}{}  The chromatic number of joins of signed graphs.  \emph{Graphs Combin.}\ 37 (2021), 2723--2735.
\MR{4338758}.  \Zbl{1479.05138}.  \arXiv{2005.14337}.				
\annot{} $\chi(\Sigma_1 \vee_+ \Sigma_2)$ (chromatic number of all-positive join) derived from chromatic numbers and deficiencies of $\Sigma_1$ and $\Sigma_2$, with an exception that is partially characterized.
\dated{27 May 2020} 
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{20xxb}{}  Deficiency in signed graphs.  Submitted.  
\arXiv{2005.14336}.					
\annot{} Coloring as in \xref{Zaslavsky}{1982b}{Thomas Zaslavsky}, $\chi(\Sigma)$ as in \xref{M\'a\v{c}ajov\'a \etal\!}{2014a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}.  The deficiency of a proper coloration is the number of unused colors, which may be positive even for a minimal coloration.  Deficiency set $\Dfc(\Sigma) := \{$deficiencies of minimal colorations$\}$.  \Problems:  Find min, max deficiencies $m(\Sigma)$ and $M(\Sigma)$.  Solved for $\chi=2$.  Solved algorithmically for $M$, complexity unknown for $m$, if $\chi=3$.  Open for $\chi>3$.
\annotctd{} Switching deficiency of $\Sigma$: easy.
\dated{27 May 2020} 
\par
\codes{\CSG:~\CCol:~\CAlgor,~\CSw}%
\itemspace

\author{Laurence R.\ Matthews} 					%MATTHEWS
\authorspace
\bsgitem{1977a}{} Bicircular matroids.  \emph{Quart.\ J.\ Math.\ Oxford} (2) 28 (1977), 213--227.  
\MR{0505702} (58 \#21732).  \Zbl{386.05022}.					
\annot{} Thorough study of bicircular matroids, introduced by \xref{Klee}{1971a}{Victor Klee} and \xref{Sim\~oes-Pereira}{1972a}{J.M.S.\ Simoes-Pereira}.  [\Cf\ \xref{Zaslavsky}{1982d}{Thomas Zaslavsky}, \xrefd{2007a}{Thomas Zaslavsky}, \xref{Wagner}{1985a}{Donald K.\ Wagner}, \xref{Coullard, del Greco, and Wagner}{1991a}{Collette R.\ Coullard, John G.\ del Greco, \& Donald K.\ Wagner}, \xref{Shull, Shuchat, Orlin, and Lepp}{1993a}{Randy Shull, Alan Shuchat, James B.\ Orlin, \& Marianne Lepp}, \xref{Gim\'enez, de Mier, and Noy}{2005a}{Omer Gimenez, Anna de Mier, \& Marc Noy}, \xref{McNulty and Neudauer}{2008a}{Jennifer McNulty \& Nancy Ann Neudauer}, \xref{Sivaraman}{2014a}{Vaidy Sivaraman}.]
\codes{\CBic}%
\itemspace

\bsgitem{1978a}{} Properties of bicircular matroids.  
In: \emph{Probl\`emes Combinatoires et Th\'eorie des Graphes} (Colloq.\ Int., Orsay, 1976), pp.\ 289--290.  Colloques Int.\ du CNRS, 260.  Editions du C.N.R.S., Paris, 1978.  
\MR{0539994} (81a:05030).  \Zbl{427.05021}.
\annot{} Summary of \xrefd{1977a}{Laurence R.\ Matthews}.
\codes{\CBic}%
\itemspace

\bsgitem{1978b}{} Matroids on the edge sets of directed graphs.  
In: \emph{Optimization and Operations Research} (Proc.\ Workshop, Bonn, 1977), pp.\ 193--199.  Lect.\ Notes in Economics and Math.\ Systems, 157.  Springer-Verlag, Berlin, 1978.  
\MR{0511340} (80a:05103).  \Zbl{401.05031}.
\annot{} Announcement of \xrefd{1978c}{Laurence R.\ Matthews}.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1978c}{} Matroids from directed graphs.  \emph{Discrete Math.}\ 24 (1978), 47--61.  
\MR{0522733} (81e:05055).  \Zbl{388.05005}.
\annot{} Invents frame matroids of poise, modular poise, and antidirection bias on a digraph.
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{1979a}{}  Infinite subgraphs as matroid circuits.  \emph{J.\ Combin.\ Theory Ser.\ B} 27 (1979), 260--273.  
\MR{0554294} (81e:05056).  \Zbl{433.05018}.
\codes{\CBic:~\CGen}%
\itemspace

\author{Laurence R.\ Matthews \& James G.\ Oxley} 
\authorspace
\bsgitem{1977a}{}  Infinite graphs and bicircular matroids.  \emph{Discrete Math.}\ 19 (1977), 61--65.  
\MR{0498193} (58 \#16348).  \Zbl{386.05021}.  
\codes{\CBic}%
\itemspace

\author{Alexey Matveev}						%MATVEEV
\authorspace
\authref  See \hyperlink{Anton Proskurnikov}{A.~Proskurnikov}.
\itemspace

\author{Jean Fran\c{c}ois Maurras} 					%MAURRAS
\authorspace
\bsgitem{1972a}{}  Optimization of the flow through networks with gains.  \emph{Math.\ Programming} 3 (1972), 135--144.  
\MR{0314441} (47 \#2993).  \Zbl{243.90048}.
\codes{\CGN:~\CMatrd}%
\itemspace

\authorx{Mano Ram Maurya}						%MAURYA
\author{Mano Ram Maurya, Raghunathan Rengaswamy, \& Venkat Venkatasubramanian}
\authorspace
\bsgitem{2003a}{}  A systematic framework for the development and analysis of signed digraphs for chemical processes.  1.  Algorithms and analysis.  \emph{Indust.\ Eng.\ Chem.\ Res.}\ 42 (2003), 4789--4810.
\codes{\CSD:~\CSta:~\CAlgor,~\CAppl}%
\itemspace

\bsgitem{2003b}{}  A systematic framework for the development and analysis of signed digraphs for chemical processes.  2.  Control loops and flowsheet analysis.  \emph{Indust.\ Eng.\ Chem.\ Res.}\ 42 (2003), 4811--4827.
\codes{\CSD:~\CSta:~\CAlgor,~\CAppl}%
\itemspace

\bsgitem{2006a}{}  A signed directed graph-based systematic framework for steady-state malfunction diagnosis inside control loops.  \emph{Chem.\ Engineering Sci.}\ 61 (2006), 1790--1810.
\codes{\CSD:~\CAppl}%
\itemspace

\bsgitem{2007a}{}  A signed directed graph and qualitative trend analysis-based framework for incipient fault diagnosis.  \emph{Chem.\ Engineering Res.\ Design} 85 (2007), no.\ 10, 1407--1422.
\codes{\CSD:~\CAppl}%
\itemspace

\author{John S.\ Maybee} 						%MAYBEE
\authorspace
\authref  See also \hyperlink{Lowell Bassett}{L.~Bassett}, \hyperlink{Harvey J.\ Greenberg}{H.J.~Greenberg}, \hyperlink{Frank Harary}{F.~Harary}, \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}, \hyperlink{George M.\ Lady}{G.M.~Lady}, and \hyperlink{Thomas J.\ Lundy}{T.J.~Lundy}.
\authrefspace

\bsgitem{1974a}{} Combinatorially symmetric matrices.  \emph{Linear Algebra Appl.}\ 8 (1974), 529--537.  
\MR{0453583} (56 \#11845).  \Zbl{438.15021} (no rev).				
\annot{} Survey and simple proofs.
\codes{\CQM:~\Csd,~\Cgg,~\CSta}\codesctd{\CExp}%
\itemspace

\bsgitem{1980a}{} Sign solvable graphs.  \emph{Discrete Appl.\ Math.}\ 2 (1980), 57--63.  
\MR{0569486} (81g:05063).  \Zbl{439.05024}.					
\codes{\CSD:~\CQM:~\CSol}%
\itemspace

\bsgitem{1981a}{}  Sign solvability.  In:  Harvey J.\ Greenberg and John S.\ Maybee eds., \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ Sympos., Boulder, Col., 1980), pp.\ 201--257.  Discussion, p.\ 321.  Academic Press, New York, 1981.  
\MR{0617930} (82g:00016) (book).  \Zbl{495.93001} (book).
\annot{} For comments, see \xref{Lancaster}{1981a}{Kelvin Lancaster}. 
\codes{\CQM:~\CSol:~\CSD}%
\itemspace

\bsgitem{1989a}{}  Qualitatively stable matrices and convergent matrices.  
In: Fred Roberts, ed., \emph{Applications of Combinatorics and Graph Theory to the Biological and Social Sciences}, pp.\ 245--258.  IMA Vols.\ Math.\ Appl., Vol.\ 17.  Springer-Verlag, New York, 1989.   
\MR{1009379} (90h:34082).  \Zbl{708.15007}.					
\annot{} Signed (di)graphs play a role in characterizations.  See e.g.\ \S7.  Also see \xref{Roberts}{1989a}{Fred S.\ Roberts}, \S4.
\codes{\CQM,~\CSD}%
\itemspace

\author{John S.\ Maybee \& Stuart J.\ Maybee}
\authorspace
\bsgitem{1983a}{}  An algorithm for identifying Morishima and anti-Morishima matrices and balanced digraphs.  \emph{Math.\ Social Sci.}\ 6 (1983), 99--103.  
\MR{0747560} (85f:05084).  \Zbl{567.05038}.
\annot{} A linear-time algorithm to determine balance or antibalance of the undirected signed graph of a signed digraph.  The algorithm of \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell} appears to be different.
\codes{\CSG:~\CBal,~\CPar:~\CAlgor}%
\itemspace

\author{John Maybee \& James Quirk} 
\authorspace
\bsgitem{1969a}{} Qualitative problems in matrix theory.  \emph{SIAM Rev.}\ 11 (1969), 30--51.  
\MR{0247866} (40 \#1127).  \Zbl{186.33503} (186, p.\ 335c).				
\annot{} An important early survey with new results.	
\par
\codes{\CQM,~\CSD:~\CSol,~\CSta,~\Cbal;~\CExp(in~part),~\CRef}%
\itemspace

\author{John S.\ Maybee \& Daniel J.\ Richman}
\authorspace
\bsgitem{1988a}{}  Some properties of GM-matrices and their inverses.  \emph{Linear Algebra Appl.}\ 107 (1988), 219--236.  
\MR{0960147} (89k:15039).  \Zbl{659.15021}.
\annot{} Square matrix $A$ is a GM-matrix if, for every positive and negative cycle $P$ and $N$ in its signed digraph, $V(P) \supseteq V(N)$.  Classification of irreducible GM-matrices; connections with the property that each $p \times p$ principal minor has sign $(-1)^p$; some conclusions about the inverse.
\par
\codes{\CSD:~\CQM}%
\itemspace

\bsgitem{1988b}{}  From qualitative matrices to quantitative restrictions.  \emph{Linear Multilinear Algebra} 22 (1988), no.\ 3, 229--248.
\MR{0937168} (89e:15032).  \Zbl{647.15001}.  
\codes{\CQM:~\CSD}%
\itemspace

\author{John S.\ Maybee \& Gerry M.\ Weiner}
\authorspace
\bsgitem{1987a}{}  $L$-functions and their inverses.  \emph{SIAM J.\ Algebraic Discrete Methods} 8 (1987), 67--76.  
\MR{0872057} (88a:26021).  \Zbl{613.15005}.					
\annot{} An $L$-function is a nonlinear generalization of a qualitative linear function.  Signed digraphs play a small role.
\codes{\CQM,~\CSD}%
\itemspace

\author{Stuart J.\ Maybee} 						%MAYBEE
\authorspace
\authref   See \hyperlink{John S.\ Maybee}{J.S.~Maybee}.
\itemspace

\authorx{W.\ Mayeda}			 			%MAYEDA
\author{W.\ Mayeda \& M.E.\ Van Valkenburg}
\authorspace
\bsgitem{1965a}{}  Properties of lossy communication nets.  \emph{IEEE Trans.\ Circuit Theory} CT-12 (1965), 334--338.  
\codes{\CGN}%
\itemspace

\author{Dillon Mayhew}						%MAYHEW
\authorspace
\authref  See also \hyperlink{Tara Fife}{T.~Fife} and \hyperlink{Daryl Funk}{D.~Funk}.
\authrefspace

\bsgitem{2005a}{}  Inequivalent representations of bias matroids.  \emph{Combin.\ Probab.\ Comput.}\ 14 (2005), 567--583.
\MR{2160419} (2006j:05040).  \Zbl{1081.05021}.					
\annot{} The number of inequivalent representations of a frame matroid over a fixed finite field is bounded, if the matroid does not have a free swirl $\bfF(2C_n,\eset)$ as a minor.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{Dillon Mayhew, Geoff Whittle, \& Stefan H.M.\ van Zwam}
\authorspace
\bsgitem{2011a}{}  An obstacle to a decomposition theorem for near-regular matroids.  \emph{SIAM J.\ Discrete Math.}\ 25 (2011), no.\ 1, 271--279.
\MR{2801229} (2012d:05097).  \Zbl{1290.05057}.  \arXiv{0905.3252}.			
\codes{\CSG:~\CMatrd}%
\itemspace

\author{R.\ Maynard} 						%MAYNARD
\authorspace
\authref  See \hyperlink{J.C.\ Angles d'Auriac}{J.C.~Angles~d'Auriac}, \hyperlink{Francisco Barahona}{F.~Barahona}, and \hyperlink{I.\ Bieche}{I.~Bieche}.
\itemspace

\author{J\'an Maz\'ak}						%MAZAK
\authorx{Jan Mazak}
\authorspace
\authref  See \hyperlink{Edita Macajova}{E.~M\'a\v{c}ajov\'a}.
\itemspace

\author{M.H.\ McAndrew}						%MCANDREW
\authorspace
\authref  See \hyperlink{D.R.\ Fulkerson}{D.R.~Fulkerson}.
\itemspace

\author{Richard McBride}						%MCBRIDE R
\authorspace
\authref  See \hyperlink{Heather Jordon}{H.~Jordon}.
\itemspace

\author{Richard D.\ McBride} 						%MCBRIDE RD
\authorspace
\authref  See also \hyperlink{Gerald G.\ Brown}{G.G.~Brown}.
\authrefspace

\bsgitem{1985a}{}  Solving embedded generalized network problems.  \emph{European J.\ Operational Res.}\ 21 (1985), 82--92. 
\Zbl{565.90038}.  
\annot{} Introducing the algorithm ``EMNET'', which employs embedded gene\-ral\-ized-network matrices (i.e., incidence matrices of real multiplicative gain graphs) with side constraints (i.e., extra rows) to speed up linear programming.
\dated{2 Oct 2009} 
\codes{\CGN:~\CInc:~\CAlgor}%
\itemspace

\bsgitem{1998a}{}  Progress made in solving the multicommodity flow problem.  \emph{SIAM J.\ Optim.}\ 8 (1998), no.\ 4, 947--955.
\MR{1641274} (99i:90110).  \Zbl{912.90128}.					
\annot{} Employing embedded generalized-network matrices to speed up linear programming.
\dated{2 Oct 2009} 
\codes{\CGN:~\CInc:~\CAlgor:~\CExp}%
\itemspace

\author{Richard D.\ McBride \& John W.\ Mamer}
\authorspace
\bsgitem{1997a}{}  Solving multicommodity flow problems with a primal embedded network simplex algorithm.  \emph{INFORMS J.\ Comput.}\ 9 (1997), no.\ 2, 154--163.
\MR{1477311}.  \Zbl{885.90040}.						
\codes{\CGN:~\CInc:~\CAlgor}%
\itemspace

\bsgitem{2004a}{}  Implementing an LU factorization for the embedded network simplex algorithm.  \emph{INFORMS J.\ Comput.}\ 16 (2004), no.\ 2, 109--119. 
\MR{2063190}.  
\annot{} Matrix factorization to speed up the method of \xref{McBride}{1985a}{Richard D.\ McBride}.
\dated{2 Oct 2009} 
\codes{\CGN:~\CInc:~\CAlgor}%
\itemspace

\author{Richard D.\ McBride \& Daniel E.\ O'Leary}
\authorspace
\bsgitem{1997a}{}  An intelligent modeling system for generalized network flow problems: With application to planning for multinational firms.  \emph{Ann.\ Operations Res.}\ 75 (1997), 355--372. 
\Zbl{894.90060}.  
\codes{\CGN:~\CInc:~\CAlgor}%
\itemspace

\author{H.\ Gilman McCann} 						%MCCANN
\authorspace
\authref  See \hyperlink{Eugene C.\ Johnsen}{E.C.~Johnsen}.
\itemspace 

\author{William McCuaig}						%MCCUAIG
\authorspace
\authref  See also \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\authrefspace

\bsgitem{1993a}{}  Intercyclic digraphs.  In:  Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 203--245.  Contemp.\ Math., Vol.\ 147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.
\MR{1224708} (94f:05062).  \Zbl{789.05042}.					
\annot{} Characterizes the digraphs with no two disjoint cycles as well as those with no two arc-disjoint cycles.  [Since cycles do not form a linear subclass of circles, this is not a biased-graphic theorem, but it might be of use in studying biased graphs that have no two disjoint balanced circles.  See \xref{Lov\'asz}{1965a}{Laszlo Lovasz}, \xref{Slilaty}{2007a}{Daniel C.\ Slilaty}.]
\codes{\CStr}%
\itemspace

\bsgitem{2000a}{}   Even dicycles.  \emph{J.\ Graph Theory} 35 (2000), no.\ 1, 46--68.
\MR{1775794} (2001f:\-05087).  \Zbl{958.05070}.					
\codes{\CSD:~\Cpar:~\CStr}%
\itemspace

\bsgitem{2001a}{}   Brace generation.  \emph{J.\ Graph Theory} 38 (2001), no.\ 3, 124--169.
\MR{1859786} (2002h:05136).  \Zbl{991.05086}.
\annot{} Results needed for \xrefd{2004a}{William McCuaig}.
\codes{\CSD:~\Cpar}%
\itemspace

\bsgitem{2004a}{\dag}  P\'olya's permanent problem.  \emph{Electronic J.\ Combin.}\ 11 (2004), Research Paper 79, 83 pp.
\MR{2114183} (2005i:05004).  \Zbl{1062.05066}.					
\annot{} See the description of \xref{Robertson, Seymour, and Thomas}{1999a}{Neil Robertson, P.D.\ Seymour, \& Robin Thomas}, who independently prove the main theorem.
\codes{\CSD:~\Cpar:~\CStr}\codesctd{~\CSG}%
\itemspace

\bsgitem{2005a}{}  When all dicycles have the same length.  Manuscript, ca.\ 2005.
\annot{} Uses the main theorem of \xrefd{2004a}{William McCuaig} and \xref{Robertson, Seymour, and Thomas}{1999a}{Neil Robertson, P.D.\ Seymour, \& Robin Thomas} to prove: A digraph has an edge weighting in which all cycles have equal nonzero total weight iff it does not contain a ``double dicycle'': a symmetric digraph whose underlying simple graph is a circle.  There is also a structural description of such digraphs.
\codes{\CSD:~\Cpar:~\CStr}\codesctd{\CSw}%
\itemspace

\author{William McCuaig, Neil Robertson, P.D.\ Seymour, \& Robin Thomas}
\authorspace
\bsgitem{1997a}{}  Permanents, Pfaffian orientations, and even directed circuits.  Extended abstract.  
In: \emph{Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing} (STOC 97, El Paso, Tex., 1997), pp.\ 402--405.  ACM Press, New York, 1997.
\Zbl{963.68153}.  
\annot{} Extended abstract of \xref{McCuaig}{2004a}{William McCuaig} and \xref{Robertson, Seymour, and Thomas}{1999a}{Neil Robertson, P.D.\ Seymour, \& Robin Thomas}.
\codes{\CSD:~\Cpar}
\itemspace

\author{W.D.\ McCuaig \& M.\ Rosenfeld}
\authorspace
\bsgitem{1985a}{}  Parity of cycles containing specified edges.  
In: B.R.\ Alspach and C.D.\ Godsil, eds., \emph{Cycles in Graphs}, pp.\ 419--431.  
Ann.\ Discrete Math., Vol.\ 27.  North-Holland Math.\ Stud., Vol.\ 115.  North-Holland, Amserdam, 1985.  
\MR{0821542} (87g:05139).  \Zbl{583.05037}.
\annot{} In a 3-connected graph, almost any two edges are in an even and an odd circle.  [By the negative-subdivision trick this generalizes to signed graphs.]
\codes{\CPar,~\Csg:~\CBal}%
\itemspace

\author{Judith J.\ McDonald}						%MCDONALD
\authorspace
\authref  See \hyperlink{Michael S.\ Cavers}{M.~Cavers} and \hyperlink{Steve Kirkland}{S.~Kirkland}.
\itemspace

\author{David D.\ McFarland}						%MCFARLAND
\authorspace
\authref  See \hyperlink{Maureen Hallinan}{M.~Hallinan}.
\itemspace

\author{Sean McGuinness}						%MCGUINNESS
\authorspace
\bsgitem{2020a}{}  Frame matroids, toric ideals, and a conjecture of White.  \emph{Adv.\ Appl.\ Math.}\ 118 (2020), art.\ 102042, 46 pp.
\MR{4088420}.  \Zbl{1440.05056}.  \arXiv{1910.08587}.  
\par
\codes{\CGG:~\CMatrd:~\CAlgeb}%
\itemspace

\authorx{Brendan D.\ McKay}						%MCKAY
\author{Brendan D.\ McKay, Mirka Miller, \& Jozef \v{S}ir\'a\v{n}}
\authorx{Brendan D.\ McKay, Mirka Miller, \& Jozef Siran}
\authorspace
\bsgitem{1998a}{}  A note on large graphs of diameter two and given maximum degree.  \emph{J.\ Combin.\ Theory Ser.\ B} 74 (1998), 110--118.
\MR{1644043} (99c:05108).  \Zbl{911.05031}.  
\annot{} Also see \xref{\v{S}iagiov\'a}{2001a}{Jana Siagiova}.
	
\codes{\CGG:~\CCov}%
\itemspace

\authorx{James McKee}						%MCKEE J
\author{James McKee \& Chris Smyth}
\authorspace
\bsgitem{2007a}{}  Integer symmetric matrices having all their eigenvalues in the interval $[-2,2]$.  \emph{J.\ Algebra} 317 (2007), 260--290.
\MR{2360149} (2008j:15038).  \Zbl{1140.15007}.  \arXiv{0705.3599}.				
\annot{} The matrices (except those of orders 1, 2) are signed-graph ``adjacency'' matrices $A$ with diagonal entries $0, 1, -1$.  There are 3 infinite families and a few sporadic examples of maximal such signed graphs; all of which satisfy $A^2=4I$.  
The proof uses ``charged signed graphs'', i.e., a signed graph with $0$, $+1$, or $-1$ attached to each vertex (and appearing on the diagonal of the adjacency matrix).  Switching a vertex negates the charge.  Dictionary:  ``strongly equivalent'' = switching isomorphic; ``bipartite'' = switching isomorphic to its negation (negation includes negating the charges).  
[The charged signed graphs are really oriented all-negative graphs with half edges.  The adjacency matrix is not $A(\Sigma)$ but an oriented adjacency matrix $\vec A$ defined by $\vec a_{ij} =$ net in-degree of $v_iv_j$ edges at $v_i$.]  
[Elementary proof for signed simple graphs:  \xref{Hou, Tang, and Wang}{2019a}{Yaoping Hou, Zikai Tang, \& Dijian Wang}.]
\dated{27 Jun 2008}
\codes{\CSG:~\Cpar:~\Cori:~\CEig}%
\annotctd{} P.\ 265 says a signed graph (without charges) that is switching isomorphic to its negation must be a bipartite graph.  [The Petersen graph with $E^- =$ \{alternating edges of a $C_6$\} is a counterexample (Alex Schaefer).  \Problem.  Characterize $\Sigma$ such that $-\Sigma \simeq \Sigma$.]
\dated{28 May 2016}
\par
\codes{\CSG:~\CSw,~\CAut}%

\bsgitem{2012a}{}  Integer symmetric matrices of small spectral radius and small Mahler measure.  \emph{Int.\ Math.\ Res.\ Not.}\ 2012 (2012), no.\ 1, 102--136.
\MR{2874929}.  \Zbl{1243.15020}.  \arXiv{0907.0371}.				
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{2020a}{}  Symmetrizable integer matrices having all their eigenvalues in the interval $[-2, 2]$.  \emph{Algebraic Combin.}\ 3 (2020), no.\ 3, 775--789.
\MR{4113606}.  \Zbl{1446.15016}.  \arXiv{2002.06082}.  
\codes{\CEig:~\Csg,~\CSw}%
\itemspace

\bsgitem{2020b}{}  Symmetrizable matrices, quotients, and the trace problem.  \emph{Linear Algebra Appl.}\ 600 (2020), 60--81.
\MR{4090841}.  \Zbl{1442.15016}.  
\codes{\CSG:~\CAdj}%
\itemspace

\author{Terry A.\ McKee} 						%MCKEE TA
\authorspace
\bsgitem{1984a}{}  Balance and duality in signed graphs. 
Proc.\ Fifteenth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing (Baton Rouge, 1984).  \emph{Congressus Numer.}\ 44 (1984), 11--18.  
\MR{0777525} (87b:05124).  \Zbl{557.05046}.
\codes{\CSG:~\CBal:~\CDual}%
\itemspace

\bsgitem{1987a}{}  A local analogy between directed and signed graphs.  \emph{Utilitas Math.}\ 32 (1987), 175--180.  
\MR{0921647} (89a:05075).  \Zbl{642.05023}.
\codes{\CSG:~\CDual,~\CClu,~\CBal}%
\itemspace

\bsgitem{2002a}{}  Chordally signed graphs.  \emph{Discrete Appl.\ Math.}\ 119 (2002), 273--280.
\MR{1906865} (2003d:05101).  \Zbl{1003.05051}.					
\annot{} A chordally signed graph is a chordal graph signed so every positive circle $C$ of length at least 4 has a chord such that $C \cup e$ is balanced.  Characterized in various ways.
\codes{\CSG}%
\itemspace

\bsgitem{2007a}{} Chordal multipartite graphs and chordal colorings.   \emph{Discrete Math.}\ 307 (2007), 2309--2314.
\MR{2340631} (2008f:05063).  \Zbl{1123.05042}.  
\annot{} P.\ 2312:  An auxiliary graph can be treated as signed; chordal coloring is signed-graph clustering.
\dated{11 Jul 2012} 
\codes{\CSG:~\CClu}%
\itemspace

\author{Kathleen A.\ McKeon} 						%MCKEON
\authorspace
\authref   See \hyperlink{Gary Chartrand}{G.~Chartrand}.
\itemspace

\author{Jennifer McNulty}						%MCNULTY
\authorspace
\authref  See also \hyperlink{Gary Gordon}{G.~Gordon}, \hyperlink{Torina Lewis}{T.~Lewis} and \hyperlink{Nancy Ann Neudauer}{N.A.~Neudauer}.
\authrefspace

\author{Jennifer McNulty \& Nancy Ann Neudauer}
\authorspace
\bsgitem{2008a}{}  On cocircuit covers of bicircular matroids.  \emph{Discrete Math.}\ 308 (2008), 4008--4013.
\MR{2418105} (2009e:05069).  \Zbl{1148.05022}.  
\codes{\CMatrd:~\CBic}%
\itemspace

\author{Luis Medina}							%MEDINA
\authorspace
\authref  See \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Germain Pasten}{G.~Past\'en}.
\itemspace

\author{Killian Meehan}						%MEEHAN
\authorspace
\authref  See \hyperlink{Yen Duong}{Y.~Duong}.
\itemspace

\authorx{Yotsanan Meemark}						%MEEMARK
\author{Yotsanan Meemark \& Borworn Suntornpoch}
\authorspace
\bsgitem{2014a}{}  Balanced unitary Cayley sigraphs over finite commutative rings.  \emph{J.\ Algebra Appl.}\ 13 (2014), no.\ 5, art.\ 1350152, 12 pp.
\MR{3190081}.  \Zbl{1291.05086}.  
\\
\codes{\CSG:~\CAlgeb:~\CBal}%
\itemspace

\author{O.\ Megalakaki}						%MEGALAKAKI
\authorspace
\authref  See \hyperlink{Abdelhakim El Maftouhi}{A.~El Maftouhi}.
\itemspace

\author{Nimrod Megiddo}						%MEGIDDO
\authorspace
\authref  See \hyperlink{Edith Cohen}{E.~Cohen} and \hyperlink{Dorit S.\ Hochbaum}{D.~Hochbaum}. 
\itemspace

\author{Ranjit Mehatari}						%MEHATARI
\authorspace
\authref  See also \hyperlink{Santanu Mandal}{S.~Mandal}.
\authrefspace

\author{Ranjit Mehatari, M.\ Rajesh Kannan, \& Aniruddha Samanta}
\authorspace
\bsgitem{2022a}{}  On the adjacency matrix of a complex unit gain graph.  \emph{Linear Multilinear Algebra} 70 (2022), no.\ 9, 1798--1813.
\MR{4429447}.  
\arXiv{1812.03747}.							
\par
\codes{\CGG:~\CAdj,~\CEig,~\CBal}%
\itemspace

\author{Ritu Rani Meherwal}						%MEHERWAL
\authorspace
\authref  See \hyperlink{G.N.\ Purohit}{G.N.~Purohit}.
\itemspace

\authorx{Kurt Mehlhorn}						%MEHLHORN
\author{Kurt Mehlhorn \& Dimitrios Michail}
\authorspace
\bsgitem{2005a}{}  Implementing minimum cycle basis algorithms.  
In:  S.E.\ Nikoletseas, ed., \emph{Experimental and Efficient Algorithms} (4th Int.\ Workshop, WEA 2005, Santorini Island, 2005), pp.\ 32--43.  \LNCS 3503.   Springer, Berlin, 2005.
\Zbl{1121.05314}.  
\annot{} The ``signed graph $G_i$'' is a signed covering graph $\tilde\Sigma_i$.  Used to find minimum cycle basis in a positively weighted graph $\Gamma$.  $\Sigma_i$ has negative edge set $S_i$, the ``witness set''.
\dated{6 Feb 2011} 
\codes{\CSG:~\CAlgor,~\CCov}%
\itemspace

\bsgitem{2006a}{}  Implementing minimum cycle basis algorithms.  \emph{ACM J.\ Exper.\ Algorithmics} 11 (2006), 14 pp.
\MR{2306622} (2007m:05139).  \Zbl{1143.05310}.					
\annot{} See \xrefd{2005a}{Kurt Mehlhorn \& Dimitrios Michail}.
\codes{\CSG:~\CAlgor,~\CCov}%
\itemspace

\author{Martin Mehlitz}						%MEHLITZ
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\authorx{Seema Mehra}						%MEHRA
\author{Seema Mehra \& Manjeet Singh}
\authorspace
\bsgitem{2017a}{}  Single valued neutrosophic signedgraphs \sic.  \emph{Int.\ J.\ Computer Appl.}\ 157 (2017), no.\ 9, 31--34.
\annot{} \Cf\ \xref{Mishra and Pal}{2016a}{S.N.\ Mishra \& A.\ Pal}.  Same confused definitions and trivial results, but neutrosophic.
\dated{30 Jul 2022} 
\codes{\CSG,~\CVS}%
\itemspace

\author{A.\ Mehrabian}						%MEHRABIAN
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\authorx{Wenjun Mei}						%MEI WJ
\author{Wenjun Mei, Ge Chen, Noah E.\ Friedkin, \& Florian D\"orfler}
\authorx{Wenjun Mei, Ge Chen, Noah E.\ Friedkin, \& Florian Dorfler}
\authorspace
\bsgitem{2022a}{}  Structural balance and interpersonal appraisals dynamics: Beyond all-to-all and two-faction networks.  \emph{Automatica} 140 (2022), art.\ 110239, 11 pp. 
\MR{4395969}.  
\arXiv{2012.10151}.  
\par
\codes{\CSD:~\CKG:~\CBal(\CGen),~\CDyn}%
\itemspace

\author{Wenjun Mei, Pedro Cisneros-Velarde, Ge Chen, Noah E.\ Friedkin, \& Francesco Bullo}	
\authorspace
\bsgitem{2019a}{}  Dynamic social balance and convergent appraisals via homophily and influence mechanisms.  \emph{Automatica} 110 (2019), art.\ 108580, 11 pp.
\MR{4395969}.  \Zbl{1429.91269}.  \arXiv{1710.09498}.  
\codes{\CSD:~\CKG:~\CBal,~\CDyn}%
\itemspace

\author{Sylvain Meignen}						%MEIGNEN
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\authorx{Ioannis N.\ Melas}						%MELAS
\author{Ioannis N.\ Melas, Regina Samaga, Leonidas G.\ Alexopoulos, \& Steffen Klamt}
\authorspace
\bsgitem{2013a}{}  Detecting and removing inconsistencies between experimental data and signaling network topologies using integer linear programming on interaction graphs.  \emph{PLoS Comput.\ Biol.}\ 9 (2013), no.\ 9, art.\ e1003204, 19 pp.
\MR{3131677} (no rev).  
\codes{\CSD,~\CBiol:~\CFr:~\CAlgor}%
\itemspace

\authorx{O.\ Melchert}						%MELCHERT
\author{O.\ Melchert \& A.K.\ Hartmann}
\authorspace
\bsgitem{2008a}{}  Ground states of 2D $\pm J$ Ising spin glasses via stationary Fokker--Planck sampling.  \emph{J.\ Stat.\ Mech.}\ 2008 (2008), art.\ P10019.
\Zbl{1459.82329}.  
\annot{} In other words, finding the frustration index of a signed planar graph.  The titular method seems to be less efficient than others.
\dated{9 Jan 2015} 
\codes{\CSG:~\CFr,~\CAlgor,~\CPhys}%
\itemspace

\author{Miguel A.\ Mel\'endez-Jim\'enez}					%MELENDEZ-JIMENEZ
\authorx{Miguel A.\ Melendez-Jimenez}
\authorspace
\authref  See \hyperlink{Antonio Parravano}{A.~Parravano}.
\itemspace

\author{Avraham A.\ Melkman}						%MELKMAN
\authorspace
\authref  See \hyperlink{Tatsuya Akutsu}{T.~Akutsu}.
\itemspace

\authorx{C.\ Mendes Ara\'ujo}						%MENDES CM
\authorx{C.\ Mendes Araujo}
\author{C.\ Mendes Ara\'ujo \& Juan R.\ Torregrosa}
\authorx{C.\ Mendes Araujo \& Juan R.\ Torregrosa}
\authorspace
\bsgitem{2009a}{}  Sign pattern matrices that admit $M$-, $N$-, $P$- or inverse $M$-matrices.  \emph{Linear Algebra Appl.}\ 421 (2009), 724--731.
\MR{2535545} (2010e:15035).  \Zbl{1170.15008}.						
\annot{} The digraphs treated are acyclic or cycles.  See Thm.\ 3.4 on signed cycles.  Also, the ``2-cycle property'' means a negative 2-cycle.
\dated{29 Sept 2012} 
\codes{\CQM:~\Csd:~\CAdj}%
\itemspace

\bsgitem{2011a}{}  Sign pattern matrices that admit $P_{0}$-matrices.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 8, 2046--2053.
\MR{2810645} (2012e:15064).  \Zbl{1222.15034}.						
\annot{} Almost identical to \xrefd{2009a}{C.\ Mendes Araujo \& Juan R.\ Torregrosa} with ``$P_{0}$-matrices'' replacing ``$M$-, $N$-, $P$- or inverse $M$-matrices''.  Treats directed circles as well as cycles and acyclic digraphs.
\dated{29 Sept 2012} 
\codes{\CQM:~\Csd:~\CAdj}%
\itemspace

\author{J.F.F.\ Mendes}						%MENDES JFF
\authorspace
\authref  See \hyperlink{M.\ Ostilli}{M.~Ostilli}.
\itemspace

\author{Marco A.\ Mendez}						%MENDEZ
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\authorx{Miguel A.\ Mendez}						%MENDEZ
\author{Miguel A.\ M\'endez \& Jos\'e L.\ Ram{\'\i}rez}
\authorx{Miguel A.\ Mendez \& Jose L.\ Ramirez}
\authorspace
\bsgitem{2019a}{}  A new approach to the $r$-Whitney numbers by using combinatorial differential calculus.  \emph{Acta Univ.\ Sapientiae Math.}\ 11 (2019), no.\ 2, 387--418.
\MR{4072595}.  \Zbl{1472.11084}.  \arXiv{1702.06519}.  
\codes{\Cgg:~\CMatrd(\CGen):~\CInvar}%
\itemspace

\authorx{Israel Mendon\c{c}a}						%MENDONCA
\authorx{Israel Mendonca}
\author{Israel Mendon\c{c}a, Rosa Figueiredo, Vincent Labatut, \& Philippe Michelon}
\authorx{Israel Mendonca, Rosa Figueiredo, Vincent Labatut, \& Philippe Michelon}
\authorspace
\bsgitem{2015a}{}  Relevance of negative links in graph partitioning: A case study using votes from the European Parliament.  In:  \emph{2nd European Network Intelligence Conference} (ENIC, Karlskrona, Sweden, 2015), pp.\ 122--129.
\arXiv{1507.04215}.  \HAL{hal-01176090}.  
\codes{\CSG:~\CClu:~\CAlgor,~\CAppl}%
\itemspace

\bsgitem{2015b}{}  Informative Value of negative links in graph partitioning, with an application to European Parliament votes.  In:  \emph{6\'eme Conf\'erence Mod\`eles et Analyse de R\'eseaux: approches math\'ematiques et informatiques (MARAMI)} (N\^{\i}mes, France, 2015), 13 pp.
\HAL{hal-01176090}.  
\codes{\CSG:~\CClu:~\CAlgor,~\CAppl}%
\itemspace

\author{Deyuan Meng}						%MENG DY
\authorspace
\authref  See also \hyperlink{Mingjun Du}{M.J.~Du}.
\authrefspace

\bsgitem{2017a}{}  Bipartite containment tracking of signed networks.  \emph{Automatica} 79 (2017), 282--289.
\MR{3627993}.  \Zbl{1371.93010}.  
\codes{\CSD:~\CAlgor}%
\itemspace

\bsgitem{2018a}{}  Dynamic distributed control for networks with cooperative--antagonistic interactions.  \emph{IEEE Trans.\ Automatic Control} 63 (2018), no.\ 8, 2311--2326.
\MR{3845965}.  \Zbl{1423.93025}.  
\codes{\CSG:~\CDyn}%
\itemspace

\author{Deyuan Meng, Ziyang Meng, \& Yiguang Hong}
\authorspace
\bsgitem{2018a}{}  Uniform convergence for signed networks under directed switching topologies.  \emph{Automatica} 90 (2018), 8--15.
\MR{3764379}.  \Zbl{1387.93130}.  
\codes{\CSD:~\CDyn:~\CAdj}%
\itemspace

\author{Jie Meng}							%MENG JE
\authorspace
\authref  See \hyperlink{Jianer Chen}{J.-E.~Chen}.
\itemspace

\author{Jixiang Meng}						%MENG JX
\authorspace
\authref  See \hyperlink{Dan Li}{D.~Li} and \hyperlink{Lili Yuan}{L.-L.~Yuan}.
\itemspace

\author{Ziyang Meng}						%MENG ZY
\authorspace
\authref  See \hyperlink{Deyuan Meng}{D.-Y.~Meng}.
\itemspace

\author{Mircea Merca}						%MERCA
\authorspace
\bsgitem{2013a}{}  A note on the $r$-Whitney numbers of Dowling lattices.  \emph{C.R.\ Acad.\ Sci.\ Paris, Ser.\ I} 351 (2013), no.\ 17-18, 649--655.
\MR{3124320}.  \Zbl{1277.05167}.  
\annot{} \Cf\ \xref{Mez\H{o}}{2010a}{Istvan Mezo}.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\authorx{Pedro Mercado}						%MERCADO
\author{Pedro Mercado, Francesco Tudisco, \& Matthias Hein}
\authorspace
\bsgitem{2016a}{}  Clustering signed networks with the geometric mean of Laplacians.  In:  D.D.\ Lee \etal, eds., \emph{Advances in Neural Information Processing Systems 29} (NIPS 29, Barcelona, 2016), 9 pp.  
URL \url{https://papers.nips.cc/book/advances-in-neural-information-processing-systems-29-2016}
\par
\arXiv{1701.00757}.  
\codes{\CSG:~\CClu:~\CLap,~\CAlgor}%
\itemspace

\bsgitem{2019a}{}  Spectral clustering of signed graphs via matrix power means.  In:  Kamalika Chaudhuri and Ruslan Salakhutdinov, eds., \emph{Proceedings of the 36th International Conference on Machine Learning} (ICML 2019, Long Beach, Calif.), 11 pp.\ + 17 suppl.  Proc.\ Machine Learning Res., Vol.\ 97.  Machine Learning Research Press, 2019.
\arXiv{1905.06230}.  
\codes{\CSG:~\CClu:~\CAdj}%
\itemspace

\author{Valeriu Mereacre}						%MEREACRE
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\author{M.A.\ Mermet}						%MERMET
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Leanne Merrill}						%MERRILL
\authorspace
\authref  See \hyperlink{Yen Duong}{Y.~Duong}.
\itemspace

\author{Russell Merris}						%MERRIS
\authorspace
\bsgitem{1994a}{}  Laplacian matrices of graphs: a survey.  \emph{Linear Algebra Appl.}\ 197/198 (1994), 143--176.
\MR{1275613} (95e:05084).  \Zbl{802.05053}.					
\annot{} Thm.:  $\Spec \Lap(\Gamma) = \Spec \Lap(-\Gamma)$ iff $\Gamma$ is bipartite.  [\Cf\ antibalanced case of \xref{B.D.~Acharya}{1980a}{B.\ Devadas Acharya}.]
\dated{21 Jan 2012} 
\par
\codes{\CPar:~\CLap:~\CEig,~\Cbal}%
\itemspace

\bsgitem{1995a}{}  A survey of graph Laplacians.  \emph{Linear Multilinear Algebra}\ 39 (1995), no.\ 1--2, 19--31.
\MR{1374468} (97c:05104).  \Zbl{832.05081}.  
\codes{\Cpar:~\CLap:~\CEig}
\itemspace

\author{Mehran Mesbahi}						%MESBAHI
\authorspace
\authref  See \hyperlink{Mathias Hudoba de Badyn}{M.H.~de~Badyn}.
\itemspace

\author{Roy Meshulam}						%MESHULAM
\authorspace
\authref  See \hyperlink{R.\ Aharoni, R.\ Meshulam, \& B.\ Wajnryb}{R.~Aharoni} and \hyperlink{Jeff Kahn \& Roy Meshulam}{J.~Kahn}.
\itemspace

\author{Nacim Meslem}						%MESLEM
\authorspace
\authref  See \hyperlink{Nacim Ramdani}{N.~Ramdani}.
\itemspace

\author{Robert Messer}						%MESSER
\authorspace
\authref  See \hyperlink{Edward M.\ Brown}{E.M.~Brown}.
\itemspace

\author{Thomas Mestl}						%MESTL
\authorspace
\authref  See \hyperlink{Erik Plahte}{E.~Plahte}.
\itemspace

\author{G\'abor M\'esz\'aros}						%MESZAROS G
\authorx{Gabor Meszaros}
\authorspace
\authref  See \hyperlink{Teeradej Kittipassorn}{T.~Kittipassorn}.
\itemspace

\author{Karola M\'esz\'aros}						%MESZAROS K
\authorx{Karola Meszaros}
\authorspace
\bsgitem{2011a}{}  Root polytopes, triangulations, and the subdivision algebra, II.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 363 (2011), no.\ 11, 6111--6141. 
\MR{2817421} (2012g:52021).  \Zbl{1233.05216}.  \arXiv{0904.3339}.			
\annot{} A signed simple graph generates a polytope $P(\Sigma)$ whose volume is calculated.  
\dated{11 Sept 2010} 
\codes{\CSG:~\CGeom}%
\itemspace

\bsgitem{2012a}{}  Demystifying a divisibility property of the Kostant partition function.  \emph{Pacific J.\ Math.}\ 260 (2012), no.\ 1, 215--225.
\MR{3001792}.  \Zbl{1256.05015}.  \arXiv{1101.0388}.				
\annot{} [Positive edges are called ``negative'' and vice versa.]  Flows on signed graphs give combinatorial interpretations of and identities for the partition functions.  Negative (``positive'') edges are introverted, so flow goes in and disappears.
\dated{25 Mar 2013} 
\codes{\CSG:~\CFlows:~\CAppl}%
\itemspace

\author{Karola M\'esz\'aros \& Alejandro H.\ Morales}
\authorx{Karola Meszaros \& Alejandro H.\ Morales}
\authorspace
\bsgitem{2012a}{}  Flow polytopes and the Kostant partition function for signed graphs (extended abstract).  In:  \emph{24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)} (Nagoya, 2012), pp.\ 947--959.  D.M.T.C.S.\ Proc.\ Discrete Math.\ \& Theor.\ Computer Sci.\ (DMTCS), Nancy, 2012.
\MR{2964267} (volume).  \Zbl{1412.05088}.  
\annot{} See \xrefd{2015a}{Karola Meszaros \& Alejandro H.\ Morales}.
\codes{\CSG:~\CFlows,~\CGeom}%
\itemspace

\bsgitem{2015a}{}  Flow polytopes of signed graphs and the Kostant partition function.  \emph{Int.\ Math.\ Res.\ Notes} 2015 (2015), no.\ 3, 830--871.
\MR{3340339}.  \Zbl{1307.05097}.  \arXiv{1208.0140}.				
\annot{} [The edge signs are the opposite of what they ought to be.]	
\par
\codes{\CSG:~\CFlows,~\CGeom}%
\itemspace

\authorx{Fr\'ed\'eric Meunier}						%MEUNIER
\authorx{Frederic Meunier}
\author{Fr\'ed\'eric Meunier \& Andr\'as Seb\H{o}}
\authorx{Frederic Meunier \& Andras Sebo}
\authorspace
\bsgitem{2009a}{}  Paintshop, odd cycles and necklace splitting.  \emph{Discrete Appl.\ Math.}\ 157 (2009), 780--793.
\MR{2499492} (2010e:90102).  \Zbl{1163.90774}.					
\annot{} The negative circles of a signed graph and its minimal and minimum balancing sets are crucial. 
	
Dictionary:  ``signed graph'' = $(|\Sigma|,E^-)$, ``odd cycle'' = negative circle, ``odd cycle clutter'' = $\cB^c(\Sigma)$, ``transversal'' = ``uncut'' = minimal balancing set, ``BIP$(G,F)$'' = MinBalSet$(\Sigma)$ (the problem of finding a minimum balancing set), ``resigning'' = switching.
\dated{22 Sept 2010} 
\codes{\CSG:~\CFr}%
\itemspace

\author{David A.\ Meyer}						%MEYER DA
\authorspace
\authref  See \hyperlink{Dongjin Song}{D.J.~Song}.
\itemspace

\author{Seth A.\ Meyer}						%MEYER SA
\authorspace
\authref  See \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}.
\itemspace

\author{Hildegard Meyer-Ortmanns}					%MEYER-ORTMANNS
\authorspace
\authref  See \hyperlink{Filippo Radicchi}{F.~Radicchi}.
\itemspace

\author{Andrew M.\ Meyers}						%MEYERS
\authorspace
\authref  See \hyperlink{Nancy Ann Neudauer}{N.A.~Neudauer}.
\itemspace

\author{Erika Meza}							%MEZA
\authorspace
\authref  See \hyperlink{Matthias Beck}{M.~Beck}.
\itemspace

\authorx{Marc M\'ezard}						%MEZARD
\authorx{Marc Mezard}
\author{Marc M\'ezard, Giorgio Parisi, \& Miguel Angel Virasoro}
\authorx{Marc Mezard, Giorgio Parisi, \& Miguel Angel Virasoro}
\authorspace
\bsgitem{1987a}{}  \emph{Spin Glass Theory and Beyond}.  World Scientific Lect.\ Notes in Physics, Vol.\ 9.  World Scientific, Singapore, 1987.
\MR{1026102} (91k:82066).  
\annot{} Focuses on the Sherrington--Kirkpatrick model, i.e., underlying complete graph, emphasizing the Parisi-type model (see articles reprinted herein), which posits numerous metastable states, separated by energy barriers of greatly varying heights and subdividing as temperature decreases (\cf\ \xref{Kirkpatrick and Sherrington}{1978a}{Scott Kirkpatrick \& David Sherrington}).  Essentially heuristic (as noted in MR): that is, the ideas awaited [and still largely await] mathematical justification.
\annotctd{} Many original articles on Ising and vector models (both of which are based on weighted signed graphs) are reprinted herein, though few are of general signed-graphic interest.
\annotctd{} [Also see, i.a., \xref{Toulouse}{1977a}{Gerard Toulouse} \etal, \xref{Chowdhury}{1986a}{Debashish Chowdhury}, 
\xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz}, \xref{Vincent, Hammann, and Ocio}{1992a}{E.\ Vincent, J.\ Hammann, \& M.\ Ocio} for physics, \xref{Barahona}{1982a}{Francisco Barahona}, etc., \xref{Gr\"otschel, J\"unger, and Reinelt}{1987a}{M.\ Grotschel, M.\ Junger, \& G.\ Reinelt} for mathematics.]
\annotctd{} [Metastable states in the model appear to correspond to local minima of the state frustration function of the underlying weighted signed graph $(\Sigma, w)$, with ultrametric distance function $d(s, s') := \min_P \max_{s'' \in P} H(s'')$, where $P$ ranges over all paths $P : s \to s'$ in the graph of states and $H$ is the Hamiltonian of a state of $(\Sigma, w)$, equivalently the (weighted) frustration of the state.  \Problem.  Study this metric on the state space of various signed $K_N$'s and other signed graphs.  Possibly this will shed light on the physics; it will certainly be interesting for signed graphs.]  
\par
\codes{\CPhys,~\CSG:~\CFr,~\CState:~\CExp,~\CRef}%
\annotctd{} Ch.\ 0, ``Introduction'', briefly compares, in the obvious way, balance in social psychology [they neglect to mention the original paper, \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}] with frustration in spin glasses. 
\par
\codes{\CPhys,~\CPsS:~\CSG:~\CBal:~\CExp}%
\annotctd{} Pt.\ 1, ``Spin glasses'', Ch.\ II, ``The TAP approach'': pp.\ 19--20 describe $1$-vertex switching of a weighted signed graph to reduce frustration, not however necessarily producing the frustration index (minimum frustration).  \Question.  How does the valley (``basin of attraction'') of a ground state (minimum frustration) compare with the valley of a metastable state (locally minimum frustration); in particular can it be much smaller?
\dated{rev.\ 15 Aug 2018} 
\par
\codes{\CPhys:~\CSG:~\CFr,~\CSw,~\CAlgor:~\CExp}%
\itemspace

\author{Istv\'an Mez\H{o}}						%MEZO
\authorx{Istvan Mezo}
\authorspace
\bsgitem{2010a}{}  A new formula for the Bernoulli polynomials.  \emph{Results.\ Math.}\ 58 (2010), 329--335.
\MR{2728160} (2011k:11036).  \Zbl{1237.11010}.  
\annot{} Introduces the $r$-Whitney numbers [of Dowling lattices, not of geometric lattices in general].  $r=1$ are the original numbers.  
[The numbers are popular.  \Cf\ \xref{Cheon and Jung}{2012a}{Gi-Sang Cheon \& Ji-Hwan Jung}, \xref{Merca}{2013a}{Mircea Merca}, \xref{Rahmani}{2014a}{Mourad Rahmani}, \xref{Mez\H{o}}{2014a}{Istvan Mezo}, \xref{Gyimesi and Nyul}{2018a}{Eszter Gyimesi \& Gabor Nyul}, \xref{Kim and Kim}{20xxa}{Taekyun Kim \& Dae San Kim}, etc.]
\dated{rev.\ 28 May 2018} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2014a}{}  A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices.  \emph{Discrete Math.}\ 328 (2014), 88--95.
\MR{3199820}.  \Zbl{1288.05016}.  
\annot{} \Cf\ \xrefd{2010a}{Istvan Mezo}, \xref{Cheon and Jung}{2012a}{Gi-Sang Cheon \& Ji-Hwan Jung}.
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Lu Miao}							%MIAO L
\authorspace
\authref  See \hyperlink{Ruifang Liu}{R.-F.~Liu}.
\itemspace

\author{Zhengke Miao}						%MIAO ZK
\authorspace
\authref  See \hyperlink{Hongping Ma}{H.P.~Ma}, \hyperlink{Guanglong Yu}{G.L.~Yu}, and \hyperlink{Longqin Wang}{L.Q.~Wang}.
\itemspace

\authorx{Isaac B.\ Michael}						%MICHAEL IB
\author{Isaac B.\ Michael \& Mark R.\ Sepanski}
\authorspace
\bsgitem{2016a}{}  Net regular signed trees.  \emph{Australasian J.\ Combin.}\ 66 (2016), no.\ 2, 192--204.
\MR{3556127}.  \Zbl{1375.05125}.  
\codes{\CSG:~\CStr,~\CEnum}%
\itemspace

\author{T.S.\ Michael}						%MICHAEL TS
\authorspace
\bsgitem{2002a}{}  Signed degree sequences and multigraphs.  \emph{J.\ Graph Theory} 41 (2002), 101--105.
\MR{1926311} (2003g:05042).  \Zbl{1012.05052}.						
\annot{} Characterizes net degree sequences of signed graphs with fixed maximum edge multiplicity.  [See \xref{Chartrand, Gavlas, Harary, and Schultz}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz} for explanation.]
\codes{\CSGw:~\CInvar,~\CAlgor}%
\itemspace

\author{Dimitrios Michail}						%MICHAIL
\authorspace
\authref  See \hyperlink{Kurt Mehlhorn}{K.~Mehlhorn}.
\itemspace

\author{Philippe Michelon}						%MICHELON
\authorspace
\authref  See \hyperlink{Israel Mendonca}{I.~Mendon\c{c}a}.
\itemspace

\author{Manuel Middendorf}						%MIDDENDORF
\authorspace
\authref  See \hyperlink{Etay Ziv}{E.~Ziv}.
\itemspace

\author{A.\ Alan Middleton}						%MIDDLETON
\authorspace
\authref  See \hyperlink{Creighton K.\ Thomas}{C.K.~Thomas}.
\itemspace

\author{Anna de Mier}						%MIER
\authorspace
\authref  See \hyperlink{Omer Gimenez}{O.~Gim\'enez}.
\itemspace

\author{S.\ Migowsky}						%MIGOWSKY
\authorspace
\authref  See \hyperlink{T.\ Wanschura}{T.~Wanschura}.
\itemspace

\author{\v{S}tefko Miklavi\v{c}}						%MIKLAVIC
\authorx{Stefko Miklavic}
\authorspace
\authref  See \hyperlink{Istvan Kovacs}{I.~Kov\'acs}.
\itemspace

\author{Alexander R.\ Miller}						%MILLER AR
\authorspace
\bsgitem{2015a}{}  Foulkes characters for complex reflection groups.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 143 (2015), no.\ 8, 3281--3293.
\MR{3348771}.  \Zbl{1314.05225}.  
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2015b}{} Eigenspace arrangements of reflection groups.  \emph{Trans.\ Amer.\ Math.\ Soc.}\  367 (2015), no.\ 12, 8543--8578.
\MR{3403065}.  \Zbl{1333.20041}.  \arXiv{1208.1944}.				
\par
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Mirka Miller}							%MILLER M*
\authorspace
\authref  See \hyperlink{C.\ Dalf'o}{C.~Dalf'o} and \hyperlink{Brendan D.\ McKay}{B.D.~McKay}.
\itemspace

\author{Raymond E.\ Miller}						%MILLER RE
\authorspace
\authref   See \hyperlink{Richard M.\ Karp}{R.M.~Karp}.
\itemspace

\author{William P.\ Miller}						%MILLER WP
\authorspace
\authref   See also \hyperlink{Joseph E.\ Bonin}{J.E.~Bonin}.
\authrefspace

\bsgitem{1997a}{}  Techniques in matroid reconstruction.  \emph{Discrete Math.}\ 170 (1997), 173--183.
\MR{1452942} (98f:05039).  \Zbl{878.05020}.						
\annot{} Dowling geometries are reconstructible from their hyperplanes, their deletions, and their contractions.
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Geyong Min}							%MIN GY
\authorspace
\authref  See \hyperlink{Fei Hao}{F.~Hao}.
\itemspace

\author{Maya Mincheva}						%MINCHEVA
\authorspace
\authref  See also \hyperlink{Georg R.\ Walther}{G.R.~Walther}.
\authrefspace

\author{Maya Mincheva \& Gheorghe Craciun}
\authorspace
\bsgitem{2008a}{}  Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks.  \emph{Proc.\ IEEE} 96 (2008), no.\ 8, 1281--1291.
\par
\codes{\CSD:~\CChem,~\CBiol:~\CDyn:~\CExp}%
\itemspace

\author{Edward Minieka} 						%MINIEKA
\authorspace
\bsgitem{1972a}{}  Optimal flow in a network with gains.  \emph{INFOR} 10 (1972), 171--178.  
\Zbl{234.90012}.  
\codes{\CGN:~\CMatrd(\Cindep),~\CBal}%
\itemspace

\bsgitem{1978a}{}  \emph{Optimization Algorithms for Networks and Graphs}.  Marcel Dekker, New York and Basel, 1978.  
\MR{0517268} (80a:90066).  \Zbl{427.90058}.
\annot{} \S4.6:  ``Flows with gains,'' pp.\ 151--174.  Also see pp.\ 80--81.
\par
\codes{\CGN:~\CBal,~\CSw,~\Cmatrd(\Cindep):~\CExp}%
\itemspace

\bsgitem{1981a}{}  \emph{Algoritmy Optimizatsii na Setyakh i Grafakh}.  
Transl.\ M.B.\ Katsnel'son and M.I.\ Rubinshte\u\i n; ed.\ E.K.\ Maslovski\u\i.  Mir, Moskva, 1981.
\MR{0641852} (83f:\-90118).  \Zbl{523.90058}.
\annot{} Russian translation of \xrefd{1978a}{Edward Minieka}.
\codes{\CGN:~\CBal,~\CSw,~\Cmatrd(\Cindep):~\CExp}%
\itemspace

\author{Maryam Mirzakhah}						%MIRZAKHAH
\authorspace
\authref  See also \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Mohammadreza Jooyandeh}{M.~Jooyandeh}.
\authrefspace

\author{M.\ Mirzakhah \& D.\ Kiani}
\authorspace
\bsgitem{2010a}{}  The Sun graph is determined by its signless Laplacian spectrum.  \emph{Electronic J.\ Linear Algebra} 20 (2010), 610--620.
\MR{2735977} (2011j:05209).  \Zbl{1205.05149}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  Some results on signless Laplacian coefficients of graphs.  \emph{Linear Algebra Appl.}\ 437 (2012), 2243--2251.
\MR{2954486}.  \Zbl{1247.05141}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Gr\'egoire Misguich}						%MISGUICH
\authorx{Gregoire Misguich}
\author{Gr\'egoire Misguich and Claire Lhuillier}
\authorx{Gregoire Misguich and Claire Lhuillier}
\authorspace
\bsgitem{2004a}{}  Two-dimensional quantum antiferromagnets.  In:  H.T.\ Diep, ed., \emph{Frustrated Spin Systems}, Ch.\ 5, pp.\ 229--306.  World Scientific, Hackensack, N.J., 2004.
\annot{} I.a., details of ground-state spin alignments in XY and Heisenberg models ($S^2$ and $S^3$ spins) on simple periodic lattices.
\dated{13 Aug 2018} 
\codes{\CSG,~\CPhys:~\CFr:~\CExp,~\CRef}%
\itemspace

\authorx{S.N.\ Mishra}						%MISHRA SN
\author{S.N.\ Mishra \& A.\ Pal}
\authorspace
\bsgitem{2016a}{}  Intuitionistic fuzzy signed graphs.  \emph{Int.\ J.\ Pure Appl.\ Math.}\ 106 (2016), no.\ 6, 113--122.
\annot{} Edges and possibly vertices of fuzzy graph are signed.  Definitions are confused and incomplete.  Trivial results.  [More in \xref{Mehra and Singh}{2017a}{Seema Mehra \& Manjeet Singh} and \xref{Moudgalya and Permi}{2022a}{Madhwesha Moudgalya R \& Kavita Permi}.]
\dated{30 Jul 2022} 
\par
\codes{\CSG,~\CVS,~\CFr}%
\itemspace

\author{V.\ Mishra} 							%MISHRA V
\authorspace
\bsgitem{1974a}{}  \emph{Graphs Associated With $(0,+1,-1)$ Arrays.}  Doctoral thesis, Indian Institute of Technology, Bombay, 1974.
\annot{} The arrays are matrices.	
\annotctd{$\otimes$} Defines tensor product $\Sigma_1 \otimes \Sigma_2$ to have an edge $(v_1,v_2)(w_1,w_2)$ iff $v_1w_1$ and $v_2w_2$ are edges, with sign $\sigma((v_1,v_2)(w_1,w_2)):=\sigma_1(v_1w_1)\sigma_2(v_2w_2)$ (\cf\ \xref{Sinha and Garg}{2014a}{Deepa Sinha \& Pravin Garg}).
\dated{23 Nov 2014} 
\codes{\CSG}%
\itemspace

\author{U.K.\ Misra}							%MISRA UK
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Lon H.\ Mitchell}						%MITCHELL
\authorspace
\authref  See \hyperlink{Nancy Matar}{N.~Matar}.
\itemspace

\author{G.\ Mitra}							%MITRA G
\authorspace
\authref  See \hyperlink{N.\ Gulpinar}{N.~G\"ulpinar}.
\itemspace

\author{S.\ Mitra}							%MITRA S
\authorspace
\bsgitem{1962a}{}  Letter to the editors.  \emph{Behavioral Sci.}\ 7 (1962), 107.
\annot{} Treats signed simple graphs via the \xref{Abelson--Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} structure matrix $R$.  Observes that balance holds iff $R = rr\transpose$ for some vector $r \in \{p,n\}^{V}$; also, asserts that frustration index $l(\Sigma) =$ minimum number of negative edges over all switchings of $\Sigma$.  [Proved in \xref{Barahona, Maynard, Rammal, and Uhry}{1982a}{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}.]  
Asserts an algorithm for computing $l(\Sigma)$: switch vertices whose negative degree exceeds positive degree, one at a time, until no such vertices remain [incorrect: consider $K_6$, all positive except a negative $C_6$].
\dated{corr.\ 20 Jan 2010}
\par
\codes{\Csg:~\Ckg:~\CAdj,~\Csw,~\CFr}%
\itemspace

\author{Sovan Mitra}							%MITRA S
\authorspace
\authref  See \hyperlink{V.\ Vasanthi}{V.~Vasanthi}.
\itemspace

\author{Michael Mitzenmacher}						%MITZENMACHER
\authorspace
\authref  See \hyperlink{Charalampos E.\ Tsourakakis}{C.E.~Tsourakakis}.
\itemspace

\author{Valia Mitsou}							%MITSOU
\authorspace
\authref  See \hyperlink{Francois Dross}{F.~Dross} and \hyperlink{Florent Foucaud}{F.~Foucaud}.
\itemspace

\author{Seiji Miyashita}						%MIYASHITA
\authorspace
\authref  See \hyperlink{Ojiro Nagai}{O.~Nagai}.
\itemspace

\authorx{Hirobumi Mizuno}						%MIZUNO
\author{Hirobumi Mizuno \& Iwao Sato}
\authorspace
\bsgitem{1997a}{}  Enumeration of finite field labels on graphs.  \emph{Discrete Math.}\ 176 (1997), 197--202.
\MR{1477289} (98e:05059).  \Zbl{893.05015}.					
\annot{} Isomorphism types, under the action of a subgroup of $\Aut\Gamma$, of coboundaries of 1-chains $f: V \to \bbF_q^+$ in $-\Gamma$.  (In other words, the edge labels are $\delta f(uv) = f(u)+f(v)$.)  
[\Question.  Does it generalize to signed graphs?  The subgroup would be of $\Aut\Sigma$, or one can count isomorphism types of switching classes under a subgroup of $\Aut[\Sigma]$.]
\dated{16 Jan 2012}
\par
\codes{\Cpar:~\Cincid}%
\itemspace

\bsgitem{2000a}{}  Zeta functions of graph coverings.  \emph{J.\ Combin.\ Theory Ser.\ B} 80 (2000), 247--257.
\MR{1794694}.  \Zbl{1023.05114}.  
\codes{\CGG}%
\itemspace

\bsgitem{2004a}{}  Weighted zeta functions of graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 91 (2004), 169--183.
\MR{2064866}.  \Zbl{1048.05044}.  
\codes{\CGG}%
\itemspace

\bsgitem{2010a}{}  Weighted scattering matrices of regular coverings of graphs.  \emph{Linear Multilinear Algebra} 58 (2010), no.\ 7, 927--940.
\MR{2742326} (2011k:05145).  \Zbl{1231.05170}.  
\par
\codes{\CGGw:~\CCov:~\CInvar,~\CAdj:~\CGen}%
\itemspace

\author{[Swathyprabhu Mj]}						%MJ
\authorspace
\authref  See \hyperlink{Swathyprabhu Mj}{Swathyprabhu} under `S'.
\itemspace

\author{Iain Moffatt}							%MOFFATT
\authorspace
\authref  See also \hyperlink{Joanna A.\ Ellis-Monaghan}{J.A.~Ellis-Monaghan}, \hyperlink{Thomas Krajewski}{T.~Krajewski}, and \hyperlink{Martin Loebl}{M.~Loebl}.
\authrefspace

\bsgitem{2010a}{}  Partial duality and Bollob\'as and Riordan's ribbon graph polynomial.  \emph{Discrete Math.}\ 310 (2010), no.\ 1, 174--183.
\MR{2558980} (2011b:05112).  \Zbl{1229.05123}.  \arXiv{0809.3014}.			
\codes{\CSG:~\CTop,~\CInvar,~\CDual}%
\itemspace

\bsgitem{2011a}{}  Unsigned state models for the Jones polynomial.  \emph{Ann.\ Combin.}\ 15 (2011), no.\ 1, 126--146.
\MR{2785760} (2012b:05087).  \Zbl{1235.05072}.  \arXiv{0710.4152}.			
\annot{} Vertex models (graphs with vertices labelled $\pm1$) and Potts models (edges labelled $\pm1$) can be replaced by unsigned models by converting an edge-labelled graph into an orientable ribbon graph.  A limited parametrized rank-corank polynomial appears (in the standard way) as the Potts partition function.
\dated{23 Apr 2009}
\codes{\CSGc:~\CInvar}%
\itemspace

\bsgitem{2013a}{}  Separability and the genus of a partial dual.  \emph{European J.\ Combin.}\ 34 (2013), 355--378.
\MR{2994404}.  \Zbl{1254.05047}.  \arXiv{1108.3526}.				
\codes{\Csg:~\CTop,~\CDual}%
\itemspace

\bsgitem{2016a}{}  Ribbon graph minors and low-genus partial duals.  \emph{Ann.\ Combin.}\ 20 (2016), 373--378.
\MR{3505325}.  \Zbl{1339.05377}.  \arXiv{1502.00269}.				
\codes{\CSG:~\CTop,~\CDual}%
\itemspace

\author{A.R.\ Moghaddamfar}						%MOGHADDAMFAR
\authorspace
\authref  See \hyperlink{Y.\ Bagheri}{Y.~Bagheri}.
\itemspace

\author{Javad Mohajeri}						%MOHAJERI
\authorspace
\authref  See \hyperlink{Siamak Fayyaz Shahandashti}{S.~Fayyaz Shahandashti}.
\itemspace


\author{Bojan Mohar}						%MOHAR
\authorspace
\authref  See also \hyperlink{Gary Greaves}{G.~Greaves}, \hyperlink{Krystal Guo}{K.~Guo}, and \xref{Kawarabayashi and Mohar}{2006a}{Ken-ichi Kawarabayashi \& Bojan Mohar}.
\authrefspace

\bsgitem{1989a}{}  An obstruction to embedding graphs in surfaces.  \emph{Discrete Math.}\ 78 (1989), 135--142.
\MR{1020656} (90h:05046).  \Zbl{686.05019}.
\annot{} The ``overlap matrix'' of a signed graph with respect to a rotation system and a spanning tree provides a lower bound on the demigenus that sometimes improves on that from Euler's formula.
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{2016a}{}  Hermitian adjacency spectrum and switching equivalence of mixed graphs.  \emph{Linear Algebra Appl.}\ 489 (2016), 324--340.
\MR{3421853}.  \Zbl{1327.05215}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges [as in \xref{Liu and Li}{2015a}{Jianxi Liu \& Xueliang Li}].  Switching.  Adjacency cospectrality, including spectrally unique families.  \Cf\ \xref{Guo and Mohar}{2017a}{Krystal Guo \& Bojan Mohar}.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CSw,~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  A new kind of Hermitian matrices for digraphs.  \emph{Linear Algebra Appl.}\ 584 (2020), 343--352.
\MR{4013179}.  \Zbl{1426.05009}.  \arXiv{1909.10878}.  
\annot{} The graphs are treated as (and are) gain graph with gains $\sqrt[6]{1}$ in the edge direction, the inverse in the opposite direction.  This is said to be more natural than gains $\sqrt[4]{1}$ as in \xrefd{2016a}{Bojan Mohar}, \xref{Guo and Mohar}{2017a}{Krystal Guo \& Bojan Mohar}.  
[\Cf\ \xref{Li and Yu}{2022a}{Shuchao Li \& Yuantian Yu}.]
\dated{23 Mar 2022} 
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Bojan Mohar \& Svatopluk Poljak}
\authorspace
\bsgitem{1993a}{}  Eigenvalues in combinatorial optimization.  
In: Richard A.\ Brualdi, Shmuel Friedland, and Victor Klee, eds., \emph{Combinatorial and Graph-Theoretical Problems in Linear Algebra}, pp.\ 107--151.  
IMA Vols.\ Math.\ Appl., 50.  Springer-Verlag, New York, 1993. 
\MR{1240959} (95e:90003).  \Zbl{806.90104}.
\annot{} Switching of a weight function on an unsigned graph (p.\ 119), from C.~Delorme and S.~Poljak, Combinatorial properties and the complexity of a max-cut approximation, Tech.\ Rep.\ 91687, Inst.\ Diskrete Math., Universit\"at Bonn, 1991.
\dated{13 Apr 2009}
\codes{\CSw}%
\itemspace

\author{Bojan Mohar \& Paul D.\ Seymour}
\authorspace
\bsgitem{2002a}{}  Coloring locally bipartite graphs on surfaces.  \emph{J.\ Combin.\ Theory Ser.\ B} 84 (2002), no.\ 2, 301--310.
\MR{1889261} (2003b:05059).  \Zbl{1079.05027}.  
\annot{} 
[Also see \xref{Nakamoto, Negami, and Ota}{2002a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}, \xrefd{2004a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}.]
\codes{\Csg:~\CTop:~\Csw}%
\itemspace

\author{Bojan Mohar \& Carsten Thomassen}
\authorspace
\bsgitem{2001a}{}  \emph{Graphs on Surfaces}.  Johns Hopkins Stud.\ Math.\ Sci.  Johns Hopkins Univ.\ Press, Baltimore, 2001.
\MR{1844449} (2002e:05050).  \Zbl{979.05002}.					
\annot{} \S3.3, ``Embedding schemes'', surveys rotation systems and edge signatures for embedding in nonorientable surfaces.  \Cf\ \xref{Ringel}{1977a}{Gerhard Ringel} and \xref{Stahl}{1978a}{Saul Stahl}.  
\S4.1, ``Embeddings combinatorially'':    Detailed treatment of embeddings from rotation systems and optionally an edge signature.  
[\xref{Lins}{1982a}{Sostenes Lins}, \xrefd{1985a}{Sostenes Lins}, \xref{\v{S}ir\'a\v{n} and \v{S}koviera}{1991a}{Jozef Siran \& Martin Skoviera}, \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}, \xrefd{1993a}{Thomas Zaslavsky}, \etal\ are regrettably never mentioned in this valuable book.]
\par
\codes{\Csg:~\CTop}%
\itemspace

\author{Afshin Moin}							%MOIN
\authorspace
\authref  See \hyperlink{Anne-Marie Kermarrec}{A.-M.~Kermarrec}.
\itemspace

\author{G.H.\ Mokashi}						%MOKASHI
\authorspace
\authref  See \hyperlink{Kenchappa S.\ Betageri}{K.S.~Betageri}.
\itemspace

\authorx{Kartik Chandra Mondal}					%MONDAL
\author{Kartik Chandra Mondal, Valeriu Mereacre, George E.\ Kostakis, Yanhua Lan, Christopher E.\ Anson, Ion Prisecaru, Oliver Waldmann, \& Annie K.\ Powell}
\authorspace
\bsgitem{2015a}{}  A strongly spin-frustrated Fe${}^\textrm{III}{}_7$ complex with a canted intermediate spin ground state of $S=7/2$ or $9/2$.  \emph{Chem.\ European J.}\ 2015 (2015), no.\ 21, 10835--10842.
\annot{} Novel arrangements of frustrated signs with $S^1$ spins in a ground state.  Especially see Figs.\ 1, 4.
\dated{20 Mar 2016} 
\codes{\CPhys:~\Csg:~\CState,~\CFr}%
\itemspace

\author{[Keivan Hassani Monfared]}					%MONFARED
\authorspace
\authref  See \hyperlink{Keivan Hassani Monfared}{K.~Hassani Monfared} (under `H').
\itemspace

\author{G.\ Monroy}							%MONROY
\authorspace
\authref  See \hyperlink{Antonio Coniglio}{A.~Coniglio}.
\itemspace

\author{Marco Montalva}						%MONTALVA
\authorspace
\authref  See also \hyperlink{Julio Aracena}{J.~Aracena}.
\authrefspace

\author{Marco Montalva, Julio Aracena, \& Anah{\'\i} Gajardo}
\authorspace
\bsgitem{2008a}{}  On the complexity of feedback set problems in signed digraphs.  IV Latin-American Algorithms, Graphs, and Optimization Sympos.\ (Puerto Varas, Chile, 2007).  \emph{Electronic Notes Discrete Math.}\ 30 (2008), 249--254.
\MR{2570648} (no rev).  \Zbl{1341.05112}.					
\annot{} Complexity of finding a minimum set of vertices, or arcs, that covers all positive, or negative, cycles in a signed digraph.  All are NP-complete, by polynomial-time reduction to the existence problems Even Cycle and Odd Cycle in the positive and negative problems, respectively.  [Directed frustration index and directed vertex frustration number are the negative-cycle cover problems, which are said to be easier than the positive-cycle cover problems.]
\dated{20 July 2009} 
\par
\codes{\CSD:~\CFr:~\CGen,~\CAlgor}%
\itemspace

\author{Mickael Montassier}						%MONTASSIER
\authorspace
\authref  See \hyperlink{Christopher Duffy}{C.~Duffy}.
\itemspace

\authorx{Amanda Montejano}						%MONTEJANO
\author{Amanda Montejano, Pascal Ochem, Alexandre Pinlou, Andr\'e Raspaud, \& \'Eric Sopena}
\authorx{Amanda Montejano, Pascal Ochem, Alexandre Pinlou, Andre Raspaud, \& Eric Sopena}
\authorspace
\bsgitem{2008a}{}  Homomorphisms of 2-edge-colored graphs.  IV Latin-American Algorithms, Graphs, and Optimization Symposium.  \emph{Electronic Notes Discrete Math.}\ 30 (2008), 33--38.
\MR{2570612} (no rev).  \Zbl{1341.05085}.  \HAL{lirmm-00196757}.  
\annot{} Preliminary version of \xrefd{2010a}{Amanda Montejano, Pascal Ochem, Alexandre Pinlou, Andre Raspaud, \& Eric Sopena}.
\codes{\CSGc:~\CHom}%
\itemspace

\bsgitem{2010a}{}  Homomorphisms of 2-edge-colored graphs.  \emph{Discrete Appl.\ Math.}\ 158 (2010), no.\ 12, 1365--1379.
\MR{2652013}.  \Zbl{1341.05085}.  \HAL{lirmm-00433039}.  
\par
\codes{\CSGc:~\CHom}%
\itemspace

\author{James D.\ Montgomery}					%MONTGOMERY
\authorspace
\bsgitem{2009a}{}  Balance theory with incomplete awareness.  \emph{J.\ Math.\ Sociology} 33 (2009), 69--96.
\Zbl{1169.91438}.  
\annot{} Signed digraphs with possible multiple arcs of different sign, with two types of vertices (``actors'' having positive and possibly negative loops, and ``objects'' having no loops), and with extra ``awareness'' arcs between actor vertices.  Emphasis on directionality of arcs.   
``Boolean multiplication'' [Boolean Hadamard product] of separate positive, negative, and awareness adjacency matrices to form mixed adjacency matrices.  
Assumption:  Over time the signed digraph evolves towards sign-transitive closure constrained by the awareness arcs, whose absence impedes transitive closure.  Four specific ``mechanisms'' are postulated for the evolution, of which two are essential (Lemma 1).  Propositions present conclusions (no surprises) about intermediate and final (i.e., constrained sign-transitively closed) signed digraphs.  
Dictionary:  ``balance closure'' = sign-transitive closure, i.e., arc-transitive closure with positive triple sign.  
[The idea of constrained closure is mathematically intriguing, though the notation is heavy.]  
[For more on sign-transitive closure in signed digraphs see \xref{Doreian and Krackhardt}{2001a}{Patrick Doreian \& David Krackhardt}.]
\dated{16 Apr 2009}
\codes{\CSD,~\CPsS:~\CBal}%
\itemspace

\author{Angelo Monti}						%MONTI
\authorspace
\authref  See \hyperlink{Tiziana Calamoneri}{T.~Calamoneri}.
\itemspace

\author{Elliott W.\ Montroll}						%MONTROLL*
\authorspace
\bsgitem{1964a}{}  Lattice statistics.  In: Edwin F.\ Beckenbach, ed., \emph{Applied Combinatorial Mathematics}, Ch.\ 4, pp.\ 96--143.  Wiley, New York, 1964.
\MR{0174486} (30 \#4687) (book).  \Zbl{141.15503} (141, p.\ 155c).
\annot{} \S4.4: ``The Pfaffian and the dimer problem''.  Exemplified by the square lattice, expounds Kasteleyn's method of signing edges to make the Pfaffian term signs all positive.  Partial proofs.  \S4.7, ``The Ising problem'', pp.\ 127--129, explains application to the Ising model.  Exceptionally readable.  [Further development in, e.g., \xref{Vazirani and Yannakakis}{1988a}{Vijay V.\ Vazirani \& Mihalis Yannakakis}, \xrefd{1989a}{Vijay V.\ Vazirani \& Mihalis Yannakakis}.]
\codes{\CSG,~\CPhys:~\CExp}%
\itemspace

\author{J.W.\ Moon} 							%MOON JW
\authorspace
\bsgitem{1966a}{}  A note on approximating symmetric relations by equivalence relations.  \emph{SIAM J.\ Appl.\ Math.}\ 14 (1966), no.\ 2, 226--227.
\MR{0205865} (34 \#5691).  \Zbl{66.00802} (66, 8b).					
\annot{} Observes that $l_{\clu}(K_n,\sigma)/\binom{n}{2}\leq\frac12$.  Thm.:   Let $\epsilon>0$.  When $n\gg0$, $l_{\clu}(K_n,\sigma)/\binom{n}{2} \geq \frac12-\epsilon$ for almost all $\sigma$.  Sequel to \xref{Zahn}{1964a}{C.T.\ Zahn, Jr.}.
\dated{10 Nov 2017} 
\codes{\Csg:~\CClu}%
\itemspace

\author{J.W.\ Moon \& L.\ Moser}
\authorspace
\bsgitem{1966a}{}  An extremal problem in matrix theory.  \emph{Mat.\ Vesnik} N.S.\ 3(18) (1966), 209--211.  
\MR{0207570} (34 \#7385).  \Zbl{146.01401} (146, p.\ 14a).
\annot{} Studies the maximum frustration index of a signed $K_{r,s}$.  
\codes{\Csg:~\CFr}%
\itemspace

\author{Suck Joong Moon}						%MOON SJ
\authorspace
\authref  See also \hyperlink{Hideo Kosako}{H.~Kosako}.
\authrefspace

\author{Suck-Joong Moon \& Hideo Kosako}
\authorspace
\bsgitem{1993a}{}  The variable-signed graph and its application.  In:  \emph{TENCON '93: IEEE Region 10 Conference on Computer, Communication, Control and Power Engineering} (Proc., 1993), Vol.\ 4, pp.\ 522--525.
\annot{} See \xref{Kosako, Moon, \etal{}}{1993a}{Hideo Kosako, Suck Joong Moon, Katsumi Harashima, \& Takeo Ikai}
	
\codes{\CSG,~\CVS:~\CSw,~\Cfr,~\CDual,~\CInc}%
\itemspace

\author{M.A.\ Moore} 						%MOORE
\authorspace
\authref  See \hyperlink{A.J.\ Bray}{A.J.~Bray}.
\itemspace

\author{G.\ Eric Moorhouse}						%MOORHOUSE
\authorspace
\bsgitem{1995a}{}  Two-graphs and skew two-graphs in finite geometries.  \emph{Linear Algebra Appl.}\ 226/\-228 (1995), 529--551.
\MR{1344584} (96f:51012).  \Zbl{839.05024}.					
\annot{} Two-graphs = switching classes of signed $K_n$'s (\cf\ \xref{Seidel}{1976a}{J.J.\ Seidel}).  Skew two-graphs = switching classes of $\bbZ_3$-gain graphs on $K_n$ (\cf\ \xref{Cameron}{1977b}{Peter J.\ Cameron}, \S8; [``two-digraphs'' in \xref{Cheng and Wells}{1984a}{Ying Cheng \& Albert L.\ Wells, Jr.}, \xref{Cheng}{1986a}{Ying Cheng}]).  Applied to construct invariants of structures in finite geometry.
\dated{5 Aug 2018} 
\par
\codes{\Csg,~\Cgg:~\Ckg:~\CTG,~\CTG(\CGen),~\CAdj,~\CSw,~\CInvar:~\CGeom}%
\itemspace

\author{Alejandro H.\ Morales}						%MORALES AH
\authorspace
\authref  See \hyperlink{Karola Meszaros}{K.~M\'esz\'aros}.
\itemspace

\author{Katherine Tapia Morales}					%MORALES KT
\authorspace
\authref  See \hyperlink{Maria Robbiano}{M.~Robbiano}.
\itemspace

\author{A.\ Moreira}							%MOREIRA
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Susan Morey}						%MOREY
\authorspace
\authref  See \hyperlink{Louiza Fouli}{L.~Fouli}.
\itemspace

\author{Aki Mori}							%MORI
\authorspace
\authref  See \hyperlink{Takayuki Hibi}{T.~Hibi}.
\itemspace

\author{Michio Morishima} 						%MORISHIMA
\authorspace
\bsgitem{1952a}{}  On the laws of change of the price-system in an economy which contains complementary commodities.  \emph{Osaka Economic Papers} 1 (1952), 101--113.  
\annot{} \S4: ``Alternative expression of the assumptions (1),'' can be interpreted with hindsight as proving that, for a signed $K_n$, every triangle is positive iff the signature switches to all positive.  (Everything is done with sign-symmetric matrices, not graphs, and switching is not mentioned in any form.)  
\codes{\Csg:~\Ckg:~\Cbal,~\Csw}%
\itemspace

\author{Robert Morris}						%MORRIS R
\authorspace
\authref  See \hyperlink{Henry Liu}{H.~Liu}.
\itemspace

\author{Timothy Morris}						%MORRIS T
\authorspace
\authref  See \hyperlink{Arthur H.\ Busch}{A.H.~Busch}.
\itemspace

\author{Julian O.\ Morrissette}						%MORRISSETTE
\authorspace
\bsgitem{1958a}{} An experimental study of the theory of structural balance.  \emph{Human Relations} 11 (1958), 239--254.  
\annot{} Proposes that edges have strengths between $-1$ and $+1$ instead of pure signs.  The \xref{Cartwright--Harary degree of balance}{1956a}{Dorwin Cartwright \& Frank Harary}, computed from circles, is modified to take account of strength.  In addition, signed graphs are allowed to have edges of two types, say $U$ and $A$, and only short mixed-type circles enter into the degree of balance.  This is said to be more consistent with the experimental data reported herein.
\par
\codes{\CPsS,~\CSG,~\CGen:~\CFr}%
\itemspace

\author{Julian O.\ Morrissette \& John C.\ Jahnke} 
\authorspace
\bsgitem{1967a}{} No relations and relations of strength zero in the theory of structural balance.  \emph{Human Relations} 20 (1967), 189--195.
\annot{} Reports an experiment; then discusses problems with and alternatives to the \xref{Cartwright--Harary}{1956a}{Dorwin Cartwright \& Frank Harary} circle degree of balance.
\codes{\CPsS:~\CFr}%
\itemspace

\author{Julian O.\ Morrissette, John C.\ Jahnke, \& Keith Baker}
\authorspace
\bsgitem{1966a}{}  Structural balance:  A test of the completeness hypothesis.  \emph{Behavioral Sci.}\ 11 (1966), no.\ 2, 121--125.
\annot{} Proposes to measure degree of balance by $c^+(\Sigma)/c(K_n)$ instead of $c^-(\Sigma)/c(|\Sigma|)$ as in \xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}, to overcome logical incompatibility between the latter measure, the principle of increasing balance, and an assumed tendency towards completeness in a (signed) graph of social relations; as well as for experimental reasons.
\dated{3 Sept 2013} 
\codes{\CSG,~\CPsS:~\CBal}%
\itemspace

\author{Hannes Moser}						%MOSER H
\authorspace
\authref  See \hyperlink{Jiong Guo}{J.~Guo}.
\itemspace

\author{L.\ Moser} 							%MOSER L*
\authorspace
\authref  See \hyperlink{J.W.\ Moon}{J.W.~Moon}.
\itemspace

\author{Tyler Moss}							%MOSS
\authorspace
\authref  See \hyperlink{Deborah Chun}{D.~Chun}.
\itemspace

\author{Sebastiano Mosterts} 						%MOSTERTS
\authorspace
\authref   See \hyperlink{Ellis L.\ Johnson}{E.L.~Johnson}.
\itemspace

\author{Satish V.\ Motammanavar}					%MOTAMMANAVAR
\authorspace
\authref  See \hyperlink {H.B.\ Walikar} {H.B.~Walikar}.
\itemspace

\authorx{Fatemeh Motialah}						%MOTIALAH
\author{Fatemeh Motialah \& Mohammad Hassan Shirdareh Haghighi}
\authorspace
\bsgitem{2019a}{}  Laplacian spectral characterization of signed sun graphs.  \emph{Theory Appl.\ Graphs} 6 (2019), no.\ 2, art.\ 3, 9 pp.
\MR{4042255}.  \Zbl{1429.05130}.  
\annot{} Sun = circle with one pendant edge at each vertex.  Determined by Laplacian spectrum iff unbalanced or has odd girth.
\dated{15 Dec 2020} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\authorx{Madhwesha Moudgalya R}					%MOUDGALYA
\author{Madhwesha Moudgalya R \& Kavita Permi}
\authorspace
\bsgitem{2022a}{}  Connectivity indices of labeled signed graphs using domination degree.  \emph{J.\ Algebraic Stat.}\  13 (2022), no.\ 3, 3280 - 3287
\annot{} \Cf\ \xref{Mishra and Pal}{2016a}{S.N.\ Mishra \& A.\ Pal}.  Same confused definitions, with badly defined ``topological indices''; obvious results.
\dated{30 Jul 2022} 
\codes{\CSG,~\CVS}%
\itemspace

\author{C.F.\ Moukarzel}						%MOUKARZEL
\authorspace
\authref  See \hyperlink{M.J.\ Alava}{M.J.~Alava}.
\itemspace

\author{Gisele Moura}						%MOURA
\authorspace
\authref  See \hyperlink{Rosa Figueiredo}{R.~Figueiredo}.
\itemspace

\author{Nazanin Movarrae}						%MOVARRAE
\authorspace
\authref  See \hyperlink{Pascal Ochem}{P.~Ochem}.
\itemspace

\author{[Eunice Gogo Mphako]}						%MPHAKO
\authorspace
\authref  See \hyperlink{Eunice Mphako-Banda}{E.~Mphako-Banda}.
\itemspace

\authorx{Eunice Mphako-Banda}					%MPHAKO-BANDA
\author{Eunice Mphako-Banda [Eunice Gogo Mphako]}
\authorspace
\bsgitem{2002a}{}  (as Eunice Gogo Mphako)  The component number of links from graphs.  \emph{Proc.\ Edinburgh Math.\ Soc.}\ 45 (2002), 723--730.
\MR{1933752} (2003g:05046).  \Zbl{009.05048}.  
\annot{} \S 5:  Bracket polynomials of signed matroids, after \xref{Schw\"arzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh}.
\dated{21 May 2013} 
\codes{\CSgnd(\CMatrd):~\CInvar}%
\itemspace

\bsgitem{2004a}{}  (as Eunice Gogo Mphako)  $H$-lifts of tangential $k$-blocks.  \emph{Discrete Math.}\ 285 (2004), 201--210.
\MR{2062843} (2005c:05046).  \Zbl{1044.05032}.  
\annot{} $H$ is a group.  ``$H$-lift'' $L_r(M)$ of $M:=\bfF(\Phi)$ means $\bfF(\Phi \cup (H\cdot S_r)^{(v_r)})$ where $\Phi$ is an $H$-gain graph with $V(\Phi) = \{v_1,\ldots,v_{r-1}\}$ and $S_r$ is the star with center $v_r$ and pendant vertices $v_1,\ldots,v_{r-1}$.  Main result:  Thm.\ 3.13:  $p_{L_r(M)}(\lambda) = (\lambda-1)p_{M}(\lambda-\#H)$.  
\S5, ``$H$-lifts of tangential blocks'': $L_r(M\full)$ is a tangential block if $M\full:=\bfF(\Phi\full)$ is one.
[This lift is not the lift matroid.]
\dated{20 Oct 2020} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\bsgitem{2015a}{}  $H$-Trees, restrictions of Dowling group geometries.  \emph{Bull.\ Korean Math.\ Soc.}\ 52 (2015), no.\ 3, 955--962.
\MR{3353304}.  \Zbl{1328.05038}.  
\annot{} $H$ is a finite group.  ``Complete $H$-tree'', ``-path'', ``-star'', ``-cycle'', ``fan'', ``wheel'' = full frame matroid $\bfF(H\cdot\Delta\full)$ where $\Delta$ is a tree, path, etc.  [The stated definition of $H$-tree is incomplete.]  Spanning submatroids are called ``$H$-trees'', etc.  Props.\ 3.1, 5.1 list simple properties.  Thm.\ 3.2 evaluates $\#E(H\cdot\Delta\full)$.  Thms.\ give the characteristic polynomials [they are special cases of \xref{Zaslavsky}{1995b}{Thomas Zaslavsky}, \S\S4, 5].
\dated{20 Oct 2020} 
\par
\codes{\Cgg:~\CMatrd,~\CInvar}%
\itemspace

\author{Andrzej Mr\'oz}						%MROZ
\authorspace
\authref  See \hyperlink{Jesus Arturo Jimenez Gonzalez}{J.A.~Jimenez~G}.
\itemspace

\author{Andrej Mrvar}  						%MRVAR
\authorspace
\authref   See also \hyperlink{Michael Brusco}{M.~Brusco}, \hyperlink{Patrick Doreian}{P.~Doreian} and \hyperlink{Wouter de Nooy}{W.~de~Nooy}.
\authrefspace

\author{Andrej Mrvar \& Patrick Doreian}
\authorspace
\bsgitem{2009a}{}  Partitioning signed two-mode networks.  \emph{J.\ Math.\ Sociology} 33 (2009), no.\ 3, 196--221. 
\Zbl{1168.91511}.  
\annot{} \S2, ``Formalization of block-modeling signed two-mode data'':  A signed two-mode network is a signed simple bipartite graph with color classes $V_1, V_2$.  The objective is partitions $\pi_1,\pi_2$ of $V_1, V_2$ that minimize a ``criterion function'' $P := \alpha i_- + (1-\alpha)i_+$; usually $\alpha = .5$.  $k_1:=\#\pi_1$ and $k_2:=\#\pi_2$, or other restrictions, may be specified.  
Definitions:  $\pi_i := \{V_{i1},\ldots,V_{ik_i}\}.$  A ``block'' is a nonvoid set $E(V_{1i},V_{2j})$.  Its sign is the sign of the majority of edges, $+$ if a draw.  $e$ is ``consistent'' with $(\pi_1,\pi_2)$ if it is in a block of sign $\sigma(e)$.  $i_\epsilon :=$ number of inconsistent edges of sign $\epsilon$.  
\dated{17 Aug 2009}
\codes{\CSG:~\CClu,~\CPsS}%
\itemspace

\authorx{Guohong Mu}						%MU GH
\author{Guohong Mu, Lulu Li, \& Xiaodi Li}	
\authorspace
\bsgitem{2020a}{}  Quasi-bipartite synchronization of signed delayed neural networks under impulsive effects.  \emph{Neural Networks}\ 129 (2020), 31--42.
\Zbl{1478.93658}.  
\annot{} Balanced signed digraphs, not switched to all positive [which would be simpler].
\dated{19 Oct 2020} 
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\authorx{Lili Mu}							%MU LL
\author{Lili Mu \& Richard P.\ Stanley}
\authorspace
\bsgitem{2015a}{}  Supersolvability and freeness for $\psi$-graphical arrangements.  \emph{Discrete Comput.\ Geom.}\ 53 (2015), 965--970.
\MR{3341588}.  \Zbl{1314.05076}.  \arXiv{1501.07612}.				
\annot{} \Cf\ \xref{Stanley}{2015a}{Richard P.\ Stanley}.
\dated{16 Nov 2018} 
\codes{\Cgg:~\CGeom}%
\itemspace

\author{G.\ Muciaccia}						%MUCIACCIA
\authorspace
\authref  See \hyperlink{R.\ Crowston, G.\ Gutin, M.\ Jones, \& G.\ Muciaccia}{R.~Crowston}.
\itemspace

\author{Antara Mukherjee}						%MUKHERJEE
\authorspace
\authref  See \hyperlink{Matthew Blair}{M.~Blair}.
\itemspace

\author{Raffaella Mulas}						%MULAS
\authorspace
\authref  See also \hyperlink{Aida Abiad}{A.~Abiad}, \hyperlink{Eleonora Andreotti}{E.~Andreotti}, and \hyperlink{Jurgen Jost}{J.~Jost}.
\authrefspace

\bsgitem{20xxa}{}  Sharp bounds for the largest eigenvalue of the normalized hypergraph Laplace operator.  Submitted. \emph{}\ 
\arXiv{2004.02154}.  
\codes{\CSH:~\CLap:~\CEig}%
\itemspace

\bsgitem{20xxb}{}  Spectral classes of hypergraphs.  Submitted.  \emph{}\ 
\arXiv{2007.04273}.  
\par
\codes{\CSH,~\Clg:~\CAdj,~\CLap,~\CIncid:~\CEig}%
\itemspace

\author{Raffaella Mulas, Christian Kuehn, \& J\"urgen Jost}
\authorspace
\bsgitem{2020a}{}  Coupled dynamics on hypergraphs: Master stability of steady states and synchronization.  \emph{Phys.\ Rev.\ E} 101 (2020), art.\ 062313, 6 pp.
\MR{4121260}.  \arXiv{2003.13775}.  
\codes{\CSH:~\CLap:~\CEig}%
\itemspace

\author{Raffaella Mulas \& Nathan Reff}
\authorspace
\bsgitem{20xxa}{}  Spectra of complex unit hypergraphs.  Submitted.  \emph{}\ 
\arXiv{2011.10458}.  
\par
\codes{\CSH(\CGen):~\CAdj,~\CLap,~\CIncid:~\CEig}%
\itemspace

\author{Raffaella Mulas, Rub\'en J.\ S\'anchez-Garc{\'\i}a, and Ben D.\ MacArthur}
\authorspace
\bsgitem{20xxa}{}  Hypergraph automorphisms.  Submitted.  \emph{}\ 
\arXiv{2010.01049}.  
\par
\codes{\CSH:~\CAut,~\CLap:~\CEig}%
\itemspace

\author{Raffaella Mulas \& Zoran Stani\'c}
\authorx{Raffaella Mulas \& Zoran Stanic}
\authorspace
\bsgitem{2022a}{}  Star complements for $\pm2$ in signed graphs.  \emph{Special Matrices} 10 (2022), 258--266.
\MR{4381526}.  \Zbl{1484.05136}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Raffaella Mulas \& Dong Zhang}
\authorspace
\bsgitem{2021a}{}  Spectral theory of Laplace operators on oriented hypergraphs.  \emph{Discrete Math.}\ 344 (2021), no.\ 6, art.\ 112372, 22 pp.
\MR{4236395}.  
\arXiv{2004.14671}.  
\par
\codes{\CSH:~\CLap:~\CEig}%
\itemspace

\author{Haiko M\"uller}						%MULLER
\authorx{Haiko Muller}
\authorspace
\authref  See \hyperlink{Ton Kloks}{T.~Kloks}.
\itemspace

\authorx{Ganesh Mundhe}						%MUNDHE
\author{Ganesh Mundhe, Y.M.\ Borse, \& K.V.\ Dalvi}
\authorspace
\bsgitem{2022a}{}  On graphic elementary lifts of graphic matroids.  \emph{Discrete Math.}\ 345 (2022), no.\ 10, art.\ 113014.
\MR{4435894}.  \Zbl{1491.05048}.  \arXiv{1910.05689}.  
\annot{} Lifts via binary coextensions.  The 6 forbidden minors for graphic matroids all of whose binary coextensions are graphic.
\dated{20 Jun 2022} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Akihiro Munemasa}						%MUNEMASA
\authorspace
\authref  See also \hyperlink{Alexander L.\ Gavrilyuk}{A.L.~Gavrilyuk}, \hyperlink{Gary Greaves}{G.~Greaves}, \hyperlink{Masaaki Harada}{M.~Harada}, and \hyperlink{Hye Jin Jang}{H.J.~Jang}.
\authrefspace

\author{Akihiro Munemasa, Yoshio Sano, \& Tetsuji Taniguchi}
\authorspace
\bsgitem{2014a}{}  Fat Hoffman graphs with smallest eigenvalue at least $-1-\tau$.  \emph{Ars Math.\ Contemp.}\  7 (2014), no.\ 1, 247--262.
\MR{3084550}.  \Zbl{1301.05221}.  \arXiv{1111.7284}.				
\codes{\CSG:~\CEig}%
\itemspace

\authorx{Luigi Muracchini}				 		%MURACCHINI
\author{Luigi Muracchini \& Anna Maria Ghirlanda}
\authorspace
\bsgitem{1965a}{} Sui grafi segnati ed i grafi commutati.  \emph{Statistica} (Bologna) 25 (1965), 677--680.  
\MR{0199122} (33 \#7272).  
\annot{} A partially successful attempt to use unoriented signed graphs to define a line graph of a digraph.  [See \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky} for the correct signed-graph approach.]  The Harary--Norman line digraph is also discussed.
\codes{\CSG:~\CBal,~\CLG}%
\itemspace

\author{Kunio Murasugi}						%MURASUGI
\authorspace
\bsgitem{1988a}{}  On the signature of a graph.  \emph{C.R.\ Math.\ Rep.\ Acad.\ Sci.\ Canada} 10 (1988), 107--111.
\MR{0933223} (89h:05056).  
\annot{} The signature of a sign-colored graph (see \xrefd{1989a}{Kunio Murasugi}) is an invariant of the sign-colored graphic matroid.
\codes{\CSGc:~\CInc,~\Cmatrd}%
\itemspace

\bsgitem{1989a}{}  On invariants of graphs with applications to knot theory.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 314 (1989), 1--49.  
\MR{0930077} (89k:57016).  \Zbl{726.05051}.						
\annot{} Studies a dichromatic form, $P_\Sigma(x,y,z)$, of \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} Tutte polynomial of a sign-colored graph.  The deletion-contraction parameters are $a_\epsilon = 1$, $b_\epsilon = x^\epsilon$ for $\epsilon = \pm1$; the initial values are such that $P_\Sigma(x,y,z) = y\inv Q_\Sigma(a,b; y,z)$ of \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}.  The polynomial is shown to be, in effect, an invariant of the sign-colored graphic matroid.
\annotctd{} Much unusual graph theory is in here.
A special focus is the degrees of the polynomial.  First Main Thm.\ 3.1: Formulas for the maximum and minimum combined degrees of $P_\Sigma(x,y,z)$.  
\S7, ``Signature of a graph'', studies the signature ($\sigma$ in the paper, $s$ here) of the Laplacian matrix $\Lap(\Sigma)$ ($B_\Sigma$ in the paper) obtained by changing the diagonal of $A(\Sigma)$ so the row sums are $0$.  Prop.\ 7.2 is a matrix-tree theorem [entirely different from that of \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}].  The Second Main Thm.\ 8.1 bounds the signature: $\#V - 2\beta_0(\Sigma^-) + 1 \leq s \leq \#V - 2\beta_0(\Sigma^+) + 1$ ($\beta_0 = c(\Sigma)$), with equality characterized.  The Laplacian matrix is further examined later on.
\S9, ``Dual graphs'': Differing from most studies, here the dual of a sign-colored plane graph is the planar dual with same edge signs [however, negating all colors is a triviality].
\S10, ``Periodic graphs'':  These graphs might be called branched covering graphs of signed gain graphs with finite cyclic gain group.  [Thus they generalize the periodic graphs of \xref{Collatz}{1978a}{L.\ Collatz} and others.]
\S\S12--15 concern applications to knot theory.
\par
\codes{\CSGc:~\CInvar,~\CInc,~\CGG(\CCov),~\CDual,~\CKnot}%
\itemspace

\bsgitem{1991a}{}  Invariants of graphs and their applications to knot theory.  
In:  S.\ Jackowski, B.\ Oliver, and K.\ Pawa{\l}owski, eds., \emph{Algebraic topology Pozna\'n 1989} (Proc., Pozna\'n, 1989), pp.\ 83--97.  \LNM 1474.  Springer-Verlag, Berlin, 1991.
\MR{1133894} (92m:57015).  \Zbl{751.57007}.
\annot{} \S\S1--3 expound results from \xrefd{1989a}{Kunio Murasugi} on the dichromatic polynomial and the signature of a sign-colored graph and knot applications.  \S5 discusses the signed Seifert graph of a link diagram.
\par
\codes{\CSGc:~\CInvar,~\CInc,~\CKnot:~\CExp}%
\itemspace

\bsgitem{1993a}{}  \emph{Musubime riron to sono \macron{o}n\macron{o}}.  [{Knot Theory and Its Applications.}]  (In Japanese.)  1993.  
\annot{} See \xrefd{1996a}{Kunio Murasugi}.
\codes{\CSGc:~\CKnot}%
\itemspace

\bsgitem{1996a}{}  \emph{Knot Theory and Its Applications}.  Birkh\"auser, Boston, 1996.
\MR{1391727} (97g:57011).  \Zbl{864.57001}.
\annot{} Updated translation of \xrefd{1993a}{Kunio Murasugi} by Bohdan Kurpita.  Pp.\ 36--37:  Construction of signed plane graph from link diagram, and conversely.
\par
\codes{\CSGc:~\CKnot}%
\itemspace

\author{Kunio Murasugi \& Jozef H.\ Przytycki}
\authorspace
\bsgitem{1993a}{}  \emph{An Index of a Graph with Applications to Knot Theory}.  Mem.\ Amer.\ Math.\ Soc., Vol.\ 106, No.\ 158.  Amer.\ Math.\ Soc., Providence, R.I., 1993.
\MR{1171835} (94d:57025).  \Zbl{792.05047}.
\annot{} Ch.\ I, ``Index of a graph''.  The ``index'' is the largest number of ``independent'' edges, where ``independent'' has a complicated recursive definition (unrelated to matchings), one of whose requirements is that the edges be ``singular'' (= simple).  The positive or negative index of a sign-colored graph is similar except that the independent edges must all be positive or negative.  [The general notion is that of the index of a graph-subgraph pair.  The signs pick out complementary subgraphs.]  
Thm.\ 2.4:  Each of these indices is additive on blocks of a bipartite graph.  The main interest, because of applications to knot theory, is in bipartite plane graphs.  Ch.\ II, ``Link theory'':  Pp.\ 26--27 define the sign-colored Seifert graph of an oriented link diagram and apply the graphical index theory.
\codes{\CSGc:~\CInvar,~\CDual,~\CKnot}%
\itemspace

\author{Tadao Murata} 						%MURATA
\authorspace
\bsgitem{1965a}{}  Analysis of lossy communication nets by modified incidence matrices.  
In: M.E.\ Van Valkenburg, ed., \emph{Proceedings, Third Annual Allerton Conference on Circuit and System Theory} (Monticello, Ill., 1965), pp.\ 751--761.  Dept.\ of Electrical Eng.\ and Coordinated Sci.\ Lab., University of Illinois, Urbana, Ill.; and Circuit Theory Group, Inst.\ of Electrical and Electronics Engineers, [1965].  
\par
\codes{\CGN:~\CInc}%
\itemspace

\author{Antoine Musitelli}						%MUSITELLI
\authorspace
\authref  See also \hyperlink{Alberto Del Pia}{A.~Del~Pia}.
\authrefspace

\bsgitem{2007a}{}  \emph{Recognition of Generalized Network Matrices}.  Doctoral thesis, \'Ecole Polytechique Fed\'eral Lausanne, 2007.
\arXiv{0807.3541}.  
\annot{} A polynomial-time algorithm for recognizing binet matrices in time $O(n^6 \#E)$.  See \xrefd{2010a}{Antoine Musitelli}.
\dated{15 January 2013} 
\par
\codes{\CSG:~\COri:~\CInc,~\CAlgor}%
\itemspace

\bsgitem{2010a}{}  Recognizing binet matrices.  \emph{Math.\ Programming} 124 (2010), no.\ 1-2, 349--381.
\MR{2679995} (2011g:68121).  \Zbl{1206.68149}.					
\annot{} \Cf\ \xref{Kotnyek}{2002a}{Balazs Kotnyek}.  Description of the algorithm of \xrefd{2007a}{Antoine Musitelli}, which involves reduction to the cases of ``cyclic'' and ``bicyclic'' matrices.  These are the incidence matrices of bidirected graphs with, respectively, one or two components that are without half edges.
\dated{15 January 2013, rev 16 Oct 2017} 
\codes{\CSG:~\COri:~\CInc,~\CAlgor}%
\itemspace

\author{Mohammed A.\ Mutar}						%MUTAR
\authorspace
\authref  See also \hyperlink{Arthur H.\ Busch}{A.H.~Busch}.
\authrefspace

\bsgitem{2017a}{}  \emph{Hamiltonicity in Bidirected Signed Graphs and Ramsey Signed Numbers}.  M.S.\ thesis, Wright State University, 2017.
\annot{} Necessary and sufficient conditions for coherent Hamiltonian cycles in oriented $\pm K_n$, $\pm K_{n,n}$, $-K_{n,n}$.  \Cf\ \xref{Busch--Mutar--Slilaty}{2022a}{Arthur Busch, Mohammed A.\ Mutar, \& Daniel Slilaty}.
\par
\codes{\CSG:~\COri:~\CCycles}
\annot{} Some values of $r^*(n,m) := \min\{r: \forall (K_r,\sigma) \supseteq [+K_n] \text{ or } [-K_m]\}$, opening a new Ramsey theory.
\dated{29 Jun 2022} 
\codes{\CSG:~\textbf{Ramsey}}%
\itemspace

\authorx{A.\ Muthaiyan}						%MUTHAIYAN
\author{A.\ Muthaiyan \& A.\ Nesamathi}
\authorspace
\bsgitem{2016a}{}  Some new face and total face signed product cordial graphs.  \emph{IJISET Int.\ J.\ Innovative Sci.\ Engin.\ Tech.}\ 3 (2016), no.\ 8, 399--407.
\annot{} More examples as in \xref{Rozario Raj and Manoharan}{2016a}{P.\ Lawrence Rozario Raj \& R.\ Lawrence Joseph Manoharan}.
\par
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\author{P.\ M\"utzel}							%MUTZEL
\authorx{P.\ Mutzel}
\authorspace
\authref  See \hyperlink{C.\ De Simone}{C.~De~Simone}.
\itemspace


\label{LETTER N} \hypertarget{LETTER N}{}	% LETTER  N


\authorx{Shokry Nada}						%NADA
\author{Shokry Nada, Amani Elrayes, Ashraf Elrokh, \& Aya Rabie}
\authorspace
\bsgitem{2019a}{}  Signed product cordial of the sum and union of two fourth power of paths and cycles.  \emph{Machine Learning Res.}\ 4 (2019), no.\ 4, 45--50.
\annot{} \Cf\ \xref{Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\dated{12 Aug 2021} 
\par
\codes{\CLab:~\CVS,~\CSG}%
\itemspace

\author{S.\ Nada, A.\ Elrokh, A.\ Elrayes, \& A.\ Rabie}
\authorspace
\bsgitem{2019a}{}  The signed product cordial for corona of paths and fourth power of paths.  \emph{Int.\ J.\ Math.\ Combin.}\ 4 (2019), 102--111
\annot{} \Cf\ \xref{Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\dated{26 Sept 2022} 
\par
\codes{\CLab:~\CVS,~\CSG}%
\itemspace

\author{[Sudev Naduvath]}						%NADUVATH
\authorspace
\authref  See \hyperlink{Naduvath K.\ Sudev}{N.K.~Sudev} (under `S').
\itemspace

\author{Mohamed Nafea}						%NAFEA
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Ojiro Nagai}							%NAGAI
\authorspace
\authref  See also \hyperlink{Hung T.\ Diep}{H.T.~Diep}.
\authrefspace

\author{Ojiro Nagai, Tsuyoshi Horiguchi, \& Seiji Miyashita}
\authorspace
\bsgitem{2004a}{}  Properties and phase transitions in frustrated Ising systems.  In:  H.T.\ Diep, ed., \emph{Frustrated Spin Systems}, Ch.\ 2, pp.\ 59--105.  World Scientific, Hackensack, N.J., 2004.
\Zbl{1104.82002} (book).  
\annot{} Physics questions on various periodic signed lattice graphs, $\dim=2,3$.  [\Question.  Do  the phenomena treated here for periodic signed lattices suggest interesting mathematics, possibly for more general $\Sigma$?]
\annotctd{}  \S 2.5, ``Ising model with large $S$ on antiferromagnetic triangular lattice'':  Spin $1/2$ generalizes to (integral) spin $S$ = spin $\in \{-S, -(S-1), \ldots, 0, \ldots, S-1, S\}$ [as in coloring $\Sigma$ with $2S+1$ colors] on all-negative triangular lattice.  Large $S$ gives new kinds of ground state (min energy).  [\Question.  Are these new signed-graph coloring problems?]  
\S2.6, ``Ising model with infinite-spin on antiferromagnetic triangular lattice'':  ``Infinite spin'' means $S\to\infty$.  
\annotctd{} Dictionary:  ``local gauge transformation'' = switching, ``spin $1/2$'' (often assumed in physics) = Ising spins $\pm1$. 
\dated{9 Aug 2018}
\par
\codes{\CSG,~\CFr,~\CSw:~\CPhys:~\CExp,~\CRef}%
\itemspace

\author{Ojiro Nagai, Koichi Nishino, Jong-Jae Kim, and Yuuzi Yamada}
\authorspace
\bsgitem{1988a}{}  Magnetic properties of a three-dimensional Ising crystal with zero-point entropy.  \emph{Phys.\ Rev.\ B} 37 (1988), no.\ 10, 5448--5451.
\annot{} [Partly from previous works cited?]  Cubic lattice with $E^- = \{(i,j,k)(i+1,j,k): j+k \text{ even}\}$.  Edge weights (``bond strengths'') $a, 1, 1$ in $x,y,z$ directions, $a>0$.  Ising ground states $\psi: V \to \{\pm1\}$ (i.e., least weight of unsatisfied edges):  For $a<2$, each $yz$-plane is satisfied ($\psi$ is constant).  For $a>2$, each $x$ line is satisfied (constant $\psi$ on all-$+$ lines, alternating on all-$-$ lines).  For $a=2$, both are ground states.  Some results are for multivalued spins $-S,\ldots,S-1,S$.  Consequence:  At $a=2$ there are ``free''' spins ($\psi(v)=\pm1$ arbitrarily in ground states) at some vertices.  [\Question.  Does this suggest interesting mathematics, possibly for more general $\Sigma$?]
\dated{14 Aug 2018} 
\codes{\CPhys:~\CSG:~\CFr}%
\itemspace

\author{O.\ Nagai, Y.\ Yamada, \& H.T.\ Diep}
\authorspace
\bsgitem{1985a}{}  Linear-chain-like excitations in a three-dimensional Ising lattice with frustration: Monte Carlo simulations.  \emph{Phys.\ Rev.\ B} 32 (1985), no.\ 1, 480--483.
\annot{} Computer simulations on a cubic lattice with $E^- = \{(i,j,k)(i+1,j,k): j \text{ even}\}$.  In Ising ground states $\psi: V \to \{\pm1\}$ (i.e., fewest unsatisfied edges), each $z$-line has constant $\psi$ along the line; in 1/4 of them the spin constant varies with time.  [\Question.  Does this suggest interesting mathematics for more general $\Sigma$?]
\dated{14 Aug 2018} 
\par
\codes{\CPhys:~\CSG:~\CFr}%
\itemspace

\author{Atulya K.\ Nagar}						%NAGAR
\authorspace
\authref  See \hyperlink{C.\ Perina}{C.~Perina} and \hyperlink{V.\ Vasanthi}{V.~Vasanthi}.
\itemspace

\author{K.M.\ Nagaraja}						%NAGARAJA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Yuuki Nagazawa}						%NAGAZAWA
\authorspace
\authref  See \hyperlink{Tetsushi Matsui}{T.~Matsui}.
\itemspace

\authorx{P.\ Nageswari}						%NAGESWARI
\author{P.\ Nageswari \& P.B.\ Sarasija}
\authorspace
\bsgitem{2014a}{}  Seidel energy and its bounds.  \emph{Int.\ J.\ Math.\ Anal.}\ 8 (2014), no.\ 58, 2869--2871.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Tatiana Nagnibeda}						%NAGNIBEDA
\authorspace
\authref  See \hyperlink{Vadim Bugaenko}{V.~Bugaenko}.
\itemspace

\author{Mina Nahvi}							%NAHVI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{T.A.\ Naikoo}							%NAIKOO
\authorspace
\authref  See \hyperlink{S.\ Pirzada}{S.~Pirzada}.
\itemspace

\author{Takeshi Naitoh}						%NAITOH
\authorspace
\authref  See \hyperlink{Kazutoshi Ando}{K.~Ando}.
\itemspace

\author{Kazuo Nakajima} 						%NAKAJIMA
\authorspace
\authref  See \hyperlink{Hyeong-ah Choi}{H.~Choi}.
\itemspace

\author{Atsuhiro Nakamoto}						%NAKAMOTO
\authorspace
\authref  See also \hyperlink{Dan Archdeacon}{D.~Archdeacon}.
\authrefspace

\author{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}
\authorspace
\bsgitem{2002a}{}  Chromatic numbers and cycle parities of quadrangulations on nonorientable closed surfaces.  Ninth Quadrennial Int.\ Conf.\ Graph Theory, Combinatorics, Algorithms Appl.  \emph{Electronic Notes Discrete Math.}\ 11 (2002), 509--518.
\MR{2155788} (no rev).  \Zbl{1075.05532}.  
\annot{} A ``cycle parity'' on surface $S$ = homomorphism $\rho: \pi(S) \to \bbZ_2 \cong \signs$, equivalently $\rho: H_1(S;\bbZ_2 \to \bbZ_2 \cong \signs$.  $\rho$ implies a signature (actually, a switching class) of any embedded graph $\Gamma$.  There are one nontrivial type of cycle parity on an orientable surface and three on a nonorientable surface $N_d$, different for odd and even $d$, except two on $N_2$ and one on  $N_1$.  
If $\Gamma \embeds S$ so every face boundary is even (``even embedding''), then $\rho(W) = \#W \mod 2$ for closed walks is a cycle parity.  
Thm.\ 9:  For three of the six types on $N_d$'s, there is a negative cut that opens $N_d$ to an orientable surface.  
[Also see \xref{Mohar and Seymour}{2002a}{Bojan Mohar \& Paul D.\ Seymour}.]
\dated{11 Jun 2012} 
\codes{\Csg:~\CTop:~\Csw}%
\itemspace

\bsgitem{2004a}{}  Chromatic numbers and cycle parities of quadrangulations on nonorientable closed surfaces.  \emph{Discrete Math}\ 285 (2004), 211--218.
\MR{2062844} (2005k:05101).  \Zbl{1044.05034}.  
\codes{\Csg:~\CTop:~\Csw}%
\itemspace

\authorx{Daishin Nakamura}						%NAKAMURA D
\author{Daishin Nakamura \& Akihisa Tamura}
\authorspace
\bsgitem{1998a}{}  The generalized stable set problem for claw-free bidirected graphs.  
In: Robert E.\ Bixby, E.~Andrew Boyd, and Roger Z.\ R\'{\i}os-Mercado, eds., \emph{Integer Programming and Combinatorial Optimization}  (6th Int.\ IPCO Conf., Houston, 1998, Proc.), pp.\ 69--83.  \LNCS 1412.  Springer, Berlin, 1998.
\MR{1726336} (2000h:05209).  \Zbl{907.90272}.						
\annot{} The problem of the title is solvable in polynomial time.  See \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg}, \xref{Tamura}{1997a}{Akihisa Tamura} for definitions.  They reduce to simple graphs, transitively bidirected with no sink or introverted edge (called ``canonical'' bidirected graphs).
\codes{\Csg:~\COri:~\CGeom,~\CSw,~\CAlgor}%
\itemspace

\bsgitem{1998b}{}  Generalized stable set problems for claw-free bi-directed graphs.  (In Japanese.)  
Theory and Applications of Mathematical Optimization (Kyoto, 1998).  \emph{S\macron{u}ri\-kai\-seki\-kenky\macron{u}sho K\macron{o}ky\macron{u}\-roku} No.\ 1068 (1998), 100--109.
\Zbl{939.05506} (no rev).  
\par
\codes{\Csg:~\COri:~\CGeom,~\CSw,~\CAlgor}
\itemspace

\bsgitem{2000a}{}  A linear time algorithm for the generalized stable set problem on triangulated bidirected graphs.  
New Trends in Mathematical Programming (Kyoto, 1998).  \emph{J.\ Operations Res.\ Soc.\ Japan} 43 (2000), 162--175.
\MR{1768393} (2001c:90093).  \Zbl{1138.90494}.						
\codes{\Csg:~\COri:~\CGeom:~\CAlgor}
\itemspace

\author{M.\ Nakamura}						%NAKAMURA M
\authorspace
\authref  See \hyperlink{M.\ Hachimori}{M.\ Hachimori}.
\itemspace

\authorx{Tota Nakamura}						%NAKAMURA T
\author{Tota Nakamura, Shin-ichi Endoh, \& Takeo Yamamoto}
\authorspace
\bsgitem{2003a}{}  Weak universality of spin-glass transitions in three-dimensional $\pm J$ models.  \emph{J.\ Phys.\ A} 36 (2003), 10895--10906.
\MR{2025232} (no rev).  \Zbl{1075.82508}.  
\annot{} Physics of Ising, XY, and Heisenberg spin-glass models on a signed square lattice graph with 3-dimensional spin vectors.  The Hamiltonian of state $S: V \to \bbS^2$ (the sphere) is $\sum_{uv\in E} \sigma(uv) S_u \cdot S_v$.
\dated{17 Jun 2012} 
\codes{\CPhys:~\CSG}%
\itemspace

\author{Bunpei Nakano}						%NAKANO
\authorspace
\authref  See \hyperlink{Takehiro Inohara}{T.~Inohara}.
\itemspace

\authorx{Norihiro Nakashima}						%NAKASHIMA
\author{Norihiro Nakashima \& Shuhei Tsujie}
\authorspace
\bsgitem{2021a}{}  Enumeration of flats of the extended Catalan and Shi arrangements with species. \emph{J.\ Integer Seq.}\ 24 (2021), no.\ 9, art.\ 21.9.2, 21 pp.
\MR{4336094}.  \Zbl{1489.52016}.  \arXiv{1904.09748}.  
\codes{\CGG:~\CGeom:~\CInvar}%
\itemspace

\author{Preetum Nakkiran}						%NAKKIRAN
\authorspace
\authref  See \hyperlink{Charalampos E.\ Tsourakakis}{C.E.~Tsourakakis}.
\itemspace

\author{Vasileios Nakos}						%NAKOS
\authorspace
\authref  See \hyperlink{Charalampos E.\ Tsourakakis}{C.E.~Tsourakakis}.
\itemspace

\author{Aur\'elien Naldi}						%NALDI
\authorx{Aurelien Naldi}						%NALDI
\authorspace
\authref  See also \hyperlink{Jean-Paul Comet}{J.-P.~Comet}.
\authrefspace

\author{Aur\'elien Naldi, Elisabeth Remy, Denis Thieffry, \& Claudine Chaouiya}
\authorx{Aurelien Naldi, Elisabeth Remy, Denis Thieffry, \& Claudine Chaouiya}
\authorspace
\bsgitem{2011a}{}  Dynamically consistent reduction of logical regulatory graphs.  \emph{Theor.\ Computer Sci.}\ 412 (2011), no.\ 21, 2207--2218,
\MR{2809505} (2012a:92077).  \Zbl{1211.92024}.					
\codes{\CSD,~\CDyn}%
\itemspace

\author{Soumen Nandi}						%NANDI
\authorspace
\authref  See \hyperlink{Julien Bensmail}{J.~Bensmail} and \hyperlink{Sandip Das}{S.~Das}.
\itemspace

\author{L.\ Nanjundaswamy} 						%NANJUNDASWAMY
\authorspace
\authref  See \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\author{Assaf Naor}							%NAOR
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{Joseph (Seffi) Naor}						%NAOR
\authorspace
\authref  See \hyperlink{Dorit S.\ Hochbaum}{D.~Hochbaum}.
\itemspace

\author{Vito Napolitano}						%NAPOLITANO
\authorspace
\authref  See \hyperlink{Marien Abreu}{M.~Abreu}.
\itemspace

\author{Sivaram K.\ Narayan}						%NARAYAN
\authorspace
\authref  See \hyperlink{Nancy Matar}{N.~Matar}.
\itemspace

\author{Ramasuri Narayanam}						%NARAYANAM
\authorspace
\authref  See \hyperlink{Priyanka Agrawal}{P.~Agrawal}.
\itemspace

\author{Jonathan Narboni}						%NARBONI
\authorspace
\authref  See \hyperlink{Frantisek Kardos}{F.~Kardo\v{s}}.
\itemspace

\author{Reza Naserasr}						%NASERASR
\authorspace
\authref  See also \hyperlink{Laurent Beaudou}{L.~Beaudou}, \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}, \hyperlink{Chiara Cappello}{C.~Cappello}, \hyperlink{Clement Charpentier}{C.~Charpentier}, and \hyperlink{Florent Foucaud}{F.~Foucaud}.
\authrefspace

\bsgitem{20xxa}{}  Not $\{\pm1,\pm2\}$-colorable planar signed graphs.  Submitted.  
\annot{} A simpler, smaller proof of \xref{Kardo\v{s} and Narboni}{2021a}{Frantisek Kardos \& Jonathan Narboni}, with remarks and questions on related homomorphisms.
\dated{17 Nov 2019} 
\par
\codes{\CSG:~\CCol,~\CHom}%
\itemspace

\author{Reza Naserasr \& Lan Anh Pham}
\authorspace
\bsgitem{2022a}{}  Complex and homomorphic chromatic number of signed planar simple graphs.  \emph{Graphs Combin.}\ 38 (2022), art.\ 58, 22 pp.
\MR{4393983}.  
\HAL{hal-03000542}.						
\annot{} Color set $C_{k,l} = \{\pm1, \ldots, \pm k, \pm1\mathrm{i}, \ldots, \pm l\mathrm{i}\}$, $\mathrm{i}=\sqrt{-1}$.  Coloration $\kappa$ is proper when $|\kappa(u)| \neq |\kappa(v)|$ or $\sgn\kappa(u)\cdot\sgn\kappa(v) \neq \sigma(uv)$, where $\sgn z := z/|z|$.
\dated{25 Oct 2020} 
\codes{\CSG:~\CCol,~\CHom}%
\itemspace

\author{Reza Naserasr, Lan Anh Pham, \& Zhouningxin Wang}
\authorspace
\bsgitem{2022a}{}  Density of $C_{-4}$-critical signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 153 (2022), 81--104.
\MR{4345281}.  \Zbl{1481.05086}.  \arXiv{2101.08612}.  \HAL{hal-03000545}.		
\annot{} $C_{-4}$ = negative quadrilateral $C_4$.
\codes{\CSG:~\CHom}%
\itemspace

\author{Reza Naserasr, Edita Rollov\'a, \& \'Eric Sopena}
\authorx{Reza Naserasr, Edita Rollova, \& Eric Sopena}
\authorspace
\bsgitem{2013a}{}  Homomorphisms of planar signed graphs to signed projective cubes.  \emph{Discrete Math.\ Theor.\ Computer Sci.}\ 15 (2013), no.\ 3, 1--12.
\MR{3119638}.  \Zbl{1283.05186}.  \HAL{hal-00876298}.  
\codes{\CSG:~\CHom}%
\itemspace

\bsgitem{2013b}{}  On homomorphisms of planar signed graphs to signed projective cubes.  In:  Jaroslav Ne\v{s}et\v{r}il and Marco Pellegrini, eds., \emph{The Seventh European Conference on Combinatorics, Graph Theory and Applications} (EuroComb 2013, Pisa), pp.\ 271--276.  CRM Ser., Vol.\ 16.  Edizioni della Normale, Scuola Normale Superiore Pisa, Pisa, Italy, 2013.
\MR{3185818} (no rev).  \Zbl{06302999}.  \HAL{hal-00829117}.  
\codes{\CSG:~\CHom}%
\itemspace

\bsgitem{2013c}{}  Homomorphisms of signed bipartite graphs.  In:  Jaroslav Ne\v{s}et\v{r}il and Marco Pellegrini, eds., \emph{The Seventh European Conference on Combinatorics, Graph Theory and Applications} (EuroComb 2013, Pisa), pp.\ 345--350.  CRM Ser., Vol.\ 16.  Edizioni della Normale, Scuola Normale Superiore Pisa, Pisa, Italy, 2013.
\MR{3185829} (no rev).  \Zbl{1292.05191}.  \HAL{hal-00829116}.  
\codes{\CSG:~\CHom}%
\itemspace

\bsgitem{2015a}{}  Homomorphisms of signed graphs.  \emph{J.\ Graph Theory} 79 (2015), no.\ 3, 178--212.
\MR{3346138}.  \Zbl{1322.05069}.  \HAL{hal-00991796}.  
\codes{\CSG:~\CHom}%
\itemspace

\author{Reza Naserasr, Sagnik Sen, \& \'Eric Sopena}
\authorx{Reza Naserasr, Sagnik Sen, \& Eric Sopena}
\authorspace
\bsgitem{2021a}{}  The homomorphism order of signed graphs.  \emph{J.\ Combin.\ Math.\ Combin.\ Computing} 116 (2021), 169--182.
\MR{4260818}.  \Zbl{1466.05088}.  \HAL{hal-02969878}.  
\codes{\CSG:~\CHom}%
\itemspace

\author{Reza Naserasr, Sagnik Sen, \& Qiang Sun}
\authorspace
\bsgitem{2016a}{}  Walk-powers and homomorphism bounds of planar signed graphs.  \emph{Graphs Combin.}\ 32 (2016), no.\ 4, 1505--1519.
\MR{3514981}.  \Zbl{1342.05059}.  
\par
\codes{\CSG:~\CHom,~\CTop}%
\itemspace

\author{Reza Naserasr, Riste \v{S}krekovski, Zhouningxin Wang, \& Rongxing Xu}
\authorspace
\bsgitem{20xxa}{}  Mapping sparse signed graphs to $(K_{2k},M)$.  Submitted.  \emph{}\ 
\arXiv{2101.08619}.  \HAL{hal-03109532}.						
\codes{\CSG:~\CHom}%
\itemspace

\author{Reza Naserasr, \'Eric Sopena, \& Thomas Zaslavsky}
\authorx{Reza Naserasr, Eric Sopena, \& Thomas Zaslavsky}
\authorspace
\bsgitem{2021a}{}  Homomorphisms of signed graphs:  An update.  \emph{European J.\ Combin.}\ 91 (2021), art.\ 103222, 20 pp.
\MR{4161815}.  \Zbl{1458.05098}.  \arXiv{1909.05982}.  \HAL{hal-02429415}.		
\annot{} Several aspects of homorphisms.  Aims to set correct terminology, give proofs of fundamental properties, raise open questions.
\dated{16 Sept 2019} 
\codes{\CSG,~\CSGc:~\CHom:~\CExp}%
\itemspace

\author{Reza Naserasr \& Zhouningxin Wang}
\authorspace
\bsgitem{2021a}{}  Bounding signed series-parallel graphs and cores of signed $K_4$-subdivisions.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 116 (2021), 27--51.
\MR{4260812}.  \Zbl{1466.05089}.  \HAL{hal-02969873}.  
\codes{\CSG:~\CHom}%
\itemspace

\author{Reza Naserasr, Zhouningxin Wang, \& Xuding Zhu}
\authorspace
\bsgitem{2021a}{}  Circular chromatic number of signed graphs.  \emph{Electronic J.\ Combin.}\ 28(2) (2021), no.\ 2, art.\ P2.44, 40 pp.
\MR{4281210}.  \Zbl{1466.05090}.  \arXiv{2010.07525}.  \HAL{hal-02969872}.		
\annot{} 
[\Cf\ \xref{Pan and Zhu}{2022a}{Zhishi Pan \& Xuding Zhu}.]
\codes{\CSG:~\CCol}%
\itemspace

\author{Reza Naserasr \& Weiqiang Yu}
\authorspace
\bsgitem{20xxa}{}  Packing signatures in signed graphs.  Submitted.  \emph{}\ 
\annot{} ``Signature packing number'' or negative-edge-set packing number $\rho(\Sigma) =$ maximum number of disjoint sets $E^-(\Sigma^\zeta)$ of switchings of $\Sigma$.  Connections to homomorphisms and the 4-Color Theorem.
Dictionary: ``signature'' = negative edge set.
\dated{15 May 2021} 
\par
\codes{\CSG:~\CSw:~\CHom,~\CCol,~\CAlgor}%
\itemspace

\bsgitem{20xxb}{}  Separating signatures in signed planar graphs.  Submitted.  \emph{}\ 
\annot{} Generalizes \xrefd{20xxa}{Reza Naserasr \& Weiqiang Yu} to disjoint sets of switchings of several arbitrary signatures.  
Dictionary: ``signature'' = negative edge set.
\dated{2 Apr 2022} 
\codes{\CSG:~\CSw,~\CCol,~\CHom}%
\itemspace

\author{C.St.J.A.\ Nash-Williams} 					%NASH-WILLIAMS
\authorspace
\bsgitem{1960a}{} On orientations, connectivity, and odd-vertex-pairings in finite graphs.  \emph{Canad.\ J.\ Math.}\ 12 (1960), 555--567.  
\MR{0118684} (22 \#9455).  \Zbl{096.38002} (96, p.\ 380b).				
\itemspace

\bsgitem{1969a}{} Well-balanced orientations of finite graphs and unobtrusive odd-vertex-pairings.  In:  W.T.~Tutte, ed., \emph{Recent Progress in Combinatorics}  (Proc.\ Third Waterloo Conf., 1968), pp.\ 133--149.  Academic Press, New York, 1969.  
\MR{0253933} (40 \#7146).  \Zbl{209.55701} (209, p.\ 557a).				
\itemspace

\author{Nagarajan Natarajan}						%NATARAJAN
\authorspace
\authref  See \hyperlink{Kai-Yang Chiang}{K.-Y.~Chiang}.
\itemspace

\author{Nutan G.\ Nayak}						%NAYAK
\authorspace
\bsgitem{2014a}{}  Equienergetic net-regular signed graphs.  \emph{Int.\ J.\ Contemp.\ Math.\ Sci.}\ 9 (2014), no.\ 14, 685--693.
\annot{}  Eigenvalues of $K_{-\overline{C}_n}$.  These lead to non-cospectral pairs of net-regular signed $K_n$'s with equal adjacency energy.
\dated{20 Apr 2019} 
\par
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2016a}{}  On net-regular signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 1 (2016), 57-64.
\annot{} Elementary facts of net regularity.  Eigenvalues of $K_{-\overline{C}_n}$.
\dated{20 Apr 2019} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2017a}{}  On net-Laplacian energy of signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ 2 (2017), no.\ 1, 11--19.
\MR{3657818}.  \Zbl{386.05081}.  
\annot{} Net Laplacian $\Lap^\pm(\Sigma) := \diag(d^\pm_\Sigma) - A(\Sigma)$.
\codes{\CSG:~\CLap,~\CEig}%
\itemspace

\bsgitem{2017b}{}  Spectra and energy of signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 1 (2017), 10--21.
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Zachary Neal}							%NEAL
\authorspace
\authref  See \hyperlink{Samin Aref}{S.~Aref}.
\itemspace

\author{Roman Nedela}						%NEDELA
\authorspace
\authref  See also \hyperlink{Aleksander Malnic}{A.~Malni\v{c}}.
\authrefspace

\author{Roman Nedela \& Martin \v{S}koviera}
\authorx{Roman Nedela \& Martin Skoviera}
\authorspace
\bsgitem{1996a}{}  Regular embeddings of canonical double coverings of graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 67 (1996), 249--277.
\MR{1399678} (97e:05078).  \Zbl{856.05029}.						
\annot{} By ``canonical double covering'' of $\Gamma$ they mean the signed covering graph $\tilde\Sigma$ of $\Sigma = -\Gamma$, but without reversing orientation at the negative covering vertex [as one would do in a signed covering graph (\cf\ e.g.\ \xref{Zaslavsky}{1992a}{Thomas Zaslavsky})], because orientable embeddings of $\Gamma$ are being lifted to orientable embeddings of $\tilde\Sigma$.  [Thus these should be thought of not as signed graphs but rather as voltage (i.e., gain) graphs with $2$-element gain group.]  Instead of reversal they twist the negative-vertex rotations by taking a suitable power.  In some cases this allows classifying the orientable, regular embeddings of $\tilde\Sigma$.
\codes{\CPar:~\CCov,~\CTop,~\CAut}%
\itemspace

\bsgitem{1997a}{}  Exponents of orientable maps.  \emph{Proc.\ London Math.\ Soc.}\ (3) 75 (1997), 1--31.
\MR{1444311} (98i:05059).  \Zbl{877.05012}.						
\annot{} Main topic: the theory of twisting of rotations as in \xrefd{1996a}{Roman Nedela \& Martin Skoviera}.
\par
\codes{\CGG:~\CCov,~\CTop,~\CAut}%
\annotctd{} Portions concern double covering graphs of signed graphs.  \S7: ``Antipodal and algebraically antipodal maps''.  A map is ``antipodal'' if it is the orientable double covering of a nonorientable map; that is, as a graph it is the canonical double covering of an unbalanced signed graph.  A partial algebraic criterion for a map to be antipodal.  \S9: ``Regular embeddings of canonical double coverings of graphs''.  See \xrefd{1996a}{Roman Nedela \& Martin Skoviera}.
\par
\codes{\CPar:~\CCov,~\CTop,~\CAut}%
\itemspace

\bsgitem{1997b}{}  Regular maps from voltage assignments and exponent groups.  \emph{European J.\ Combin.}\ 18 (1997), 807--823.
\MR{1478826} (98j:05061).  \Zbl{908.05036}.						
\annot{} Cases in which the classification of \xrefd{1996a}{Roman Nedela \& Martin Skoviera} is necessarily incomplete are studied by taking larger voltage (i.e., gain) groups and twisting the rotations at covering vertices by taking a power that depends on the position of the vertex in its fiber.  Main result: the (very special) conditions on twisting under which a regular map lifts to a regular map.
\par
\codes{\CGG:~\CCov,~\CTop,~\CAut}%
\itemspace

\author{Seiya Negami}						%NEGAMI
\authorspace
\authref  See \hyperlink{Dan Archdeacon}{D.~Archdeacon} and \hyperlink{Atsuhiro Nakamoto}{A.~Nakamoto}.
\itemspace

\authorx{Peter Nelson}						%NELSON
\author{Peter Nelson \& Jorn van der Pol}
\authorspace
\bsgitem{2019a}{}  On the number of biased graphs.  \emph{SIAM J.\ Discrete Math.}\ 33 (2019), no.\ 1, 373--382.
\MR{3915407}.  \Zbl{1405.05084}.  \arXiv{1807.11383}.				
\annot{} Almost all biased simple graphs are complete graphs with Hamiltonian bias.
\dated{20 Aug 2019} 
\codes{\CGG:~\CEnum}%
\itemspace

\author{Max Nelson-Kilger}						%NELSON-KILGER
\authorspace
\authref  See \hyperlink{M.\ Hamit Fisek}{M.H.~Fi\c{s}ek}.
\itemspace

\author{Mohammad Ali Nematollahi}					%NEMATOLLAHI
\authorspace
\authref  See \hyperlink{Uzma Ahmad}{U.~Ahmad} and \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Toshio Nemoto}						%NEMOTO
\authorspace
\authref   See \hyperlink{Kazutoshi Ando}{K.~Ando}.
\itemspace

\author{Anna Nenca}							%NENCA
\authorspace
\authref  See \hyperlink{Janusz Dybizbanski}{J.~Dybizba\'nski}.
\itemspace

\author{Hanna Nencka [H.\ Nencka-Fisek]}				%NENCKA
\authorx{H.\ Nencka}
\authorspace
\authref   See \hyperlink{H.\ Nencka-Fisek}{H.~Nencka-Fisek}.
\itemspace

\author{H.\ Nencka-Fisek [H.\ Nencka]}					%NENCKA-FISEK
\authorx{H.\ Nencka-Fisek}
\authorspace
\authref   See also \hyperlink{Ph.\ Combe}{Ph.~Combe}. 
\authrefspace

\bsgitem{1982a}{}  Can frustrations arise in Ising systems with multi-spin interactions of different signs?  \emph{J.\ Appl.\ Phys.}\ 53 (1982), 7969--7970.
\codes{\CSG(\CGen),~\CPhys:~\Csh:~\CFr}%
\itemspace

\bsgitem{1984a}{}  Necessary and sufficient conditions for the overblocking effect.  In:  A.\ P\c{e}kalski and J.\ Sznajd, eds., \emph{Static Critical Phenomena in Inhomogeneous Systems} (Proc.\ XX Karpacz Winter School Theor.\ Phys., Karpacz, Poland, 1984), pp.\ 337--343.  Lect.\ Notes in Physics Vol.\ 206.  Springer-Verlag, Berlin, 1984.
\MR{0839663} (87i:82096).  
\annot{} Signs are defined for arbitrary proper subhypercubes of the hypercube $Q_d$ [thus giving a signed hypergraph].  A ``plaquette'' (this is non-standard) is a $k-1$-dimensional band around a $k$-subhypercube; its sign is the product of signs of its $k-1$-faces.  Overblocking means not all plaquettes can simultaneously be negative (``frustrated'').  The interesting proof is by the adjacency graph of 2-faces of a 3-cube in $Q_d$.  Identify opposite 2-faces to a single vertex whose sign is the product of 2-face signs; the faces of a 3-cube form a triangle whose vertices alternate in sign, if all plaquettes were negative.  Conclusion:  All 2-faces cannot be negative, if $d>2$.  
[Presumably a similar argument should be applied to plaquettes of $k-1$-faces of a $k$-cube, $k>3$, but it is not.  There would be one plaquette per dimension.  \Question.  Is there such a generalization?]
\dated{19 Jun 2012} 
\codes{\CSG(\CGen),~\CPhys:~\Csh:~\CFr}%
\itemspace

\bsgitem{1985a}{}  Topological closure as the necessary condition for frustration or phase transitions.  \emph{J.\ Math.\ Phys.}\ 26 (1985), 1597--1599.
\MR{0793301} (87f:82026).  
\annot{} A higher-dimensional Ising model with a sign attached to each ``plaquette'' (see \xrefd{1985a}{H.\ Nencka-Fisek}).  A plaquette is frustrated if the spin product (spin = $\pm1$) of its sites (vertices) fails to match the attached sign.  A necessary condition for frustration is said to be an ``umbrella'' (a topological construction, possibly a cap on the plaquette?).  An example is a triangular lattice.  The main example is $\bbZ^d$.  The theorem implies that only $k<d$ can have a frustrated plaquette [obvious, if a plaquette lives in the boundary of a subcube].  [The article seems imprecise.  The idea could be worth pursuing.]
\dated{26 Dec 2014} 
\codes{\CSG(\CGen),~\CPhys:~\Csh:~\CFr}%
\itemspace

\author{A.\ Nesamathi}						%NESAMATHI
\authorspace
\authref  See \hyperlink{A.\ Muthaiyan}{A.~Muthaiyan}.
\itemspace

\author{Jaroslav Ne\v{s}et\v{r}il}					%NESETRIL
\authorx{Jaroslav Nesetril}
\authorspace
\authref   See also \hyperlink{Jan Kratochvil}{J.~Kratochv{\'\i}l}.
\authrefspace

\author{Jaroslav Ne\v{s}et\v{r}il \& Andr\'e Raspaud}
\authorx{Jaroslav Nesetril \& Andre Raspaud}
\authorspace
\bsgitem{2000a}{}  Colored homomorphisms of colored mixed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 80 (2000), 147--155.
\MR{1778206} (2001k:05078).  \Zbl{1026.05047}.  
\codes{\Csg,~\Csd:~\CHom}%
\itemspace

\author{Nancy Ann Neudauer}						%NEUDAUER
\authorspace
\authref  See also \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}, \hyperlink{Luis Goddyn}{L.~Goddyn}, \hyperlink{Gary Gordon}{G.~Gordon}, and \hyperlink{Jennifer McNulty}{J.~McNulty}.
\authrefspace

\bsgitem{1998a}{}  \emph{The Transversal Presentations and Graphs of Bicircular Matroids}.  Doctoral dissertation, University of Wisconsin--Madison, 1998.
\MR{2697849}.  
\annot{} The matroids are $\bfF(\Gamma,\eset)$.
\codes{\CBic}%
\itemspace

\bsgitem{2002a}{}  Graph representations of a bicircular matroid.  \emph{Discrete Appl.\ Math.}\ 118 (2002), no.\ 3, 249--262.
\MR{1892972} (2003b:05047).  \Zbl{990.05025}.					
\annot{} Survey of parts of \xref{Brualdi and Neudauer}{1997a}{Richard A.\ Brualdi \& Nancy Ann Neudauer}, \xref{Wagner}{1985a}{Donald K.\ Wagner}, and \xref{Coullard, del Greco, and Wagner}{1991a}{Collette R.\ Coullard, John G.\ del Greco, \& Donald K.\ Wagner}, with supplementary results on nice graphs whose bicircular matroid, $\bfF(\Gamma,\eset)$, equals $M$.
\codes{\CBic}%
\itemspace

\author{Nancy Ann Neudauer, Andrew M.\ Meyers, \& Brett Stevens}
\authorspace
\bsgitem{2001a}{}  Enumeration of the bases of the bicircular matroid on a complete graph.  
Proc.\ Thirty-second Southeastern Int.\ Conf.\ Combinatorics, Graph Theory and Computing (Baton Rouge, La., 2001).  
\emph{Congressus Numer.}\ 149 (2001), 109--127.
\MR{1887396} (2002m:05054).  \Zbl{1003.05031}.					
\annot{} Counts bases and connected bases.  Very complicated formulas.   [The results count labelled simple $1$-trees and $1$-forests.  A \emph{$1$-tree} is a tree with one extra edge forming a circle.  A \emph{$1$-forest} is a disjoint union of $1$-trees.  A connected basis of the bicircular matroid $\bfF(K_n,\eset)$ for $n \geq 3$ is a labelled simple $1$-tree; a basis is a labelled simple $1$-forest.  \xref{Riddell}{1951a}{R.J.\ Riddell} has a less complicated formula for $1$-trees.]
\codes{\CBic:~\CInvar(\CBases)} %
\itemspace

\author{Nancy Ann Neudauer \& Brett Stevens}
\authorspace
\bsgitem{2001a}{}  Enumeration of the bases of the bicircular matroid on a complete bipartite graph.  \emph{Ars Combin.}\ 66 (2003), 165--178.
\MR{1961484} (2004a:05034).  \Zbl{1075.05510}.					
\annot{} Bases are counted and their structure compared to the spanning trees of the graph.  [A basis is a simple, labelled 1-forest (\cf\ \xref{Neudauer, Meyers, and Stevens}{2001a}{Nancy Ann Neudauer, Andrew M.\ Meyers, \& Brett Stevens}) whose circles are even.]
\codes{\CBic:~\CInvar(\CBases)}%
\itemspace

\author{Nancy Ann Neudauer \& Daniel Slilaty}	% {20xxa}
\authorspace
\bsgitem{2017a}{}  Bounding and stabilizing realizations of biased graphs with a fixed group.  \emph{J.\ Combin.\ Theory Ser.\ B} 122 (2017), 149--166. 
\MR{3575200}.  \Zbl{1350.05073}.  
\codes{\CGG}%
\itemspace

\author{A.\ Neumaier} 						%NEUMAIER
\authorspace
\bsgitem{1982a}{}  Completely regular twographs.  \emph{Arch.\ Math.\ (Basel)} 38 (1982), 378--384.  
\MR{0658386} (83g:05066).  \Zbl{475.05045}.						
\annot{} In the signed graph $(K_n, \sigma)$ of a two-graph (see \xref{D.E.~Taylor}{1977a}{D.E.\ Taylor}), a ``clique'' is a vertex set that induces an antibalanced subgraph.  A two-graph is ``completely regular'' if every clique of size $i$ lies in the same number of cliques of size $i + 1$, for all $i$.  Thm.\ 1.4 implies there is only a small finite number of completely regular two-graphs.
\codes{\CTG}%
\itemspace

\author{Michael Neumann}						%NEUMANN
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{V\'{\i}ctor Neumann-Lara}					%NEUMANN-LARA*
\authorx{Victor Neumann-Lara}
\authorspace
\authref  See \hyperlink{Italo J.\ Dejter}{I.J.~Dejter}.
\itemspace

\author{Bryan Nevarez}						%NEVAREZ
\authorspace
\authref  See \hyperlink{Matthias Beck}{M.~Beck}.
\itemspace

\author{T.M.\ Newcomb}						%NEWCOMB
\authorspace
\authref  See also \hyperlink{K.O.\ Price}{K.O.~Price}.
\authrefspace

\bsgitem{1968a}{}  Interpersonal balance.  In: R.P.~Abelson \etal, eds., \emph{Theories of Cognitive Consistency: A Sourcebook}.  Rand-McNally, Chicago, Ill., 1968.
\codes{\CPsS}%
\itemspace

\author{G.F.\ Newell}							%NEWELL
\authorspace
\bsgitem{1950a}{}  Crystal statistics of a two-dimensional triangular Ising lattice.  \emph{Phys.\ Rev.}\ (2) 79 (1950), 876--882.
\MR{0039631} (12, 576i).  \Zbl{038.13902} (38, p.\ 139b).				
\annot{} The same physics conclusions as \xref{Houtappel's}{1950a}{R.M.F.\ Houtappel}, \xrefd{1950b}{R.M.F.\ Houtappel} for a signed, weighted triangular lattice.
[Also see \xref{I.\ Sy\^ozi}{1950a}{Itiro Syozi}, \xref{Wannier}{1950a}{G.H.\ Wannier}.]
\dated{20 Jun 2012} 
\codes{\CPhys,~\CWG,~\Csg:~\CFr}%
\itemspace

\author{Alantha Newman}						%NEWMAN A
\authorspace
\authref  See \hyperlink{Nir Ailon}{N.~Ailon}.
\itemspace

\author{Charles M.\ Newman}						%NEWMAN CM
\authorspace
\authref  See also \hyperlink{Federico Camia}{F.~Camia} and \hyperlink{Alberto Gandolfi}{A.~Gandolfi}.
\authrefspace

\author{Charles M.\ Newman \& Daniel L.\ Stein}
\authorspace
\bsgitem{1997a}{}  Metastate approach to thermodynamic chaos.  \emph{Phys.\ Rev.\ E} (3) 55 (1997), no.\ 5, part A, 5194--5211.
\MR{1448389} (98k:82098).  \arXiv{cond-mat/9612097}.  
\annot{} A technical paper supporting \xrefd{1998a}{Charles M.\ Newman \& Daniel L.\ Stein}.
\dated{26 Aug 2012} 
\par
\codes{\CPhys:~\Csg:~\CState,~\Cfr}%
\itemspace

\bsgitem{1998a}{}  Simplicity of state and overlap structure in finite-volume realistic spin glasses.  \emph{Phys.\ Rev.\ E} 57 (1998), no.\ 2, 1356--1366.
\MR{1606015} (99b:82057).  \arXiv{cond-mat/9711010}.  
\codes{\CPhys}%
\itemspace

\author{Mike Newman}						%NEWMAN M
\authorspace
\authref  See \hyperlink{Daryl Funk}{D.~Funk}.
\itemspace

\bsgitem{1998a}{}  Thermodynamic chaos and the structure of short-range spin glasses.  
In:  Anton Bovier and Pierre Picco, eds.,  \emph{Mathematical Aspects of Spin Glasses and Neural Networks}, pp.\ 243--287.  Progress in Prob., Vol.\ 41.  Birkh\"auser, Boston, 1998.
\MR{1601751} (99b:82056).  \Zbl{896.60078}.  
\annot{} See especially \S3, ``The standard SK picture''.  The Hamiltonian $H_\sigma(s)=-\sum_{vw \in E} \sigma(vw)s(v)s(w)$ is standard.  Criticizes the typical physics application of randomly signed (and possibly weighted) $K_n$ (Sherrington--Kirkpatrick model) to $\bbZ^d$-lattice graphs by limits of finite (cubical) subgraphs.  Raises the question of a ``pure state'' (\cf\ \xref{M\'ezard, Parisi, and Virasoro}{1987a}{Marc Mezard, Giorgio Parisi, \& Miguel Angel Virasoro} \etal) of a signed $K_n$, where a state is $s: V \to \signnos$ and a pure state is apparently a linear combination of or probability distribution on states, especially in the $\bbZ^d$ limit.  A pure state is not well defined but is related to states of low frustration (and high probability).  
[\Question.  Is there a graphical meaning of a pure state, based on the (ambiguous) physics definition?  It should involve states with low frustration, because they dominate the partition function $Z(\sigma)=\sum_s e^{H_\sigma(s)}$, and on the qualities desired for computing quantities of physical interest, especially in terms of $H$ and $Z$.]  
\annotctd{} A ``metastate'' is a measure on states, essentially a linear combination with explicit coefficients.  Pure states on $\bbZ^d$ should be metastates.  See \xrefd{1997a}{Charles M.\ Newman \& Daniel L.\ Stein}.  
[\Question.  Is there a graph-theory meaning to all this?  Does it lead to a definition of frustration in an infinite signed (or gain) graph?]
\dated{26 Aug 2012} 
\codes{\CPhys:~\Csg:~\CState,~\Cfr:~\CExp,~\CRef}%
\itemspace

\bsgitem{2010a}{}  Distribution of pure states in short-range spin glasses.  \emph{Int.\ J.\ Modern Phys.\ B} 24 (2010), no.\ 14, 2091--2106.
\MR{2659908} (2011g:82055).  \Zbl{1195.82093}.  
\annot{} Further development of \xrefd{1997a}{Charles M.\ Newman \& Daniel L.\ Stein}; \cf\ \xrefd{1998a}{Charles M.\ Newman \& Daniel L.\ Stein}.
\dated{26 Aug 2012} 
\par
\codes{\CPhys:~\Csg:~\CState,~\Cfr}%
\itemspace

\author{Sang Nguyen} 						%NGUYEN
\authorspace
\authref   See \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}.
\itemspace

\author{Aidin Niaparast}						%NIAPARAST
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Andr\'e Nichterlein}						%NICHTERLEIN
\authorspace
\authref  See \hyperlink{Falk Huffner}{F.~H\"uffner}.
\itemspace

\author{Robert Nickel}						%NICKEL
\authorspace
\authref  See \hyperlink{Winfried Hochstattler}{W.~Hochst\"attler}.
\itemspace

\author{Rolf Niedermeier}						%NIEDERMEIER
\authorspace
\authref  See \hyperlink{Falk Huffner}{F.~H\"uffner}.
\itemspace

\author{F.\ Nieto}							%NIETO
\authorspace
\authref  See \hyperlink{A.J.\ Ramirez-Pastor}{A.J.~Ram{\'\i}rez-Pastor} and \hyperlink{F.\ Roma}{F.~Rom\'a}.
\itemspace

\author{Juhani Nieminen} 						%NIEMINEN
\authorspace
\bsgitem{1976a}{}  Weak balance: A combination of Heider's theory and cycle and path-balance.  \emph{Control Cybernet.}\ 5 (1976), 69--73.  
\MR{0429628} (55 \#2639).  
\annot{} $S^c$ is the ``signed closure'' of a signed digraph $S$.  $S$ is ``weakly balanced'' if in $S^c$ all directed digons and all induced transitive triangles are positive.  Thm.:  $S$ is weakly balanced iff it is path- and cycle-balanced.  Also, the degree of weak balance.   
\par
\codes{\CSD:~\CBal}\codesctd{\CSD:~\CFr:~\CAlgor}%
\itemspace

\author{Peter Nijkamp} 						%NIJKAMP
\authorspace
\authref   See \hyperlink{Floor Brouwer}{F.~Brouwer}.
\itemspace

\author{Vladimir Nikiforov}						%NIKIFOROV
\authorspace
\authref  See also \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~de~Abreu}, \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de~Freitas}, and \hyperlink{Leonardo Silva de Lima}{L.S.~de~Lima}.
\authrefspace

\bsgitem{2008a}{}  A spectral condition for odd cycles in graphs.  \emph{Linear Algebra Appl.}\ 428 (2008), no.\ 7, 1492--1498.
\MR{2388633} (2008k:05130).  \Zbl{1152.05045}.  \arXiv{0707.4499}.			
\annot{} For order $n\gg0$, if $-\Gamma$ has least eigenvalue $< \sqrt{\lfloor n^2/4 \rfloor}$, then it has negative (i.e., odd) circles of all lengths $\leq n/320$.  
[\Question.  Does this property generalize to signed graphs as: eigenvalue bound $\implies$ all negative circles of lengths $\leq$ upper limit?]
\dated{20 Sept 2015} 
\par
\codes{\Csg:~\Cpar:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2014a}{}  Maxima of the $Q$-index: degenerate graphs.  \emph{Electronic J.\ Linear Algebra} 27 (2014), art.\ 15, 250--257.
\MR{3194954}.  \Zbl{1288.05164}.  \arXiv{1309.4837}.				
\annot{} Assume $\Gamma$ is ``$k$-degenerate'': every (induced) subgraph has a vertex of degree $\leq k$.  Thm.\ 1.2:  $\lambda_1(-\Gamma) \leq \lambda_1(-(K_k \vee \bar K_{n-k}))$, $=$ iff $\Gamma = K_k \vee \bar K_{n-k}$.  ($\lambda_1=$ max eigenvalue of $\Lap(\Sigma) :=$ Laplacian matrix.)  Thm.\ 1.3:  $\lambda_1(-\Gamma) \leq$ function of $n$, $\#E$, $\Delta$, and $\delta$.  
\dated{20 Jan 2015} 
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2014b}{}  An asymptotically tight bound on the $Q$-index of graphs with forbidden cycles.  \emph{Publ.\ Inst.\ Math.\ (Beograd)} (N.S.)\ 95(109) (2014), 189--199.
\MR{3221226}.  \Zbl{1433.05204}.  \arXiv{1310.1430}.				
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Vladimir Nikiforov \& Xiying Yuan}
\authorspace
\bsgitem{2014a}{}  Maxima of the $Q$-index: graphs without long paths.  \emph{Electronic J.\ Linear Algebra} 27 (2014), art.\ 32, 504--514.
\MR{3266163}.  \Zbl{1320.05078}.  \arXiv{1308.4341}.				
\annot{} Thm.\ 1.4:   For large $n$:  (i) $\Gamma \not\supseteq P_{2k+1} \implies \Gamma=K_k \vee \bar K_{n-k}$ or $\lambda_1(-\Gamma) < \lambda_1(-(K_k \vee \bar K_{n-k}))$.  (ii)  $\Gamma \not\supseteq P_{2k+2} \implies \Gamma=K_k \vee \bar K_{n-k} \cup e$ or $\lambda_1(-\Gamma) < \lambda_1(-(K_k \vee \bar K_{n-k} \cup e))$.  ($P_l=$ path of length $l$.  $\lambda_1=$ max eigenvalue of $\Lap(\Sigma)$.)
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2015a}{}  Maxima of the $Q$-index: forbidden even cycles.  \emph{Linear Algebra Appl.}\ 471 (2015), 636--653.  
\MR{3314357}.  \Zbl{1307.05149}.  \arXiv{1410.2142}.				
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Yuri Nikolayevsky}						%NIKOLAYEVSKY
\authorspace
\authref  See \hyperlink{Grant Cairns}{G.~Cairns}.
\itemspace

\author{Zoran Nikoloski}						%NIKOLSKI
\authorspace
\authref  See \hyperlink{Natasa Kejzar}{N.~Kej\v{z}ar}.
\itemspace

\authorx{Wenjie Ning}						%NING WJ
\author{Wenjie Ning, Hao Li, \& Mei Lu}
\authorspace
\bsgitem{2013a}{}  On the signless Laplacian spectral radius of irregular graphs.  \emph{Linear Algebra Appl.}\ 438 (2013), no.\ 5, 2280--2288.
\MR{3005290}.  \Zbl{1258.05074}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Koichi Nishino}						%NISHINO
\authorspace
\authref  See \hyperlink{Ojiro Nagai}{O.~Nagai}.
\itemspace

\author{Kenta Nishiyama}						%NISHIYAMA
\authorspace
\authref  See \hyperlink{Takayuki Hibi}{T.~Hibi}.
\itemspace

\author{M.\ Nogala}							%NOGALA
\authorspace
\authref  See \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\author{Kenta Noguchi}						%NOGUCHI
\authorspace
\bsgitem{2017a}{}  Even embeddings of the complete graphs and their cycle parities.  \emph{J.\ Graph Theory} 85 (2017), no.\ 1, 187--206.
\MR{3634482}.  \Zbl{1377.05042}.  
\annot{} Surface embedding of $-\Gamma$ so every face boundary is positive.	
[\Cf\ \xref{Archdeacon, Hutchinson, \etal\!}{2001a}{Dan Archdeacon, Joan Hutchinson, Atsuhiro Nakamoto, Seiya Negam [Seiya Negami], \& Katsuhiro Ota}, \xref{Nakamoto, Negami, and Ota}{2002a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}, \xrefd{2004a}{Atsuhiro Nakamoto, Seiya Negami, \& Katsuhiro Ota}.]
\codes{\Csg:~\CTop:~\Csw}
\itemspace

\authorx{J.D.\ Noh}							%NOH
\author{J.D.\ Noh, H.\ Rieger, M.\ Enderle, \& K.\ Knorr}
\authorspace
\bsgitem{2002a}{}  Critical behavior of the frustrated antiferromagnetic six-state clock model on a triangular lattice.  \emph{Phys.\ Rev.\ E} 66 (2002), art.\ 026111, 7 pp.
\MR{1922222} (2003f:82030).  \arXiv{cond-mat/0204087}.  
\annot{} The all-negative triangular lattice with 6th-root-of-unity spins.
	
\dated{24 Mar 2013} 
\codes{\CPhys,~\Csg:~\CPar:~\CFr}%
\itemspace

\author{Rafidah MD Noor}						%NOOR
\authorspace
\authref  See \hyperlink{Saeed Reza Shahriary}{S.R.~Shahriary}.
\itemspace

\author{Wouter de Nooy}						%NOOY	%DE NOOY
\authorspace
\bsgitem{1999a}{}  The sign of affection: Balance-theoretic models and incomplete signed digraphs.  \emph{Social Networks} 21 (1999), 269--286.
\annot{} Vertex ranking (a partial ordering) based on arc signs.  Thm.\ 3 characterizes equality of rank.  Thm.\ 6 characterizes strict inequality.	
\dated{11 Sept 2010} 
\codes{\CSD:~\CPsS,~\CBal,~\CClu}%
\itemspace

\bsgitem{2008a}{}  Signs over time: statistical and visual analysis of a longitudinal signed network.  \emph{J.\ Social Structure} 9 (2008), art.\ 1, 32 pp.
\codes{\CSG:~\CFr,~\CPsS:~\CDyn}%
\itemspace

\author{Wouter de Nooy, Andrej Mrvar, \& Vladimir Batagelj}
\authorspace
\bsgitem{2005a}{}  \emph{Exploratory Social Network Analysis with Pajek}.  
Structural Anal.\ Soc.\ Sci., No.\ 27.  Cambridge Univ.\ Press, Cambridge, Eng., 2005.
\annot{} Pajek is a computer package that analyzes networks, i.e., graphs, including signed graphs.  
Ch.\ 4: ``Sentiments and friendship.''  Computation of balance and clusterability of signed (di)graphs.   \S4.2: ``Balance theory.''  Introductory.  
\S4.4: ``Detecting structural balance and clusterability.''  How to use Pajek to optimize clustering.  \S4.5: ``Development in time.''  Pajek can look for evolution towards balance or clusterability.  
\annotctd{} \S10.3: ``Triadic analysis.''  Types of balance and clusterability, with the triads (order-3 induced subgraphs) that do or do not occur in each.  Table 16, p.\ 209, ``Balance-theoretic models'', is a chart of 6 related models.  \S\S10.7, 10.10: ``Questions'' and ``Answers.''  Some are on balance models.  \S10.9: ``Further reading.''
\dated{28 Apr 2009}
\par
\codes{\CSG,~\CSD,~\CPsS:~\CBal,~\CClu,~\CAlgor:~\CExp}%
\itemspace

\author{Robert Z.\ Norman} 						%NORMAN RZ
\authorspace
\authref   See also \hyperlink{M.\ Hamit Fisek}{M.H.~Fi\c{s}ek} and \hyperlink{Frank Harary}{F.~Harary}.
\authrefspace

\author{Robert Z.\ Norman \& Fred S.\ Roberts} 
\authorspace
\bsgitem{1972a}{}  A derivation of a measure of relative balance for social structures and a characterization of extensive ratio systems.  
\emph{J.\ Math.\ Psychology} 9 (1972), 66--91.  
\MR{0293041} (45 \#2121).  \Zbl{233.92006}.
\annot{} Circle (``cycle'') indices of imbalance: the proportion of circles that are unbalanced, with circles weighted nonincreasingly according to length.
\par
\codes{\CSG:~\CFr(\CCircles)}%
\itemspace

\bsgitem{1972b}{}  A measure of relative balance for social structures.  In:  Joseph Berger, Morris Zelditch, Jr., and Bo Anderson, eds., \emph{Sociological Theories in Progress, Vol.\ Two}, Ch.\ 14, pp.\ 358--391.  Houghton Mifflin, Boston, 1972.  
\annot{} Exposition and application of \xrefd{1972a}{Robert Z.\ Norman \& Fred S.\ Roberts}.
\codes{\CSG:~\CFr(\CCircles):~\CExp,~\CPsS}
\itemspace

\author{Mathilde Noual}						%NOUAL
\authorspace
\authref  See also \hyperlink{Julio Aracena}{J.~Aracena}, \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, and \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\authrefspace

\author{Mathilde Noual, Damien Regnault, \& Sylvain Sen\'e}
\authorx{Mathilde Noual, Damien Regnault, \& Sylvain Sene}
\authorspace
\bsgitem{2013a}{}  About non-monotony in Boolean automata networks.  \emph{Theor.\ Computer Sci.}\ 504 (2013), 12--25.
\MR{3107548}.  \Zbl{1297.68179}.  
\codes{\CSD:~\CDyn}%
\itemspace

\authorx{Beth Novick}						%NOVICK
\author{Beth Novick \& Andr\'as Seb\"o}
\authorx{Beth Novick \& Andras Sebo}
\authorspace
\bsgitem{1995a}{}  On combinatorial properties of binary spaces.  In: Egon Balas and Jens Clausen, eds., \emph{Integer Programming and Combinatorial Optimization} (4th Int.\ IPCO Conf., Copenhagen, 1995, Proc.), pp.\ 212--227.  \LNCS 920.  Springer-Verlag, Berlin, 1995.
\MR{1367983} (96h:05039).  
\annot{} The clutter of negative circuits of a signed binary matroid $(M,\sigma)$.  Important are the lift and extended lift matroids, $\bfL(M,\sigma)$ and $\bfL_0(M,\sigma)$, defined as in signed graph theory.  An elementary result: the clutter is signed-graphic iff $\bfL_0(M,\sigma)/e_0$ is graphic (which is obvious).  There are also more substantial but complicated results.  [See \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}, \S8.4.]	
\codes{\CSgnd(\CMatrd),~\CSG:~\CMatrd}%
\itemspace

\bsgitem{1996a}{}  On ideal clutters, metrics and multiflows.  In: William H.\ Cunningham, S.\ Thomas McCormick, and Maurice Queyrann, eds., \emph{Integer Programming and Combinatorial Optimization} (5th Int.\ IPCO Conf., Vancouver, 1996, Proc.), pp.\ 275--287.  \LNCS 1084.  Springer-Verlag, Berlin, 1996.
\MR{1441807} (98i:90075).  
\codes{\CSgnd(\CMatrd):~\CMatrd}%
\itemspace

\author{Marc Noy}							%NOY
\authorspace
\authref  See \hyperlink{Omer Gimenez}{O.~Gim\'enez}.
\itemspace

\author{Cyriel van Nuffelen} 						%NUFFELEN
\authorspace
\bsgitem{1973a}{} On the rank of the incidence matrix of a graph.  
Colloque sur la Theorie des Graphes (Bruxelles, 1973).  \emph{Cahiers Centre Etudes Rech.\ Oper.}\ 15 (1973), 363--365.  
\MR{0347660} (50 \#162).  \Zbl{269.05116}.					
\annot{} Theorem restated: the unoriented incidence matrix has rank $\rk \bfF(-\Gamma)$.  
[Because the matrix represents $\bfF(-\Gamma)$: see \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}.  In retrospect, partially implicit in \xref{Stewart}{1966a}{B.M.\ Stewart} and completely so in \xref{Stanley}{1973a}{Richard P.\ Stanley}.]
\codes{\Cpar:~\CInc,~\Cecyc}%
\itemspace

\bsgitem{1976a}{}  On the incidence matrix of a graph.  \emph{IEEE Trans.\ Circuits Systems} CAS-23 (1976), 572.  
\MR{0441791} (56 \#186).  
\annot{} Summarizes \xrefd{1973a}{Cyriel van Nuffelen}.	
\codes{\Cpar:~\CInc,~\Cecyc}%
\itemspace

\author{Koji Nuida}							%NUIDA
\authorspace
\authref  See also \hyperlink{Takuro Abe}{T.~Abe}.
\authrefspace

\bsgitem{2010a}{}  A characterization of signed graphs with generalized perfect elimination orderings.  \emph{Discrete Math.}\ 310 (2010), no.\ 4, 819--831.
\MR{2574831} (2011a:05140.  \Zbl{1209.05119}.  \arXiv{0712.4118}.			
\codes{\CSG:~\CStr,~\CGeom}%
\itemspace

\author{Yasuhide Numata}						%NUMATA
\authorspace
\authref  See \hyperlink{Takuro Abe}{T.~Abe}.
\itemspace

\author{G\'abor Nyul}							%NYUL
\authorspace
\authref  See \hyperlink{Eszter Gyimesi}{E.~Gyimesi}.
\itemspace


\label{LETTER O} \hypertarget{LETTER O}{}	% LETTER  O


\author{Suil O}							%O SI
\authorspace
\authref  See \hyperlink{Gary Greaves}{G.~Greaves}.
\itemspace

\author{Jan Obdr\v{z}\'alek}						%OBDRZALEK
\authorx{Jan Obdrzalek}
\authorspace
\authref  See \hyperlink{Robert Ganian, Petr Hlineny, \& Jan Obdrzalek}{R.~Ganian}.
\itemspace

\author{G.\ Obermair}						%OBERMAIR
\authorspace
\authref  See \hyperlink{K.\ Jungling}{K.~J\"ungling}.
\itemspace

\author{Mohammad Reza Oboudi}					%OBOUDI
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}.
\authrefspace

\bsgitem{2016a}{}  Energy and Seidel energy of graphs.  \emph{MATCH Commun. Math. Comput. Chem.}\ 75 (2016), 291--303.
\MR{3496590}.  \Zbl{1461.05133}.  
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Seidel energy of complete multipartite graphs.  \emph{Special Matrices} 2021 (2021), no.\ 9, 212--216.
\MR{4241490}.  \Zbl{1476.05124}.  
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\authorx{Cian O'Brien}						%OBRIEN	%O'BRIEN
\author{Cian O'Brien, Kevin Jennings, \& Rachel Quinlan}
\authorspace
\bsgitem{2020a}{}  Alternating signed bipartite graphs and difference-1 colourings.  \emph{Linear Algebra Appl.}\ 604 (2020), 370--398.
\MR{4122676}.  \Zbl{1447.05098}.  \arXiv{1911.10869}.  
\codes{\CSG:~\CECol}%
\itemspace

\author{Pascal Ochem}						%OCHEM
\authorspace
\authref  See also \hyperlink{Francois Dross}{F.~Dross} and \hyperlink{Amanda Montejano}{A.~Montejano}.
\authrefspace

\author{Pascal Ochem \& Nazanin Movarrae}
\authorspace
\bsgitem{2017a}{}  Oriented, 2-edge-colored, and 2-vertex-colored homomorphisms.  \emph{Inform.\ Processing Lett.}\ 123 (2017), 42--46.
\MR{3635310}.  \Zbl{1405.05120}.  \HAL{irmm-02083721}.  
\codes{\CSGc:~\CHom}%
\itemspace

\author{Pascal Ochem, Alexandre Pinlou, \& Sagnik Sen}
\authorspace
\bsgitem{2014a}{}  Homomorphisms of signed planar graphs.  Manuscript, 2014.  
\arXiv{1401.3308}.  
\par
\codes{\CSGc:~\CHom}%
\itemspace

\bsgitem{2017a}{}  Homomorphisms of 2-edge-colored triangle-free planar graphs.  \emph{J.\ Graph Theory} 85 (2017), no.\ 1, 258--277.
\MR{3634486}.  \Zbl{1365.05097}.  \HAL{irmm-01376130}.  
\codes{\CSG:~\CHom}
\itemspace

\author{M.\ Ocio}							%OCIO
\authorspace
\authref  See \hyperlink{E.\ Vincent}{E.~Vincent}.
\itemspace

\author{B.\ O'Connor}						%O'CONNOR
\authorspace
\authref  See \hyperlink{A.\ Krieger}{A.~Krieger}.
\itemspace

\author{Damien Octeau}						%OCTEAU
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Hidefumi Ohsugi}						%OHSUGI
\authorspace
\authref  See also \hyperlink{Tetsushi Matsui}{T.~Matsui}.
\authrefspace

\bsgitem{1999a}{}  Unimodular regular triangulations of (0,1)-polytopes associated with finite \linebreak graphs.  In:  Chrystopher L.\ Nehaniv \etal, eds., \emph{Algebraic Engineering} (Proc.\ Int.\ Workshop Formal Languages Computer Sys., Kyoto, 1997, and First Int.\ Conf.\ Semigroups Algebraic Engineering, Aizu, 1997), pp. 159--171.  World Scientific, Singapore, 1999.
\Zbl{1027.52008}.  
\codes{\Csg:~\Cpar:~\CGeom,~\CAlgeb}%
\itemspace

\bsgitem{2000a}{}  Compressed polytopes, initial ideals and complete multipartite graphs.  In:  \emph{Proceedings of the Third Symposium on Algebra, Languages and Computation} (Osaka, 1999), pp.\ 45--54.  Shimane Univ., Matsue, 2000.
\MR{1774200} (no rev).  \Zbl{943.13016}.  
\annot{} Extended abstract of \xref{Ohsugi and Hibi}{2000a}{Hidefumi Ohsugi \& Takayuki Hibi}
\dated{24 Jan 2016} 
\codes{\Csg:~\CPar:~\CGeom,~\CAlgeb}%
\itemspace

\author{Hidefumi Ohsugi \& Takayuki Hibi}
\authorspace
\bsgitem{1998a}{}  Normal polytopes arising from finite graphs.  \emph{J.\ Algebra} 207 (1998), 409--426.
\MR{1644250} (2000a:13010).  \Zbl{926.52017}.					
\annot{} The odd-cycle condition of \xref{Fulkerson, Hoffman, and McAndrew}{1965a}{D.R.\ Fulkerson, A.J.\ Hoffman, \& M.H.\ McAndrew} is employed in polynomial algebra.  ``Graph polytope'' [later named ``edge polytope''] $P_{-\Gamma} := \conv\mathbf{x}(E(-\Gamma))$, where $\mathbf{x}(E(-\Gamma)) =$ \{columns of incidence matrix $\Eta(-\Gamma)$\}.  
Dictionary:  ``$\cP_G$'' = $P_{-\Gamma}$.
[This is antibalanced.  \Problem.  Generalize to signed graphs, including balanced graphs.]
\dated{30 May 2011} 
\codes{\Csg:~\CPar:~\CGeom,~\CAlgeb}%
\itemspace

\bsgitem{1999a}{}  Toric ideals generated by quadratic binomials.  \emph{J.\ Algebra} 218 (1999), 509--527.
\MR{1705794} (2000f:13055).  \Zbl{0943.13014}.  
\annot{} \S1, ``Binomial ideals arising from finite graphs'':  The edge ring of $-\Gamma$ (negative edges because the analysis is antibalanced---even and odd graph circles are different) is $K[x_ix_j: ij \in E(\Gamma)]$.  Thm.\ 1.2:  The toric ideal $I_\Gamma$ is generated by quadratic binomials iff in $-\Gamma$, each positive circle has certain chords, each contrabalanced tight handcuff has an edge between its circles, and each two disjoint negative circles are joined by at least 2 edges [i.e., antibalanced criteria].  [\Problem.  Understand this via signed graphs.]
\S4:  Edge polytope $P_{-\Gamma}$ properties such as minimal volume.  
\S5, ``Simple edge polytopes'':  E.g., Cor.\ 5.4:  $P_{-\Gamma}$ is simple iff $\Gamma=K_{p,q}$.  
[\Problem.  Generalize to signed graphs, including ordinary graphs $\Gamma$ (i.e., all positive).  The edge ring would be bidirected: $K[\mathbf{x}^{\vec e_{ij}}: \vec e_{ij} \in E(\Beta)]$ for a bidirection $\Beta$ of $\Sigma$.  E.g., one expects $P_{+\Gamma}$ to be simple iff $\Gamma=+K_n$.]
\dated{5 Oct 2014, 3 Jun 2015} 
\par
\codes{\Csg:~\CPar:~\CAlgeb,~\CGeom}%
\itemspace

\bsgitem{1999b}{}  A normal $(0, 1)$-polytope none of whose regular triangulations is unimodular.  \emph{Discrete Comput.\ Geom.}\ 21 (1999), 201--204.
\MR{1668090} (99k:52023).  \Zbl{927.52018}.  
\annot{} The polytope is the edge polytope $\conv\Eta(-\Gamma)$ (\cf\ \xrefd{1998a}{Hidefumi Ohsugi \& Takayuki Hibi}) where $\Gamma = C_5$ with a triangle on each edge.
\dated{18 Aug 2018} 
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\bsgitem{2000a}{}  Compressed polytopes, initial ideals and complete multipartite graphs.  \emph{Illinois J.\ Math.}\ 44 (2000), no.\ 2, 391--406.
\MR{1775328} (2001e:05092).  \Zbl{943.13016}.  
\annot{} Edge polytope $P_{-K_{n_1,\ldots,n_k}}$ and edge ring.
[This is antibalanced.  \Problem.  Generalize to signed graphs, including balanced graphs.]
\dated{5 Oct 2014} 
\codes{\Csg:~\CPar:~\CGeom,~\CAlgeb}%
\itemspace

\bsgitem{2003a}{}  Normalized volumes of configurations related with root systems and complete bipartite graphs.  \emph{Discrete Math.}\ 268 (2003), 217--242.
\MR{1983280} (2004m:\-52018).  \Zbl{1080.14059}.					
\annot{} A configuration consists of the vectors representing an acyclic orientation of a complete signed bipartite graph.  The volume is that of the pyramid over the configuration with apex at the origin.  (Successor to \xref{Fong}{2000a}{Wungkum Fong}.)
[\Question.  Is there a connection with the chromatic polynomial?]
\codes{\Csg:~\CGeom:~\CInvar}%
\itemspace

\bsgitem{2008a}{}  Simple polytopes arising from finite graphs.  In:  Matthias Dehmer, Michael Drmota, and Frank Emmert-Streib, eds., \emph{Proceedings of the 2008 International Conference on Information Theory and Statistical Learning} (ITSL 2008, Las Vegas, 2008), pp.\ 73--79. CSREA Press, 2008.  URL \url{https://dblp1.uni-trier.de/db/conf/itsl/itsl2008}
\arXiv{0804.4287}.  
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\author{Hidefumi Ohsugi \& Kazuki Shibata}
\authorspace
\bsgitem{2012a}{}  Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part.  \emph{Discrete Comput.\ Geom.}\ 47 (2012), 624--628.
\MR{2891253}.  \Zbl{1238.52005}.  \arXiv{1109.0791}.				
\codes{\Csg:~\Cpar:~\CGeom}%
\itemspace

\author{Nacim Oijid}							%OIJID
\authorspace
\authref  See \hyperlink{Julien Bensmail}{J.~Bensmail}.
\itemspace

\author{Ayao Okiji}							%OKIJI
\authorspace
\authref  See \hyperlink{Yasuhiro Kasai}{Y.~Kasai}.
\itemspace

\author{E.\ Olaru} 							%OLARU
\authorspace
\authref  See \hyperlink{St.\ Antohe}{St.~Antohe}.
\itemspace

\author{Mari\'an Olej\'ar}						%OLEJAR
\authorx{Marian Olejar}
\authorspace
\authref  See \hyperlink{Jozef Siran}{J.~\v{S}ir\'a\v{n}}.
\itemspace

\author{D.D.\ Olesky}						%OLESKY
\authorspace
\authref   See also \hyperlink{B.D.\ Bingham}{B.D.~Bingham}, \hyperlink{Thomas Britz}{T.~Britz}, \hyperlink{Minerva Catral}{M.~Catral}, \hyperlink{G.J.\ Culos}{G.J.~Culos}, \hyperlink{D.A.\ Grundy, D.D.\ Olesky, \& P.\ van den Driessche}{D.A.~Grundy}, \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}, and \hyperlink{Keivan Hassani Monfared}{K.~Hassani Monfared}.
\authrefspace

\author{D.D.\ Olesky, M.J.\ Tsatsomeros, \& P.\ van den Driessche}
\authorspace
\bsgitem{2012a}{}  Sign patterns with a nest of positive principal minors.  \emph{Linear Algebra Appl.}\ 436 (2012), 4392--4399.
\MR{2917416}.  \Zbl{1244.15022}.  
\codes{\CQM:~\CSD}%
\itemspace[

\authorx{Aroldo Oliveira}						%OLIVEIRA A
\author{Aroldo Oliveira, Leonardo Silva de Lima, \& Nair Maria Maia de Abreu}
\authorspace
\bsgitem{2012a}{}  On the spread and the chromatic number of a graph.  Proc.\ Forty-Third Southeastern Int.\ Conf.\ Combinatorics, Graph Theory and Computing.  \emph{Congressus Numer.}\ 212 (2012), 57--64.
\MR{3024414}.  \Zbl{1278.05105}.  
\annot{} For $\Lap(-\Gamma)$, spread = (largest $-$ smallest eigenvalue) $\leq \chi(\Gamma)$.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Carla Silva Oliveira}						%OLIVEIRA CS
\authorspace
\authref  See also \hyperlink{Leonardo Silva de Lima}{L.S.~de~Lima}.
\authrefspace

\author{Carla Oliveira \& Leonardo de Lima}
\authorspace
\bsgitem{2016a}{}  A lower bound for the sum of the two largest signless Laplacian eigenvalues.  14th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW'16, Gargnano, Italy, 2016).  \emph{Electronic Notes Discrete Math.}\ 55 (2016), 173--176.
\Zbl{1356.05082}.  \arXiv{1412.0323}.\MR none 7/19.  				
\annot{} A degree bound for $\Lap(-\Gamma)$.  \Cf\ \xref{Li and Tian}{2015a}{Shuchao Li \& Yi Tian}.
\dated{8 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, \& Pierre Hansen}
\authorspace
\bsgitem{2010a}{}  Bounds on the index of the signless Laplacian of a graph.  \emph{Discrete Appl.\ Math.}\ 158 (2010), no.\ 4, 355--360.
\MR{2588119} (2011d:05228).  \Zbl{1225.05174}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, \& Steve Kirkland}
\authorspace
\bsgitem{2010a}{}  Bounds on the $Q$-spread of a graph.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 9, 2342--2351.
\MR{2599864} (2011k:05146).  \Zbl{1214.05082}.					
\annot{} \Cf\ \xref{M.H.~Liu and Liu}{2010a}{Muhuo Liu \& Bolian Liu}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{F.\ Oloomi}							%OLOOMI
\authorspace
\authref  See \hyperlink{R.\ Masoumi}{R.~Masoumi}.
\itemspace

\author{Stig W.\ Omholt}						%OMHOLT
\authorspace
\authref  See \hyperlink{Erik Plahte}{E.~Plahte}.
\itemspace

\author{G.R.\ Omidi}							%OMIDI
\authorspace
\authref  See also \hyperlink{F.\ Ayoobi}{F.~Ayoobi} and \hyperlink{Willem H.\ Haemers}{W.H.~Haemers}.
\authrefspace

\bsgitem{2009a}{}  On a signless Laplacian spectral characterization of $T$-shape trees.  \emph{Linear Algebra Appl.}\ 431 (2009), no.\ 9, 1607--1615.
\MR{2555062} (2010m:05181).  \Zbl{1169.05351}.					
\annot{} A subdivided $K_{1,3}$ is determined by $\Spec \Lap(-\Gamma)$.  [Continued in \xref{Omidi and Vatandoost}{2010a}{Gholam R.\ Omidi \& Ebrahim Vatandoost} and \xref{Bu and Zhou}{2012a}{Changjiang Bu \& Jiang Zhou}.]
\dated{28 Nov 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Gholam R.\ Omidi \& Ebrahim Vatandoost}
\authorspace
\bsgitem{2010a}{}  Starlike trees with maximum degree 4 are determined by their signless Laplacian spectra.  \emph{Electronic J.\ Linear Algebra} 20 (2010), 274--290.
\MR{2653539} (2011c:05205).  \Zbl{1205.05151}.					
\annot{} A subdivided $K_{1,4}$ is determined by $\Spec \Lap(-\Gamma)$.  Continuation of \xref{Omidi}{2009a}{G.R.\ Omidi}.  [Continued in \xref{Bu and Zhou}{2012a}{Changjiang Bu \& Jiang Zhou}.]
\dated{28 Nov 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Kenji Onaga} 						%ONAGA
\authorspace
\bsgitem{1966a}{}  Dynamic programming of optimum flows in lossy communication nets.  \emph{IEEE Trans.\ Circuit Theory} CT-13 (1966), 282--287.  
\codes{\CGN}%
\itemspace

\bsgitem{1967a}{}  Optimal flows in general communication networks.  \emph{J.\ Franklin Inst.}\ 283 (1967), 308--327.  
\MR{0218100} (36 \#1189).  \Zbl{203.22402} (203, p.\ 224b).
\codes{\CGN}%
\itemspace

\author{Shmuel Onn}						%ONN
\authorspace
\authref   See also \hyperlink{Peter Kleinschmidt}{P.~Kleinschmidt}.
\authrefspace

\bsgitem{1997a}{}  Strongly signable and partitionable posets.  \emph{European J.\ Combin.}\ 18 (1997), 921--938.
\MR{1485377} (99d:06007).  \Zbl{887.06003}.						
\annot{} For ``signability'' see \xref{Kleinschmidt and Onn}{1995a}{Peter Kleinschmidt \& Shmuel Onn}.  A strong signing is an exact signing that satisfies a recursive condition on lower intervals.
\par
\codes{\CS,~\CGeom}%
\itemspace

\author{Rikio Onodera} 						%ONODERA
\authorspace
\bsgitem{1968a}{}  On signed tree-graphs and cotree-graphs.  \emph{RAAG Res.\ Notes} (3) No.\ 133 (1968), ii + 29 pp. 
\MR{0237383} (38 \#5671).  \Zbl{182.58201} (182, p.\ 582a).					
\annot{} The adjacency graph of trees of a graph is signed from a vertex signature and is shown to be balanced.  [Trivial.]
\dated{24 July 2010} 
\par
\codes{\CSG:~\CBal}%
\itemspace

\author{The Open University} 						%OPEN UNIVERSITY
\authorspace
\bsgitem{1981a}{}  Graphs and Digraphs.  Unit 2 in Course TM361:  Graphs, Networks and Design.  
The Open Univ.\ Press, Walton Hall, Milton Keynes, England, 1981.
\annot{} Social sciences (pp.\ 21--23).  Signed digraphs (pp.\ 50--52).  [Published version: see \xref{Wilson and Watkins}{1990a}{Robin J.\ Wilson \& John J.\ Watkins}.]
\codes{\CSG,~\CPsS,~\CSD:~\CExp}%
\itemspace

\authorx{Peter Orlik} 							%ORLIK
\author{Peter Orlik \& Louis Solomon}
\authorspace
\bsgitem{1980a}{}  Unitary reflection groups and cohomology.  \emph{Invent.\ Math.}\ 59 (1980), 77--94.  
\MR{0575083} (81f:32017).  \Zbl{452.20050}.					
\annot{} Thm.\ (4.8):  The characteristic polynomials of the Dowling lattices and jointless Dowling lattices of $\bbZ_r$, computed via group theory as part of the general treatment of finite unitary reflection groups.
\codes{\Cgg:~\Cmatrd,~\CGeom}%
\itemspace

\bsgitem{1982a}{}  Arrangements defined by unitary reflection groups.  \emph{Math.\ Ann.}\ 261 (1982), 339--357.  
\MR{0679795} (84h:14006).  \Zbl{491.51018}.						
\annot{} In the intersection lattice of reflection hyperplanes of a finite unitary reflection group, the characteristic polynomial of an upper interval has an integral factorization.  The proofs involve detailed study of the group actions on $\bbC^l$. 
Dictionary:   $\cA_l(r)$ and $\cA_l^k(r)$ are the arrangements corresponding to the rank-$l$ Dowling lattices and partially jointless Dowling lattices of $\bbZ_r$.  
Relevant results:  \S2: ``Monomial groups'': Cor.\ (2.4) counts the flats, Prop.\ (2.5) and Cor.\ (2.7) gives the polynomials for $\cA_l(r)$ [all known from \xref{Dowling}{1973b}{Thomas A.\ Dowling}].  Cor.\ (2.10) counts the flats, Prop.\ (2.13) gives the polynomial of $\cA_l^k(r)$, Prop.\ (2.14) notes that proper upper intervals are Dowling lattices [all fairly obvious via gain graphs and coloring (\xref{Zaslavsky}{1995b}{Thomas Zaslavsky})].
\codes{\Cgg:~\Cmatrd,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{1983a}{}  Coxeter arrangements.  In:  Peter Orlik, ed., \emph{Singularities} (Arcata, Calif., 1981), Part 2, pp.\ 269--291.  
Proc.\ Sympos.\ Pure Math., Vol.\ 40.  Amer.\  Math.\ Soc., Providence, R.I., 1983.  
\MR{0713255} (85b:32016).  
\codes{\Cgg:~\Cmatrd,~\CGeom,~\CInvar}%
\itemspace

\author{James B.\ Orlin} 						%ORLIN
\authorspace
\authref  See also \hyperlink{Ravindra K.\ Ahuja}{R.K.~Ahuja}, \hyperlink{Muralidharan Kodialam}{M.~Kodialam}, and \hyperlink{Randy Shull}{R.~Shull}.
\authrefspace

\bsgitem{1984a}{}  Some problems on dynamic/periodic graphs.  In: \emph{Progress in combinatorial optimization} (Proc.\ Conf., Waterloo, Ont., 1982), pp.\ 273--293.  Academic Press, Toronto, 1984.
\MR{0771882} (86m:90058).  \Zbl{547.05060}.
\annot{} Problems on $1$-dimensional periodic graphs (i.e., covering (di)graphs of $\bbZ$-gain graphs $\Phi$) that can be solved in $\Phi$: connected components, strongly connected components, directed path from one vertex to another, Eulerian trail (directed or not), bicolorability, and spanning tree with minimum average cost.
\par
\codes{\CGG,~\CGD:~\CCov:~\CPaths,~\CCircles,~\CCol:~\CAlgor}%
\itemspace

\bsgitem{1985a}{}  On the simplex algorithm for networks and generalized networks.  \emph{Math.\ Programming Study} 24 (1985), 166--178.  
\MR{0820998} (87k:90102).  \Zbl{592.90031}.
\par
\codes{\CGN:~\CMatrd(\CBases):~\CAlgor}%
\itemspace

\authorx{Charles E.\ Osgood}				 		%OSGOOD
\author{Charles E.\ Osgood \& Percy H.\ Tannenbaum}
\authorspace
\bsgitem{1955a}{}  The principle of congruity in the prediction of attitude change.  \emph{Psychological Rev.}\ 62 (1955), 42--55.  
\codes{\CVS:~\CPsS}
\itemspace

\author{Eiji O'Shima} 						%O'SHIMA
\authorspace
\authref   See \hyperlink{Masao Iri}{M.~Iri} and \hyperlink{J.\ Shiozaki}{J.~Shiozaki}.
\itemspace

\author{M.A.\ Osorio}							%OSORIO
\authorspace
\authref  See \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\author{Patric R.J.\ \"Osterg\r{a}rd}					%OSTERGARD
\authorspace
\authref  See \hyperlink{Ferenc Szollosi}{F.~Sz\"oll\H{o}si}.
\itemspace

\authorx{M.\ Ostilli}							%OSTILLI
\author{M.\ Ostilli \& J.F.F.\ Mendes}
\authorspace
\bsgitem{2009a}{}  Small-world of communities: communication and correlation of the meta-network.  \emph{J.\ Stat.\ Mech.}\ 2009 (2009), art.\ L08004.
\arXiv{0812.0608}.  
\annot{} Short version of \xrefd{2009b}{M.\ Ostilli \& J.F.F.\ Mendes}.
\codes{\CPhys:~\CFr}%
\itemspace

\bsgitem{2009b}{}  Communication and correlation among communities.  \emph{Phys.\ Rev.\ E} 80 (2009), art.\ 011142, 23 pp.
\MR{2552037} (2010h:91167).  \arXiv{0902.0888}.  
\par
\codes{\CPhys:~\CFr,~\Csg}%
\itemspace

\author{Katsuhiro Ota}						%OTA
\authorspace
\authref  See \hyperlink{Dan Archdeacon}{D.~Archdeacon} and \hyperlink{Atsuhiro Nakamoto}{A.~Nakamoto}.
\itemspace

\authorx{Li Ou}							%OU L
\author{Li Ou, Yaoping Hou, \& Zhuang Xiong}
\authorspace
\bsgitem{2021a}{}  The net Laplacian spectra of signed complete graphs.  \emph{Contemp.\ Math.}\ 2 (2021), no.\
 4, 409--417.
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Sang-Il Oum}						%OUM
\authorspace
\authref  See \hyperlink{Tony Huynh}{T.~Huynh}.
\itemspace

\author{James G.\ Oxley} 						%OXLEY
\authorspace
\authref   See also \hyperlink{Thomas H.\ Brylawski [Tom Brylawski]}{T.~Brylawski}, \hyperlink{Tara Fife}{T.~Fife}, \hyperlink{Jim Geelen}{J.~Geelen}, \hyperlink{Joseph P.S.\ Kung}{J.P.S.~Kung}, and \hyperlink{Laurence R.\ Matthews}{L.R.~Matthews}.
\authrefspace

\bsgitem{1992a}{}  Infinite matroids.  
In:  Neil White, ed.,  \emph{Matroid Applications}, Ch.\ 3, pp.\ 73--90.  
Encycl.\ Math.\ Appl., Vol.\ 40.  Cambridge Univ.\ Press, Cambridge, Eng., 1992.
\MR{1165540} (93f:05027).  \Zbl{766.05016}.					
\annot{} See Exer.\ 3.20.
\codes{\CBic:~\CExp}%
\itemspace

\bsgitem{1992b}{}  \emph{Matroid Theory}.  Oxford Univ.\ Press, Oxford, 1992.
\MR{1207587} (94d:\-05033).  \Zbl{784.05002}.					
\annot{} Thm.\ 6.6.3: proof from \xref{Brylawski's}{1975a}{Thomas H.\ Brylawski [Tom Brylawski]}.
\codes{\Cgg:~\Csw:~\CExp}%
\annotctd{} \S10.3: Exer.\ 3 concerns the Dowling lattices of $\GF(q)^\times$.
\S12.2: Exer.\ 13 concerns $\bfF(\Omega)$.
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{2011a}{}  \emph{Matroid Theory}, 2nd ed.  Oxford Grad.\ Texts Math., 21.  Oxford Univ.\ Press, Oxford, 2011. 
\MR{2849819} (2012k:05002).  \Zbl{1254.05002}.					
\annot{} \S6.10, ``Dowling geometries'':  Frame (i.e., bias) matroid theory of biased graphs.  Examples: gain and signed graphs, \xref{Dowling}{1973a}{Thomas A.\ Dowling}, \xrefd{1973b}{Thomas A.\ Dowling} geometries, bicircular, even-circle (even-cycle, factor), and poise and antidirection matroids.  Representability of Dowling geometries.  \xref{Kahn and Kung's}{1982a}{Jeff Kahn \& Joseph P.S.\ Kung} varieties.  
Other mentions of Dowling geometries in Prop.\ 14.10.22, \S15.3 p.\ 590, \S15.9 p.\ 605, and Appendix, ``Some interesting matroids'', p.\ 663; of bicircular matroids in Exer.\ 10.4.12, Prop.\ 11.1.6, Exer.\ 11.1.7, Conj.\ 14.3.12, Thm.\ 14.10.19.  
\dated{21 Mar 2011} 
\annotctd{} Spikes (with tips) and swirls, important in matroid structure theory, are the lift (extended lift) and frame matroids of biased $2C_n$'s.  Spikes:  pp.\ 40--42, 72--74, 111--112, 197--202, 545--548, 568, 662, \etal\ \  Swirls:  pp.\ 552, 568, 664, \etal\ \  [The biased-graph representation could simplify some of the descriptions.]
\dated{7 Feb 2013} 
\par
\codes{\CGG:~\CMatrd,~\CBic,~\CECyc:~\CExp,~\CExr}%
\itemspace

\author{James Oxley, Dirk Vertigan, \& Geoff Whittle}
\authorspace
\bsgitem{1996a}{}  On inequivalent representations of matroids over finite fields.  \emph{J.\ Combin.\ Theory Ser.\ B} 67 (1996), 325--343.
\MR{1399683} (97d:05052).  \Zbl{856.05021}.						
\annot{} \S5: Free swirls, $\bfF(2C_n,\eset)$ ($n\geq4$), mentioning their relationship to Dowling lattices, and complete free spikes, $\bfL_0(2C_n,\eset)$.
\codes{\CGG:~\CMatrd}%
\itemspace

\authorx{Olayiwola O.\ Oyeleye}					%OYELEYE
\author{Olayiwola O.\ Oyeleye \& Mark A.\ Kramer}
\authorspace
\bsgitem{1988a}{}  Qualitative simulation of chemical process systems: Steady-state analysis.  \linebreak{}
\emph{AIChE J.}\ 34 (1988), no.\ 9, 1441--1454.
\annot{} Extends the signed digraph of \xref{Iri, Aoki, O'Shima, and Matsuyama}{1979a}{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama} ``to account for complex dynamics''.
\dated{17 Feb 2013} 
\par
\codes{\CSD,~\CVS:~\CAppl,~\CAlgor}%
\itemspace

\author{Kenta Ozeki}							%OZEKI
\authorspace
\authref  See \hyperlink{Ken-ichi Kawarabayashi \& Kenta Ozeki}{K.~Kawarabayashi} and \hyperlink{Seog-Jin Kim}{S.-J.~Kim}.
\itemspace


\label{LETTER P} \hypertarget{LETTER P}{}	% LETTER  P


\author{M.L.\ Paciello}						%PACIELLO
\authorspace
\authref  See \hyperlink{M.\ Falcioni}{M.~Falcioni}.
\itemspace

\author{Manfred W.\ Padberg} 						%PADBERG
\authorspace
\authref   See \hyperlink{Ellis L.\ Johnson}{E.L.~Johnson}.
\itemspace

\author{E.\ Padmavathy}						%PADMAVATHY
\authorspace
\authref  See \hyperlink{V.\ Jude Annie Cynthia}{V.J.A.~Cynthia}.
\itemspace

\author{Carles Padr\'o}						%PADRO
\authorspace
\authref  See \hyperlink{Amos Beimel}{A.~Beimel}.
\itemspace

\author{Steven R.\ Pagano}						%PAGANO
\authorspace
\bsgitem{1998a}{\dag}  \emph{Separability and Representability of Bias Matroids of Signed Graphs}.  Doctoral thesis, State University of New York at Binghamton, 1998.
\MR{2697393} (no rev).  
\annot{} Ch.\ 1: ``Separability''.  Graphical characterization of bias-matroid $k$-sepa\-ra\-tions of a biased graph.  Also, some results on the possibility of $k$-sepa\-ra\-tions in which one or both sides are connected subgraphs.
\par
\codes{\CGG:~\CMatrd:~\CStr}%
\annotctd{} Ch.\ 2: ``Representability''.  The frame matroid of every signed graph is representable over all fields with characteristic $\neq 2$.  For which signed graphs is it representable in characteristic $2$ (and therefore representable over $\GF(4)$, by the theorem of Geoff Whittle, A characterization of the matroids representable over $\GF(3)$ and the rationals.  \emph{J.\ Combin.\ Theory Ser.\ B} 65 (1995), 222--261.  \MR{1358987} (96m:05046).  \Zbl{835.05015}.)?  
Solved (for $3$-connected signed graphs having vertex-disjoint negative circles and hence nonregular matroid).  There are two essentially different types: (i) two balanced graphs joined by three independent unbalanced digons; (ii) a cylindrical signed graph, possibly with balanced graphs adjoined by $3$-sums.  [See notes on \xref{Seymour}{1995a}{P.D.\ Seymour} for definition of (ii) and for Lov\'asz--Slilaty's structure theorem in the case without vertex-disjoint negative circles.]
\annotctd{} Furthermore, the representations of these graphs in characteristic not $2$ are all canonical signed-graphic, while any representations over $\GF(4)$ are canonical $\bbZ_3$-gain graphic.
\codes{\CSG:~\CMatrd:~\CInc,~\CStr,~\CTop}%
\annotctd{} Ch.\ 3: ``Miscellaneous results''.
\codes{\CSG:~\CMatrd:~\CInc,~\CStr}%
	
\itemspace

\bsgitem{1999a}{}  Binary signed graphs.  Manuscript, ca.\ 1999.
\codes{\CSG:~\CMatrd:~\CInc,~\CStr}%
\itemspace

\bsgitem{1999b}{}  Signed graphic $\GF(4)$ forbidden minors.  Manuscript, ca.\ 1999.
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{1999c}{}  $\GF(4)$-representations of bias matroids of signed graphs: The $3$-connected case.  Manuscript, ca.\ 1999.
\codes{\CSG:~\CMatrd:~\CInc,~\CStr,~\CTop}%
\itemspace

\author{Igor Pak}							%PAK
\authorspace
\authref  See \hyperlink{Sergei Chmutov}{S.~Chmutov}.
\itemspace

\author{Anita Pal}							%PAL
\authorspace
\authref  See \hyperlink{Debanjan Banerjee}{D.~Banerjee} and \hyperlink{S.N.\ Mishra}{S.N.~Mishra}.
\itemspace

\authorx{Matteo Palassini}						%PALASSINI
\author{Matteo Palassini \& Sergio Caracciolo}
\authorspace
\bsgitem{2000a}{}  Monte Carlo simulation of the three-dimensional Ising spin glass.  In:  David P.\ Landau \etal, eds., \emph{Computer Simulation Studies in Condensed-Matter Physics XII} (Proc.\ Twelfth Workshop, Athens, Ga., 1999), pp.\ 162--166.  Springer Proc.\ Phys., Vol.\ 85.  Springer, Berlin, 2000.
\arXiv{cond-mat/9911449}.  
\annot{} Physical quantities on the $\pm J$ cubic lattice model, i.e., a signed cubic lattice graph.  
\dated{28 Mar 2013} 
\codes{\CPhys,~\Csg:~\CFr,~\CState}%
\itemspace

\author{Matteo Palassini, Frauke Liers, Michael Juenger, \& A.P.\ Young}
\authorspace
\bsgitem{2003a}{}  Low-energy excitations in spin glasses from exact ground states.  \emph{Phys.\ Rev.\ B} 68 (2003), art.\ 064413, 16 pp.
\arXiv{cond-mat/0212551}.  
\annot{} \S\S II--III:  Take a ground state $\zeta_0$.  Add $(\epsilon/\#E) \zeta_0(v_i)\zeta_0(v_j)$ to $J_{ij}$, raising energy of $\zeta_0$ by $\epsilon>0$ and of any other state $\zeta$ by less, the amount depending on the edges whose signs differ in $\zeta_0$ and $\zeta$ [i.e., negative edges in $\Sigma^{\zeta_0\zeta}$].  This may change the relative energies of states.  See if a near-ground state becomes a ground state.  [This interesting approach makes sense only for weighted $\Sigma$.]
\annotctd{} \S\S IV--V:  Algorithm for frustration index (equivalently, weighted frustration index) of a weighted signed graph, tested on small cubic lattice graphs.  It uses a branch-and-cut method that requires solving many linear programs.  [This part makes sense for unweighted $\Sigma$.]
\annotctd{} Dictionary:  $J_{ij}$ = signed weight of edge $e_{ij}$; ``state'' = vertex signing $\zeta$; ``energy'' of $\zeta$ = weighted frustration of $(\Sigma^\zeta,\#J)$ := total unsigned weight of $E^-(\Sigma^\zeta)$; ``ground state'' = any $\zeta$ that minimizes energy; minimum energy = energy of ground state = weighted frustration index $l(\Sigma,\#J)$; ``free'' or ``periodic'' boundary conditions = nontoroidal ([path]${}^3$) or toroidal ([circle]${}^3$).
\dated{22 Dec 2014} 
\par
\codes{\CSG,~\CWG:~\CState(\Cfr),~\CFr:~\CAlgor}%
\itemspace

\author{Jeepamol J.\ Palathingal}					%PALATHALINGAL
\authorspace
\bsgitem{2020a}{}  \emph{Studies on Graph Operators}.  Doctoral thesis, Mahatma Gandhi University, Kottayam, 2020.
\annot{} Ch.\ 4:  Signed Gallai graph.  Ch.\ 5:  Signed anti-Gallai graph.  \Cf\ \xref{Palathingal and Lakshmanan}{2017a}{Jeepamol J.\ Palathingal \& Aparna Lakshmanan S}, \xrefd{20xxa}{Jeepamol J.\ Palathingal \& Aparna Lakshmanan S}.
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\author{Jeepamol J.\ Palathingal \& Aparna Lakshmanan S}
\authorspace
\bsgitem{2017a}{}  Forbidden subgraph characterizations of extensions of Gallai graph operator to signed graph.  \emph{Ann.\ Pure Appl.\ Math.}\ 14 (2017), no.\ 3, 437--448.
\annot{} Gallai graph $\Gamma(\Sigma)$: $V(\Gamma) := E(\Sigma)$, $ef\in E(\Gamma)$ when $e,f \notin$ triangle, and three different sign functions modelled on $\Lambda_{BC}$, $\Lambda_{\times}$, $\Lambda_\bullet$.  Thms.:  All forbidden induced subgraphs for each type (but the type is not determined by forbidden induced subgraphs).
\dated{17 Dec 2019} 
\par
\codes{\CSG:~\CLG(\CGen):~\CStr}%
\itemspace

\bsgitem{20xxa}{}  Forbidden subgraph characterizations of extensions of anti-Gallai graph operator to signed graph.  Preprint.  
\annot{} Anti-Gallai graph $\Delta(\Sigma)$: $V(\Delta) := E(\Sigma)$, $ef\in E(\Delta)$ when $e,f \in$ triangle, and the three sign functions in \xrefd{2017a}{Jeepamol J.\ Palathingal \& Aparna Lakshmanan S}.  Thms.:  The same kind as in \xrefd{2017a}{Jeepamol J.\ Palathingal \& Aparna Lakshmanan S}.
\dated{17 Dec 2019} 
\codes{\CSG:~\CLG(\CGen):~\CStr}%
\itemspace

\author{Edgar M.\ Palmer} 						%PALMER
\authorspace
\authref   See \hyperlink{Frank Harary}{F.~Harary} and \hyperlink{F.\ Kharari}{F.~Kharari}.
\itemspace

\author{B.L.\ Palowitch, Jr.}						%PALOWITCH
\authorspace
\authref  See \hyperlink{Mark A.\ Kramer}{M.A.~Kramer}.
\itemspace

\author{Rong-Ying Pan}						%PAN RY
\authorspace
\authref  See \hyperlink{Ya-Hong Chen}{Y.H.~Chen}.
\itemspace

\author{Yongliang Pan}						%PAN YL
\authorspace
\authref  See \hyperlink{Yaoping Hou}{Y.P.~Hou}.
\itemspace

\authorx{Zhishi Pan}							%PAN ZS
\author{Zhishi Pan \& Xuding Zhu}
\authorspace
\bsgitem{2022a}{}  The circular chromatic numbers of signed series-parallel graphs.  \emph{Discrete Math.}\ 345 (2022), no.\ 3, art.\ 112733, 10 pp.
\MR{4349877}.  
\annot{} \Cf\ \xref{Naserasr, Wang, and Zhu}{2021a}{Reza Naserasr, Zhouningxin Wang, \& Xuding Zhu}.
\codes{\CSG:~\CCol}%
\itemspace

\author{Casian Pantea}						%PANTEA
\authorspace
\authref  See \hyperlink{David Angeli}{D.~Angeli} and \hyperlink{Gheorghe Craciun}{G.~Craciun}.
\itemspace

\author{Pietro Panzarasa}						%PANZARASA
\authorspace
\authref  See \hyperlink{Valerio Ciotti}{V.~Ciotti}.
\itemspace

\author{Giovanni Paolini}						%PAOLINI
\authorspace
\authref  See also \hyperlink{Emanuele Delucchi}{E.~Delucchi}.
\authrefspace

\bsgitem{2020a}{}  Shellability of generalized Dowling posets.  \emph{J.\ Combin.\ Theory Ser.\ A} 171 (2020), art.\ 105159, 19 pp. 
\MR{4019110}.  \Zbl{1477.06011}.  \arXiv{1811.08403}.				
\annot{} The marked Dowling posets of \xref{Bibby and Gadish}{2018a}{Christin Bibby \& Nir Gadish} and \xref{Delucchi--Girard--Paolini}{2019a}{Emanuele Delucchi, Noriane Girard, \& Giovanni Paolini}.
	
\codes{\Cgg:~\CMatrd(\CGen)}%
\itemspace

\author{P.\ Paolucci}							%PAOLUCCI
\authorspace
\authref  See \hyperlink{S.\ Cabasino}{S.~Cabasino}.
\itemspace

\author{Gyula Pap}							%PAP
\authorspace
\bsgitem{2005a}{}  Packing non-returning $A$-paths algorithmically.  In:  Stefan Felsner, ed., \emph{2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)} (Berlin, 2005), pp.\ 139--144, electronic.  Discrete Math.\ \& Theor.\ Computer Sci.\ Proceedings AE, 2005.  URL \url{http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/77}.
\Zbl{1192.05123}.  
\annot{} See \xrefd{2008a}{Gyula Pap}.
\codes{\CGG:~\CStr,~\CPaths,~\CAlgor}%
\itemspace

\bsgitem{2007a}{}  Packing non-returning $A$-paths.  \emph{Combinatorica} 27 (2007), no.\ 2, 247--251.
\MR{2321928} (2008c:05148).  \Zbl{1136.05060}.
\annot{} Given: $\Phi$ with gain group $\fS(\Omega)$, the symmetric group of a set $\Omega$, $A \subseteq V$, and $\omega: A \to \Omega$.  An $A$-path is a path $P$ with endpoints $v,w \in A$ and internally disjoint from $A$; it is ``returning'' if $\omega(v)\phi(P) = \omega(w)$.  Thm.  The largest number of disjoint returning $A$-paths equals the minimum, over all satisfied edge subsets $F$, of the maximum number of disjoint $[A \cup V(F)]$-paths in $\|\Phi\| \setminus F$.  [For ``satisfied'' edges see \xref{Zaslavsky}{2009a}{Thomas Zaslavsky}.]
	
Generalizes and simplifies \xref{Chudnovsky, Geelen, \etal{}}{2006a}{Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis Goddyn, Michael Lohman, \& Paul Seymour}, which is the case where the gains act regularly and $\omega=$ constant.  
\par
\codes{\CGG:~\CStr,~\CPaths}%
\itemspace

\bsgitem{2008a}{}  Packing non-returning $A$-paths algorithmically.  \emph{Discrete Math.}\ 308 (2008), no.\ 8, 1472--1488. 
\MR{2392063} (2009e:05160).  \Zbl{1135.05060}.					
\par
\codes{\CGG:~\CStr,~\CPaths,~\CAlgor}%
\itemspace

\author{Christos H.\ Papadimitriou} 					%PAPADIMITRIOU
\authorspace
\authref   See also \hyperlink{Esther M.\ Arkin}{E.M.~Arkin} and \hyperlink{Andrea S.\ LaPaugh}{A.S.~LaPaugh}.
\authrefspace

\author{Christos H.\ Papadimitriou \& Kenneth Steiglitz} 
\authorspace
\bsgitem{1982a}{}  \emph{Combinatorial Optimization:  Algorithms and Complexity}.  Prentice-Hall, Englewood Cliffs, N.J., 1982.  
\MR{0663728} (84k:90036).  \Zbl{503.90060}.
\annot{} Repr.\ with minor additions and corrections:  Dover Publications, Mineola, N.Y., 1998.  
\MR{1637890} (no rev).  \Zbl{944.90066}.
\annot{} See Ch.\ 10, Problems 6--7, p.\ 244, for bidirected graphs and flows in relation to the matching problem.
\codes{\Csg:~\COri:~\CFlows}%
\itemspace

\bsgitem{1985a}{}  \emph{Kombinatornaya optimiztsiya.  Algoritmy i Slozhnost'}.  Transl.\ V.B.~Alekseev.  Mir, Moskva, 1985.
\MR{0801895} (86i:90067).  \Zbl{598.90067}.
\annot{} Russian translation of \xrefd{1982a}{Christos H.\ Papadimitriou \& Kenneth Steiglitz}.
\codes{\Csg:~\COri:~\CFlows}%
\itemspace

\author{Apostolos N.\ Papadopoulos}					%PAPADOPOULOS
\authorspace
\authref  See \hyperlink{Fragkiskos D.\ Malliaros}{F.D.~Malliaros}.
\itemspace

\author{Charis Papadopoulos}						%PAPADOPOULOS
\authorspace
\authref  See \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender}.
\itemspace

\author{Konstantinos Papalamprou}					%PAPALAMPROU
\authorspace
\authref  See also \hyperlink{Gautam Appa}{G.~Appa} and \hyperlink{Leonidas S.\ Pitsoulis}{L.S.~Pitsoulis}.
\authrefspace

\bsgitem{2009a}{}  \emph{Structural and Decomposition Results for Binet Matrices, Bidirected Graphs and Signed-Graphic Matroids.}  Doctoral thesis, London School of Economics, 2009.
\MR{3301606} (no rev).  
\codes{\CSG:~\COri,~\CInc,~\CMatrd}%
\itemspace

\bsgitem{2021a}{}  On the basis pair graphs of signed-graphic matroids.  \emph{Discrete Math.\ Algorithms Appl.}\ 13 (2021), no.\ 4, art.\ 2150033, 5 pp.
\MR{4284036}.  
\codes{\CSG:~\CMatrd:~\CDual,~\CBases}%
\itemspace

\author{Konstantinos Papalamprou \& Leonidas Pitsoulis}
\authorspace
\bsgitem{2012a}{}  On the basis pair graphs of signed-graphic matroids.  In:  \emph{Combinatorial Optimization} (Second Int.\ Sympos., ISCO 2012, Athens), pp.\ 463--474.  \LNCS 7422.  Springer, Heidelberg, 2012.
\MR{3006050}.  \Zbl{06101806}.  
\codes{\CSG:~\CMatrd:~\CAlgor}
\itemspace

\bsgitem{2013a}{}  Decomposition of binary signed-graphic matroids.  \emph{SIAM J.\ Discrete Math.}\ 27 (2013), no.\ 2, 669--692.
\MR{3040957}.  \Zbl{1272.05021}.  \arXiv{1011.6497}.				
\annot{} A binary matroid is signed-graphic iff, for some copoint $H$, all the bridges of $H$ (in the sense of Tutte) are graphic aside from one that is signed-graphic (and possibly graphic).
\dated{5 Feb 2010} 
\par
\codes{\CSG:~\CMatrd,~\CStr}%
\itemspace

\bsgitem{2017a}{}  Signed-graphic matroids with all-graphic cocircuits.  \emph{Discrete Math.}\ 340 (2017), no.\ 12, 2889--2899.
\MR{3698077}.  \Zbl{1370.05032}.  
\annot{} Frame matroids $\bfF(\Sigma)$ where all copoints are graphic matroids.
\par
\codes{\CSG:~\CMatrd}%
\itemspace

\author{Konstantinos Papalamprou, Leonidas S.\ Pitsoulis, \& Eleni-Maria E.\ Vretta}
\authorspace
\bsgitem{2021a}{}  On characterizing the class of cographic signed-graphic matroids.  \emph{Discrete Appl.\ Math.}\ 296 (2021), 90--102.
\MR{4243323}.  \Zbl{1461.05027}.  
\codes{\CSG:~\CMatrd}%
\itemspace

\authorx{G.\ Pardella}						%PARDELLA
\author{G.\ Pardella \& F.\ Liers}
\authorspace
\bsgitem{2008a}{}  Exact ground states of large two-dimensional planar Ising spin glasses.  \emph{Phys.\ Rev.\ E} 78 (2008), art.\ 056705, 10 pp.
\MR{2551393} (2011c:82040).  \arXiv{0801.3143}.  
\annot{} A new algorithm for the frustration index of a planar signed graph.  \Cf\ \xref{Bieche \etal\!}{1980a}{I.\ Bieche, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry} and \xref{Barahona}{1981a}{Francisco Barahona}, \xrefd{1982a}{Francisco Barahona}.  
\dated{23 Nov 2014} 
\codes{\Csg:~\CFr:~\CAlgor}%
\itemspace

\authorx{Eduardo G.\ Pardo}						%PARDO
\author{Eduardo G.\ Pardo, Mauricio Soto, \& Christopher Thraves}
\authorspace
\bsgitem{2015a}{}  Embedding signed graphs in the line: Heuristics to solve MinSA problem.  \emph{J.\ Combin.\ Optim.}\ 29 (2015), 451--471.
\MR{3297933}.  \Zbl{1316.90058}.  
\par
\codes{\CSG:~\CGeom:~\CAlgor}%
\itemspace

\author{Ojas Parekh}							%PAREKH
\authorspace
\authref  See \hyperlink{Erik G.\ Boman}{E.G.~Boman}.
\itemspace

\author{Angelo Parente}						%PARENTE
\authorspace
\authref  See \hyperlink{Fabrizio Marinelli}{F.~Marinelli}.
\itemspace

\author{Giorgio Parisi}						%PARISI
\authorspace
\authref  See also \hyperlink{S.\ Cabasino}{S.~Cabasino}, \hyperlink{Sergio Caracciolo}{S.~Caracciolo}, \hyperlink{Barbara Coluzzi}{B.~Coluzzi}, \hyperlink{M.\ Falcioni}{M.~Falcioni}, \hyperlink{L.A.\ Fernandez}{L.A.~fndez}, \hyperlink{Enzo Marinari}{E.~Marinari}, and \hyperlink{Marc Mezard}{M.~M\'ezard}.
\authrefspace

\bsgitem{1987a}{}  Spin glass theory.  In:  R.\ Pynn and T.\ Riste, eds., \emph{Time Dependent Effects in Disordered Materials}, pp.\ 317--329.  Plenum, 1987.  
Repr.\ in Giorgio Parisi, \emph{Field Theory, Disorder and Simulations}, Ch.\ 14, pp.\ 285--297.  World Scientific, Singapore, 1992. 
\annot{} Relatively (a careful word) simple explanation of physics of ``Ising spin glasses'' = spin states on signed graphs, and of the replica method for studying them.
\dated{9 Aug 2018} 
\codes{\Csg:~\CPhys,~\CFr:~\CExp}%
\itemspace

\bsgitem{1991a}{}  On the emergence of tree-like structures in complex systems.  In:  O.T.\ Solbrig and G.\ Nicolis, eds., \emph{Perspectives on Biological Complexity}, pp.\ 77--111.  Int.\ Union of Biological Sci., 1991.  
Repr.\ in Giorgio Parisi, \emph{Field Theory, Disorder and Simulations}, Ch.\ 15, pp.\ 298--335.  World Scientific, Singapore, 1992. 
\annot{} \S 5.3, ``Spin glasses'':  Friendly treatment of balance and frustration in signed graphs with a distinct physics slant, e.g., asymptotic behaviors.
\dated{9 Aug 2018} 
\codes{\CSG:~\CBal,~\CFr,~\CPhys:~\CExp}%
\itemspace

\bsgitem{1994a}{}  $D$-dimensional arrays of Josephson junctions, spin glasses and $q$-deformed harmonic oscillators.  \emph{J.\ Phys.\ A} 27 (1994), 7555--7568.
\MR{1312271} (95m:82070).  \Zbl{844.60095}.  \arXiv{cond-mat/9410088}.		
\annot{} Physics on hypercube $Q_D$  with complex unit gains $\phi$ ($\phi$ is a ``$U(1)$ gauge field'').  Spins $\zeta(v)$ can be (i) complex units or (ii) Gaussian random complex numbers, or (iii) $\zeta$ can be a unit vector $\in \bbC^n$; mainly, (ii).  
Assumed:  each square (``plaquette'') $C_{\alpha,\beta}$ (with vertices $x,x+e_\alpha,x+e_\alpha+e_\beta,x+e_\beta,x$ for any $x \in V(Q_D)$) has gain $e^{iB\sigma_{\alpha,\beta}}$ in a fixed orientation, where $\sigma_{\alpha,\beta} \in \signnos$ determines which orientations have gains $e^{iB}$ and $e^{-iB}$.  $B=0$ gives balance; $B=\pi$ gives all plaquette gains $-1$ (full frustration).  
If $D \leq 3$, but not if $D > 3$, the choices of $\sigma$ are equivalent by switching in the gain group $\bbC^\times$.  The statistics of random $\sigma$ are investigated.
\dated{19 Jun 2012} 
\codes{\CPhys,~\Cgg}%
\itemspace

\bsgitem{1996a}{}  A mean field theory for arrays of Josephson junctions.  \emph{J.\ Math.\ Phys.}\ 37 (1996), no.\ 10, 5158--5170.
\MR{1411624} (97i:82029).  \Zbl{872.60038}.  
\annot{} Complex unit gain graphs.  The Hamiltonian is the quadratic form $\bar z A(\Phi) z$.
\dated{12 Aug 2012} 
\codes{\CGG:~\CPhys}%
\itemspace

\author{Boram Park}							%PARK B
\authorspace
\authref  See \hyperlink{JiSun Huh}{J.S.~Huh}.
\itemspace

\author{Jin-Woo Park}						%PARK JW
\authorspace
\authref  See also \hyperlink{Sang-Gu Lee}{S.-G.~Lee}.
\authrefspace

\author{Jin-Woo Park \& Sung-Soo Pyo}
\authorspace
\bsgitem{2011a}{}  Construction and nonnegativity of sign idempotent sign pattern matrices.  \emph{Linear Algebra Appl.}\ 435 (2011), 2860--2869.
\MR{2825287} (2012h:15010).  \Zbl{1230.15004}.  
\codes{\CQM:~\CSD,~\Cbal,~\Csw}%
\itemspace

\author{Se-Won Park}						%PARK SW
\authorspace
\authref  See \hyperlink{Sang-Gu Lee}{S.-G.~Lee}.
\itemspace

\authorx{Antonio Parravano}						%PARRAVANO
\author{Antonio Parravano, Ascensi\'on Andina-D\'{\i}az, \& Miguel A.\ Mel\'endez-Jim\'enez}
\authorx{Antonio Parravano, Ascension Andina-Diaz, \& Miguel A.\ Melendez-Jimenez}
\authorspace
\bsgitem{2016a}{}  Bounded confidence under preferential flip:  a coupled dynamics of structural balance and opinions.  \emph{PLoS ONE} 1 (2016), no.\ 10, art.\ 164323, 23 pp.
\arXiv{1610.02660}.  
\codes{\CSG:~\CBal,~\CDyn}%
\itemspace

\author{Leila Parsaei-Majd}						%PARSAEI-MAJD
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari} and \hyperlink{Ebrahim Ghorbani}{E.~Ghorbani}.
\authrefspace

\author{Leila Parsaei-Majd, Zoran Stani\'c, \& Behruz Tayfeh-Rezaie}
\authorx{Leila Parsaei-Majd, Zoran Stanic, \& Behruz Tayfeh-Rezaie}
\authorspace
\bsgitem{20xxa}{}  On weight-symmetric 3-coloured digraphs.  Submitted.  \emph{}\ 
\annot{} ``3-coloured digraph'' = gain graph $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$.  `1Weight-symmetric'' means $\Phi \simeq -\Phi$.  Singularity of $A(\Phi)$.  Constructions of $\Phi \simeq -\Phi$; constructions of $\Phi \not\simeq -\Phi$ with symmetric adjacency spectrum.
\dated{19 Sept 2022} 
\codes{\Cgg:~\CAdj:~\CEig;~\CSw}%
\itemspace

\author{M.\ Parvathi}							%PARVATHI
\authorspace
\bsgitem{2004a}{}  Signed partition algebras.  \emph{Commun.\ Algebra} 32 (2004), no.\ 5, 1865--1880.
\MR{2099708} (2005g:16060).  \Zbl{1081.20008}.					
\annot{} They are the special case of \xref{Bloss}{2003a}{Matthew Bloss} where $\fG = \{+,-\}$.
\dated{21 Mar 2011} 
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& M.\ Kamaraj}
\authorspace
\bsgitem{1998a}{}  Signed Brauer's algebras.  \emph{Commun.\ Algebra} 26 (1998), no.\ 3, 839--855. 
\MR{1606174} (99c:16028).  \Zbl{944.16015}.					
\annot{} The algebra is generated by multiplying two-layer signed graphs (``Brau\-er graphs'').  In the product $\Sigma_1\Sigma_2$ the bottom layer of $\Sigma_1$ cancels with the top layer of $\Sigma_2$ using edge-sign product.  (Signs are represented by arrows [!].)
\dated{5 Jun 2012} 
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2002a}{}  Matrix units for signed Brauer's algebras.  \emph{Southeast Asian Bull.\ Math.}\ 26 (2002), no.\ 2, 279--297. 
\MR{2047807} (2005b:16055).  \Zbl{1066.16014}.					
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& A.\ Joseph Kennedy}
\authorspace
\bsgitem{2004a}{}  $G$-vertex colored partition algebras as centralizer algebras of direct products.  \emph{Commun.\ Algebra} 32 (2004), no.\ 11, 4337--4361.
\MR{2102453} (2005i:16068).  \Zbl{1081.20009}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2004b}{}  Representations of vertex colored partition algebras.  \emph{Southeast Asian Bull.\ Math.}\ 28 (2004), no.\ 3, 493--518.
\MR{2084740} (2006c:16051).  \Zbl{1081.20010}.					
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2005a}{}  Extended $G$-vertex colored partition algebras as centralizer algebras of symmetric groups.  \emph{Algebra Discrete Math.}\ 2005, no.\ 2, 58--79.
\MR{2238218} (2007b:\-16068).  \Zbl{1091.20005}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& D.\ Savithri}
\authorspace
\bsgitem{2002a}{}  Representations of $G$-Brauer algebras.  \emph{Southeast Asian Bull.\ Math.}\ 26 (2002), no.\ 3, 453--468.
\MR{2047837} (2005b:16056).  \Zbl{1065.20017}.					
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& C.\ Selvararj}
\authorspace
\bsgitem{1999a}{}  Signed Brauer's algebras as centralizer algebras.  \emph{Commun.\ Algebra} 27 (1999), no.\ 12, 5985--5998. 
\MR{1726289} (2000j:16051).  \Zbl{944.16016}.					
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2004a}{}  Note on signed Brauer's algebras.  \emph{Southeast Asian Bull.\ Math.}\ 27 (2004), no.\ 5, 883--898.
\MR{2175793} (2006i:16047).  \Zbl{1071.16010}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2006a}{}  Characters of signed Brauer's algebras.  \emph{Southeast Asian Bull.\ Math.}\ 30 (2006), no.\ 3, 495--514.
\MR{2243691} (2007d:16068).  \Zbl{1150.16303}.					
\par
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& B.\ Sivakumar}
\authorspace
\bsgitem{2008a}{}  The Klein-4 diagram algebras.  \emph{J.\ Algebra Appl.}\ 7 (2008), no.\ 2, 231--262.
\MR{2417044} (2009b:16032).  \Zbl{1167.16012}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2008b}{}  R-S correspondence for $(Z_2 \times Z_2) \wr S_n$ and Klein-4 diagram algebras.  \emph{Electronic J.\ Combin.}\ 15 (2008), no.\ 1, Research Paper R98, 28 pp.
\MR{2426161} (2009i:05233).  \Zbl{1163.05300}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi, B.\ Sivakumar, \& A.\ Tamilselvi}
\authorspace
\bsgitem{2007a}{}  R-S correspondence for the hyper-octahedral group of type $B_n$---a different approach.  \emph{Algebra Discrete Math.}\ 2007 (2007), no.\ 1, 86--107.
\MR{2367517} (2008k:\-05203).  \Zbl{1164.05465}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Parvathi \& A.\ Tamilselvi}
\authorspace
\bsgitem{2007a}{}  Robinson-Schensted correspondence for the signed Brauer algebras.  \emph{Electronic J.\  Combin.}\ 14 (2007), no.\ 1, Research Paper 49, 26 pp.
\MR{2336326} (2008e:\-05143).  \Zbl{1163.05336}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{2008a}{}  Robinson-Schensted correspondence for the $G$-Brauer algebras.  
In:  S.K.\ Jain and S.\ Parvathi, eds., \emph{Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications} (Proc.\ Int.\ Conf., Chennai, 2006), pp.\ 137--150.  Contemp.\ Math., Vol.\ 456.  Amer.\ Math.\  Soc., Providence, R.I., 2008.
\MR{2416147} (2009m:16060).  \Zbl{1187.05085}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{M.\ Siva Parvathi}						%PARVATHI
\authorspace
\authref  See \hyperlink{M.\ Venkata Anusha}{M.V.~Anusha}.
\itemspace

\authorx{Germain Past\'en}						%PASTEN
\authorx{Germain Pasten}
\author{Germain Past\'en, Oscar Rojo, \& Luis Medina}
\authorx{Germain Pasten, Oscar Rojo, \& Luis Medina}
\authorspace
\bsgitem{2021a}{}  On the $A_\alpha$-eigenvalues of signed graphs.  \emph{Mathematics} 2021 (2021), no.\ 9, art.\ 1990, 14 pp.
\codes{\CSG:~\CAdj(\CGen):~\CEig}%
\itemspace

\author{Sukanta Pati}						%PATI
\authorspace
\authref  See \hyperlink{R.B.\ Bapat}{R.B.~Bapat}, \hyperlink{S.\ Barik}{S.~Barik}, and \hyperlink{Debajit Kalita}{D.~Kalita}.
\itemspace

\author{Vaia Patta}							%PATTA
\authorspace
\authref  See \hyperlink{Chris Heunen}{C.~Heunen}.
\itemspace

\author{Philippa Pattison}						%PATTISON
\authorspace
\bsgitem{1993a}{}  \emph{Algebraic Models for Social Networks}.  Structural Analysis in the Social Sciences, 7.  Cambridge Univ.\ Press, Cambridge, 1993.  
\annot{} Ch.\ 8, pp.\ 258--9:  ``The balance model.  The complete clustering model.''  They are embedded in a more general framework. 
\par
\codes{\CSG,~\CS:~\CAdj,~\CBal,~\CClu:~\CExp}%
\itemspace

\author{Laura Patuzzi}						%PATUZZI
\authorspace
\authref  See \hyperlink{Maria Aguieiras A.\ de Freitas}{M.A.A.~de~Freitas}.
\itemspace

\author{G.A.\ Patwardhan} 						%PATWARDHAN
\authorspace
\authref  See \hyperlink{B.\ Devadas Acharya}{B.D.~Acharya} and \hyperlink{Mukhtiar Kaur Gill}{\hyperlink{Mukhtiar Kaur Gill}{M.K.~Gill}}.
\itemspace

\author{Debdas Paul}						%PAUL D
\authorspace
\authref  See \hyperlink{Steve Kirkland}{S.~Kirkland}.
\itemspace

\author{Soumyajit Paul}						%PAUL S
\authorspace
\authref  See \hyperlink{Sandip Das}{S.~Das}.
\itemspace

\author{[Viji Paul]}							%PAUL V
\authorspace
\authref  See \hyperlink{Viji Paul}{Viji Paul} (under `V').
\itemspace

\authorx{Lo{\"\i}c Paulev\'e}						%PAULEVE
\authorx{Loic Pauleve}
\author{Lo{\"\i}c Paulev\'e \& Adrien Richard}
\authorx{Loic Pauleve \& Adrien Richard}
\authorspace
\bsgitem{2010a}{}  Topological fixed points in Boolean networks.  Points fixes topologiques dans les r\'eseaux bool\'eens.  \emph{C.\ R.\ Acad.\ Sci.\ Paris, Ser.\ I} 348 (2010), 825--828.
\MR{2677973} (2011g:05305).  \Zbl{1194.94209}.  \HAL{hal-00510892}.  
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2012a}{}  Static analysis of boolean networks based on interaction graphs: A survey.  \emph{Electronic Notes Theor.\ Computer Sci.}\ 284 (2012), 93--104.
\MR{2956839}.  \Zbl{1283.92044}.  \HAL{hal-00714476}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Vern I.\ Paulsen}						%PAULSEN
\authorspace
\authref  See \hyperlink{Bernhard G.\ Bodmann}{B.G.~Bodmann} and \hyperlink{Roderick B.\ Holmes}{R.B.~Holmes}.
\itemspace

\author{Payal [Payal Dabas]}						%PAYAL  DABAS
\authorx{Payal}							%PAYAL
\authorspace
\authref  See also \hyperlink{Sangita Kansal}{S.~Kansal}.
\authrefspace

\author{Payal \& Sangita Kansal}
\authorspace
\bsgitem{2022a}{}  Structural matrices for signed Petri net.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 19 (2022), no.\ 2, 102--107.
\MR{4452559}.  
\codes{\CSG:~\CInc}%

\bsgitem{20xxa}{}  Analysis of signed Petri net.  \emph{Int.\ J.\ Computing Sci.\ Math.}\ (to appear).
\par
\codes{\CSG:~\CInc}%
\itemspace

\bsgitem{20xxb}{}  Domination in signed Petri net.  Submitted.  \emph{}\ 
\arXiv{2001.04374}.  
\codes{\CSG}%
\itemspace

\bsgitem{20xxc}{}  Logic signed Petri net.  Submitted.  \emph{}\ 
\arXiv{2008.11585}.  
\codes{\CSG}%
\itemspace

\author{Charles Payan}						%PAYAN
\authorspace
\bsgitem{1983a}{}  Perfectness and Dilworth number.  \emph{Discrete Math.}\ 44 (1983), no.\ 2, 229--230.
\MR{0689816} (84e:05090).  \Zbl{518.05053}.					
\annot{} See \xref{Benzaken, Hammer, and de Werra}{1985a}{C.\ Benzaken, P.L.\ Hammer, \& D.\ de Werra}.
\codes{\CSGc}%
\itemspace

\author{Edmund R.\ Peay} 						%PEAY
\authorspace
\bsgitem{1977a}{}  Matrix operations and the properties of networks and directed graphs.  \emph{J.\ Math.\ Psychology} 15 (1977), 89--101.  
\MR{0444333} (56 \#2690).  
\codes{\CSD,~\CWD:~\CAdj:~\CGen}%
\itemspace

\bsgitem{1977b}{}  Indices for consistency in qualitative and quantitative structures.  \emph{Human Relations} 30 (1977), 343--361.
\annot{} Proposes an index of inclusterability for signed graphs and generalizes to edges weighted by a linearly ordered set. 
\codes{\CSG,~\CGen:~\CClu:~\CFr(\CGen)}%
\itemspace

\bsgitem{1980a}{}  Connectedness in a general model for valued networks.  \emph{Social Networks} 2 (1980), 385--410.
\MR{0602317} (82h:92053) (\qv).  
\annot{} Real-number edge weights; the value of a path is the minimum absolute weight.	
	
\dated{11 Sept 2010} 
\codes{\CWG}%
\itemspace

\bsgitem{1982a}{}  Structural models with qualitative values.  \emph{J.\ Math.\ Sociology} 8 (1982), 161--192.  
\MR{0655909} (83d:92107).  \Zbl{486.05060}.
\annot{} See mainly \S3:  ``Structural consistency.'' 					
\codes{\Csd:~\CGen:~\CBal,~\CClu}%
\itemspace

\author{Luke Pebody}						%PEBODY
\authorspace
\authref  See \hyperlink{Bela Bollobas}{B.~Bollob\'as}.
\itemspace

\author{Timoth\'ee Pecatte}						%PECATTE
\authorspace
\authref  See \hyperlink{Lucas Isenmann}{L.~Isenmann}.
\itemspace

\author{Elisabeth P\'ecou}						%PECOU
\authorx{Elisabeth Pecou}
\authorspace
\authref  See \hyperlink{Mirela Domijan}{M.~Domijan}.
\itemspace

\author{Yan Ru Pei}							%PEI YR
\authorspace
\authref  See also \hyperlink{Haik Manukian}{H.~Manukian}.
\authrefspace

\author{Yan Ru Pei \& Massimiliano Di Ventra}
\authorspace
\bsgitem{20xxa}{}  A finite-temperature phase transition for the Ising spin-glass in $d\geq2$.  Submitted. \emph{}\ 
\arXiv{2105.01188}.  
\codes{\CSG:~\CFr:~\CPhys,~\CState}%
\itemspace

\author{Yan Ru Pei, Haik Manukian, \& Massimiliano Di Ventra}
\authorspace
\bsgitem{2020a}{}  Generating weighted MAX-2-SAT instances with frustrated loops: an RBM case study.  \emph{J.\ Machine Learning Res.}\ 21 (2020), 1--55.
\MR{4209445}.  
\arXiv{1905.05334}.  
\annot{} Weighted MAX-2-SAT means weighted frustration index $l(\Sigma,w) := \min\{ w(D) : D \subseteq E,\ \Sigma \setm D \text{  balanced}\}$ for given weights $w: E \to \bbR$.
\par
\codes{\CSG:~\CWG:~\CFr:~\CPhys,~\CState,~\CAlgor}%
\itemspace

\author{Britta Peis}							%PEIS
\authorspace
\authref  See \hyperlink{Winfried Hochstattler}{W.~Hochst\"attler} and \hyperlink{Martin Latsch}{M.~L\"atsch}.
\itemspace

\author{David B.\ Peizer}						%PEIZER
\authorspace
\authref  See \hyperlink{Philip J.\ Runkel}{P.J.~Runkel}.
\itemspace

\author{Uri N.\ Peled} 						%PELED*
\authorspace
\authref   See \hyperlink{Srinivasa R.\ Arikati}{S.R.~Arikati}, \hyperlink{Amitava Bhattacharya}{A.~Bhattacharya}, \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}, \hyperlink{Toshihide Ibaraki}{T.~Ibaraki}, and \hyperlink{N.V.R.\ Mahadev}{N.V.R.~Mahadev}.
\itemspace

\authorx{Martin Pelikan}						%PELIKAN
\author{Martin Pelikan \& Alexander K.\ Hartmann}
\authorspace
\bsgitem{2007a}{}  Obtaining ground states of Ising spin glasses via optimizing bonds instead of spins.  (Extended abstract.)  In:  \emph{GECCO '07: Genetic and Evolutionary Computation Conference} (GECCO 2007, London), p.\ 628.  ACM, New York, 2007.
\annot{} Announcement of \xrefd{2007b}{Martin Pelikan \& Alexander K.\ Hartmann}.
\codes{\CSG,~\CPhys:~\CState(\Cfr):~\CAlgor}%
\itemspace

\bsgitem{2007b}{}  Obtaining ground states of Ising spin glasses via optimizing bonds instead of spins.  Report, Missouri Estimation of Distribution Algorithms Laboratory, Dept.\ of Mathematics and Computer Science, University of Missouri--St.\ Louis, 2007.  URL \url{http://medal-cs.umsl.edu/}
\codes{\CSG,~\CPhys:~\CState(\Cfr):~\CAlgor}%
\itemspace

\author{Marcello Pelillo}						%PELILLO
\authorspace
\authref  See \hyperlink{Roland Glantz}{R.~Glantz}.
\itemspace

\authorx{R.A.\ Pendavingh}						%PENDAVINGH
\author{R.A.\ Pendavingh \& S.H.M.\ van Zwam}
\authorspace
\bsgitem{2010a}{}  Confinement of matroid representations to subsets of partial fields.  \emph{J.\ Combin.\ Theory Ser.\ B} 100 (2010), 510--545.
\MR{2718674} (2011i:05043).  \Zbl{1231.05062}.  \arXiv{0806.4487}.			
\annot{} \xref{Dowling's}{1973b}{Thomas A.\ Dowling}, \xrefd{1973a}{Thomas A.\ Dowling} $Q_n(\GF(q)^\times)$ is an example.
\dated{1 Sept 2017} 
\codes{\CMatrd:~\Cgg}%
\itemspace

\bsgitem{2013a}{}  Skew partial fields, multilinear representations of matroids, and a matrix tree theorem.  \emph{Adv.\ Appl.\ Math.}\ 50 (2013), 201--227.
\MR{2996392}.  \Zbl{1256.05047}.  
\annot{} Introduces representation of Dowling geometries $Q_n(\fG)$ over a skew partial field (\cf\ \xref{van Zwam}{2009a}{Stefan H.M.\ van Zwam}, \S3.2).  [Continued in \xref{Vertigan}{2015a}{Dirk Vertigan}.]
\dated{28 Jan 2015} 
\codes{\CMatrd:~\Cgg:~\CInc}%
\itemspace

\author{Irena Penev}							%PENEV
\authorspace
\authref  See \hyperlink{Valerio Boncompagni}{V.~Boncompagni}.
\itemspace

\authorx{Di Peng}							%PENG D
\author{Di Peng, Xiangbai Gu, Yuan Xu, \& Qunxiong Zhu}
\authorspace
\bsgitem{2015a}{}  Integrating probabilistic signed digraph and reliability analysis for alarm signal optimization in chemical plant.  \emph{J.\ Loss Prevention Process Industries} 33 (2015), 279--288.
\codes{\CSD:~\CRand,~\CAppl}%
\itemspace

\author{Yuejian Peng}						%PENG YJ
\authorspace
\authref  See \hyperlink{Yongtao Li}{Y.-T.~Li}.
\itemspace

\author{Francisco Pereira} 						%PEREIRA
\authorspace
\authref   See \hyperlink{Alan J.\ Hoffman}{A.J.~Hoffman}.
\itemspace

\author{Mercedes P\'erez Mill\'an}					%PEREZ-MILLAN
\authorx{Mercedes Perez Millan}
\authorspace
\authref  See \hyperlink{Alicia Dickenstein}{A.~Dickenstein}.
\itemspace

\authorx{C.\ Perina}							%PERINA
\author{C.\ Perina, N.\ Buckley, \& A.K.\ Nagar}
\authorspace
\bsgitem{2017a}{}  Application of metaheuristics algorithms and signed graphs to portfolio turnover management.  \emph{Int.\ J.\ Innovation Management Tech.}\ 8 (2017), no.\ 2, 161--165.
\annot{} Same error in balance testing as in \xref{Vasanthi \etal{}}{2015a}{V.\ Vasanthi, S.\ Arumugam, Atulya K.\ Nagar, \& Sovan Mitra}.  Confusing exposition.  Genetic, simulated annealing, and ant-colony algorithms with example.
\codes{\CAppl:~\CSG:~\CBal:~\CAlgor}%
\itemspace

\author{Kavita S.\ Permi}						%PERMI
\authorspace
\authref  See \hyperlink{Madhwesha Moudgalya R}{M.~Moudgalya~R} and \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{F.\ Peruggi}							%PERUGGI
\authorspace
\authref  See \hyperlink{Antonio Coniglio}{A.~Coniglio}.
\itemspace

\author{Pawe{\l} Petecki}						%PETECKI
\authorx{Pawel Petecki}
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo}.
\itemspace

\author{M.\ Petersdorf} 						%PETERSDORF
\authorspace
\bsgitem{1966a}{}  Einige Bemerkungen \"uber vollst\"andige Bigraphen.  \emph{Wiss.\ Z.\ Techn.\ Hochsch.\ Ilmenau} 12 (1966), 257--260.  
\MR{0225682} (37 \#1275).  \Zbl{156.44302}. 
\annot{} Treats signed $K_n$'s.  Satz 1:  $\max_\sigma l(K_n,\sigma) = \lfloor (n  - 1)^2/4 \rfloor$ with equality iff $(K_n,\sigma)$ is antibalanced.  [From which follows easily the full Thm.\ 14 of \xref{Abelson and Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg}.]  Also, some further discussion of antibalanced and unbalanced cases.
[For extensions of this problem see notes on \xref{Erd\H{o}s, Gy\H{o}ri, \& Simonovits}{1992a}{P.\ Erdos, E.\ Gyori, \& M.\ Simonovits}.]
\codes{\CSG:~\CFr}%
\itemspace

\author{Ion Petre}							%PETRE
\authorspace
\authref  See \hyperlink{Artiom Alhazov}{A.~Alhazov} and \hyperlink{Tero Harju}{T.~Harju}.
\itemspace

\author{Rossella Petreschi}						%PETRESCHI
\authorspace
\authref  See also \hyperlink{Tiziana Calamoneri}{T.~Calamoneri}.
\authrefspace

\author{Rossella Petreschi \& Andrea Sterbini}
\authorspace
\bsgitem{1995a}{}  Recognizing strict 2-threshold graphs in $O(m)$ time.  \emph{Inform.\ Processing Lett.}\ 54 (1995), no.\ 4, 193--198.  
\MR{1337823} (96i:68034a).  \Zbl{0875.68453}.  
Erratum.  \emph{Ibid.}\ 56 (1995), no.\ 1, 65.
\MR{1361260} (96i:68034b).  
\annot{} Uses the auxiliary signed graph of \xref{Mahadev and Peled}{1988a}{N.V.R.\ Mahadev \& U.N.\ Peled}.
\dated{22 Mar 2017} 
\codes{\CSG:~\CAppl:~\CBal,~\CAlgor}%
\itemspace

\author{[Rossella Petreschim \& Andrea Sterbini]}				%PETRESCHIM
\authorspace
\authref  Misprint for \hyperlink{Rossella Petreschi \& Andrea Sterbini}{R.~Petreschi \& A.~Sterbini}.
\itemspace

\author{Norbert Peyerimhoff}						%PEYERIMHOFF
\authorspace
\authref  See \hyperlink{Carsten Lange}{C.~Lange} and \hyperlink{Shiping Liu}{S.-P.~Liu}.
\itemspace

\author{Nathan Pflueger}						%PFLUEGER
\authorspace
\bsgitem{2011a}{}  Graph reductions, binary rank, and pivots in gene assembly.  \emph{ Discrete Appl.\ Math.}\ 159 (2011), no.\ 17, 2117--2134.
\MR{2832336} (2012j:05421). \\ \Zbl{1237.05181}.  \arXiv{1103.4334}.			
\codes{\CSG:~\CAlgor,~\CAppl}%
\itemspace

\author{Lan Anh Pham}						%PHAM
\authorspace
\authref  See \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\authorx{Geevarghese Philip}						%PHILIP
\author{Geevarghese Philip, Ashutosh Rai, \& Saket Saurabh}
\authorspace
\bsgitem{2015a}{}  Generalized pseudoforest deletion: Algorithms and uniform kernel.  In:  Giuseppe F.\ Italiano \etal, eds., \emph{Mathematical Foundations of Computer Science 2015} (40th Int.\ Symp., MFCS 2015, Milan, 2015), Part II, pp.\ 517--528.  \LNCS 9235.  Springer, Berlin, 2015.
\MR{3419512}.  \Zbl{1400.05242}.  
\annot{} ``Pseudoforest'' [or $1$-forest] = independent set in the bicircular matroid $\bfF(\Gamma,\eset)$.  Problem: Can $\Gamma\ \setm$ ($\leq k$ vertices) be a pseudoforest?  Generally, ``$l$-pseudoforest'' = forest + $l$ edges.  Problem: Can $\Gamma\ \setm$ ($\leq k$ vertices) be an $l$-pseudoforest?
\dated{22 Dec 2017} 
\codes{\Cbic:~\CAlgor}%
\itemspace

\bsgitem{2018a}{}  Generalized pseudoforest deletion: Algorithms and uniform kernel.  \emph{SIAM J.\ Discrete Math.}\  32 (2018), no.\ 2, 882--901.
\MR{3787778}.  \Zbl{1384.05148}.  
\annot{} Extended version of \xrefd{2015a}{Geevarghese Philip, Ashutosh Rai, \& Saket Saurabh}.
\codes{\Cbic:~\CAlgor}%
\itemspace

\author{J.L.\ Phillips}							%PHILLIPS JL
\authorspace
\bsgitem{1967a}{}  A model for cognitive balance.  \emph{Psychological Rev.}\ 74 (1967), 481--495.  
\annot{} Proposes to measure imbalance of a signed (di)graph by largest eigenvalue of a matrix close to $I + A(\Sigma)$.  (\Cf\ Abelson 1967a.)  Possibly, means to treat only graphs that are complete aside from isolated vertices.  [Somewhat imprecise.]  Summary of Ph.D.\ thesis.
\codes{\CSG:~\CBal,~\CFr,~\CAdj}%
\itemspace

\author{Nancy V.\ Phillips}						%PHILLIPS NV
\authorspace
\authref   See \hyperlink{Fred Glover}{F.~Glover}.
\itemspace

\author{[Alberto Del Pia]}						%PIA
\authorspace
\authref  See \hyperlink{Alberto Del Pia}{A.~Del Pia}.
\itemspace

\authorx{Jean-Claude Picard}						%PICARD
\author{Jean-Claude Picard \& H.\ Donald Ratliff}
\authorspace
\bsgitem{1973a}{}  A graph-theoretic equivalence for integer programs.  \emph{Operations Res.}\ 21 (1973), 261--269.  
\MR{0359788} (50 \#12240).  \Zbl{263.90021}.  
\annot{} A minor application of signed switching to a weighted graph arising from an integer linear program.  
\codes{\Csg:~\Csw}%
\itemspace

\author{Th\'eo Pierron}						%PIERRON
\authorspace
\authref  See \hyperlink{Julien Bensmail}{J.~Bensmail}, \hyperlink{Francois Dross}{F.~Dross}, and \hyperlink{Florent Foucaud}{F.~Foucaud}.
\itemspace

\author{Oleg Pikhurko}						%PIKHURKO
\authorspace
\bsgitem{2001a}{}  Uniform families and count matroids.  \emph{Graphs Combin.}\ 17 (2001), 729--740.
\MR{1876580} (2003c:05120).  \Zbl{987.05060}.  
\annot{} Generalizes graphic count matroids (\xref{White and Whiteley}{1983a}{Neil White \& Walter Whiteley}, \xref{Whiteley}{1996a}{Walter Whiteley}) to $r$-hypergraphs and more general linear functions as rank bounds.
\dated{27 Jul 2022} 
\codes{\CMatrd:~\CBic:~\CGen}%
\itemspace

\author{Vincent Pilaud}						%PILAUD
\authorspace
\bsgitem{2013a}{}  Signed tree associahedra.  Manuscript, 2013, 50 pp.
\arXiv{1309.5222}.  
\annot{} A signed tree is a tree with vertices in $\pm[n]$.  Nesting structure of induced subgraphs, etc.
\dated{7 Feb 2022} 
\codes{\CVS}%
\itemspace

\bsgitem{2014a}{}  Signed tree associahedra.  In:  \emph{26th International Conference on Formal Power Series and Algebraic Combinatorics} (FPSAC 2014, Chicago), pp.\ 309--320.  DMTCS Proc., Vol.\ AT.  URL \url{https://doi.org/10.46298/dmtcs.2402}.
\MR{3466354} (book).  \Zbl{1394.52012}.  \HAL{hal-01207614}.  
\annot{} Condensed version of \xrefd{2013a}{Vincent Pilaud}.
\dated{7 Feb 2022} 
\codes{\CVS}%
\itemspace

\author{Marcin Pilipczuk}						%PILIPCZUK
\authorspace
\authref  See \hyperlink{Marek Cygan}{M.~Cygan}.
\itemspace

\author{Micha{\l} Pilipczuk}						%PILIPCZUK
\authorx{Michal Pilipczuk}
\authorspace
\authref  See \hyperlink{Marek Cygan}{M.~Cygan}.
\itemspace

\author{P.\ Pincus} 							%PINCUS
\authorspace
\authref   See \hyperlink{S.\ Alexander}{S.~Alexander}.
\itemspace

\author{Alexandre Pinlou}						%PINLOU
\authorspace
\authref  See also \hyperlink{Christopher Duffy}{C.~Duffy}, \hyperlink{Fabien Jacques}{F.~Jacques}, \hyperlink{Amanda Montejano}{A.~Montejano}, and \hyperlink{Pascal Ochem}{P.~Ochem}.
\authrefspace

\bsgitem{2016a}{}  \emph{Some Problems in Graph Coloring: Methods, Extensions and Results}.  Habilitation \'a Diriger des Recherches, Univ.\ Montpellier, 2016.
\HAL{tel-01376199}.  
\annot{} \S2.2, ``Homomorphisms of signed graphs''.
\codes{\CSG:~\CCol,~\CHom}%
\itemspace

\author{Shariefuddin Pirzada}						%PIRZADA
\authorx{S.\ Pirzada}
\authorspace
\authref  See also \hyperlink{Mushtaq A.\ Bhat}{M.A.~Bhat} and \hyperlink{Tahir Shamsher}{T.~Shamsher}.
\authrefspace

\bsgitem{2012a}{}  Signed degree sequences in signed graphs.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 179--204.
\Zbl{1300.05068}.  
\annot{} 	
Dictionary:  ``Signed degree'' = net degree $d^\pm(v)$ [\cf\ \xref{Chartrand, Gavlas, Harary, and Schultz}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz}].
\codes{\CSG:~\Cori:~\CInvar:~\CExp}\codesctd{\CSG:~\Cori:~\CInvar}%
\itemspace

\author{S.\ Pirzada \& Mushtaq A.\ Bhat}	
\authorspace
\bsgitem{2014a}{}  Energy of signed digraphs.  \emph{Discrete Appl.\ Math.}\ 169 (2014), 195--205.
\MR{3175069}.  \Zbl{1288.05166}.  \arXiv{1309.6266}.				
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\author{S.\ Pirzada \& F.A.\ Dar}
\authorspace
\bsgitem{2007a}{}  Signed degree sets in signed 3-partite graphs.  \emph{Mat.\ Vesnik} 59 (2007), no.\ 3, 121--124.
\MR{2361920} (2008k:05095).  \Zbl{1224.05222}.  
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\bsgitem{2007b}{}  Signed degree sequences in signed 3-partite graphs.  \emph{J.\ Korean Soc.\ Ind.\ Appl.\ Math.}\ 11 (2007), no.\ 1, 9--14.
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\author{S.\ Pirzada, Muhammad Ali Khan, \& E.\ Sampathkumar}
\authorspace
\bsgitem{2013a}{}  Coloring of signed graphs.  In: Kamiel Cornelissen, Ruben Hoeksma, \etal, eds., \emph{Proceedings of the 12th Cologne-Twente Workshop on Graphs and Combinatorial Optimization} (CTW-2013, Enschede, 2013), pp.\ 191--195.  Centre for Telematics and Information Technology, Univ.\ of Twente, Enschede, Neth., 2013.
\annot{} $\sigma$ bipartitions $V(\Gamma)$ as $V_\text{odd} \cup V_\text{even}$ by parity of $d^-(v)$.  Coloring = properly coloring $\Gamma_c := \Gamma{:}V_\text{odd} \cup \Gamma{:}V_\text{even}$.  [Why not an arbitrary bipartition?  Signs seem superfluous.]  Results: chromatic polynomial [the usual one of $\Gamma_c$], examples like paths, circles.
\dated{1 Apr 2019} 
\codes{\CSGc:~\CCol}%
\itemspace

\author{S.\ Pirzada, T.A.\ Naikoo, \& F.A.\ Dar}
\authorspace
\bsgitem{2007a}{}  Signed degree sets in signed graphs.  \emph{Czechoslovak Math.\ J.}\ 57 (2007), no.\ 3, 843--848.
\MR{2356284} (2008g:05088).  \Zbl{1174.05059}.  \arXiv{math/0609121}.			
\annot{} The set, as opposed to sequence, of net degrees [\cf\ \xref{Chartrand, Gavlas, Harary, and Schultz}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz}] of a signed simple graph can be any finite set of integers.  Also, the smallest order of a signed graph with given net degree set. 
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\bsgitem{2007b}{}  Signed degree sequences in signed bipartite graphs.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 4 (2007), no.\ 3, 301--312.
\MR{2384886} (no rev).  \Zbl{1143.05307}.  \arXiv{math/0609122}.			
\annot{} Characterization of net degree sequences of signed, simple, bipartite graphs.
\dated{15 Nov 2011} 
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\bsgitem{2008a}{}  A note on signed degree sets in signed bipartite graphs.  \emph{Appl.\ Anal.\ Discrete Math.}\ 2 (2008), no.\ 1, 114--117.
\MR{2396733} (2009a:05092).  \Zbl{1199.05159}.  \arXiv{math/0609122}.  
\annot{} Every finite set of integers is the net degree set of a connected signed bipartite graph.
\dated{10 Sept 2010} 
\codes{\CSG:~\Cori:~\CInvar}%
\itemspace

\author{S.\ Pirzada, Tahir Shamsher, \& Mushtaq A.\ Bhat}
\authorspace
\bsgitem{20xxa}{}  On ordering of minimal energies in bicyclic signed graphs.  Submitted.  \emph{}\ 
\annot{} Among connected signed graphs with cyclomatic number 2 and order $n$, the 20 with least energy for $n \geq 30$ and the 16 for $n\geq17$, via extensive computations.
\dated{3 Feb 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{20xxb}{}  On the eigenvalues of signed complete bipartite graphs.  Submitted.  \emph{}\ 
\arXiv{2111.07262}.  
\annot{} Interesting relations between $\Sigma^-$ structure and eigenvalues.	
\dated{23 Jan 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Toma\v{z} Pisanski}				 		%PISANSKI
\authorx{Tomaz Pisanski}
\authorspace
\authref  See also \hyperlink{Nino Basic}{N.~Basic}, \hyperlink{V.\ Batagelj \& T.\ Pisanski}{V.~Batagelj}, and \hyperlink{Francesco Belardo}{F.~Belardo}.
\authrefspace

\author{Toma\v{z} Pisanski \& Primo\v{z} Poto\v{c}nik}
\authorx{Tomaz Pisanski \& Primoz Potocnik}
\authorspace
\bsgitem{2004a}{}  Graphs on surfaces.
In: Jonathan L.\ Gross and Jay Yellen, eds., \emph{Handbook of Graph Theory}, pp.\ 611--624. 
Discrete Math.\ Appl.\ (Boca Raton).
CRC Press, Boca Raton, Fla., 2004.
\MR{2035186} (2004j:05001) (book).  \Zbl{1036.05001} (book).
\annot{} Cryptic.  Dictionary (my best guess): ``signed edge'' = oriented edge; ``signed boundary walk'' (of a face) = directed face boundary walk; ``signature'' = set of negative edges of an embedding; ``switch'' = negative (= orientation-reversing) edge of an embedding.
\codes{\Csg:~\CTop}%
\itemspace

\author{Toma\v{z} Pisanski \& Jo\v{z}e Vrabec}
\authorx{Tomaz Pisanski \& Joze Vrabec}
\authorspace
\bsgitem{1982a}{}  Graph bundles.  Preprint Ser., Dept.\ Math., University of Ljubljana, 1982.
\annot{} Definition (see \xref{Pisanski, Shawe-Taylor, and Vrabec}{1983a}{Tomaz Pisanski, John Shawe-Taylor, \& Joze Vrabec}), examples, superimposed structure, classification.
\codes{\CGG:~\CCov(\CGen)}%
\itemspace

\author{Toma\v{z} Pisanski, John Shawe-Taylor, \& Jo\v{z}e Vrabec}
\authorx{Tomaz Pisanski, John Shawe-Taylor, \& Joze Vrabec}
\authorspace
\bsgitem{1983a}{}  Edge-colorability of graph bundles.  \emph{J.\ Combin.\ Theory Ser.\ B} 35 (1983), 12--19.
\MR{0723566} (85b:05086).  \Zbl{505.05034}, (\Zbl{515.05031}).
\annot{} A graph bundle is, roughly, a covering graph with an arbitrary graph $F_v$ (the ``fibre'') over each vertex $v$, so that the edges covering $e{:}vw$ induce an isomorphism $F_v \to F_w$.
\codes{\CGG:~\CCov(\CGen):~\CECol}%
\itemspace

\author{Toma\v{z} Pisanski \& Arjana \v{Z}itnik}
\authorx{Tomaz Pisanski \& Arjana Zitnik}
\authorspace
\bsgitem{2009a}{}  Representing graphs and maps.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 8, pp.\ 151--180.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581545} (no rev).  \Zbl{1225.05185}.  
\annot{} Signed graphs are implicit in pp.\ 170--171.  Dictionary:  ``flat'' = $+$, ``twisted'' = $-$.
\dated{26 Aug 2108} 
\codes{\Csg:~\CTop:~\CExp}%
\itemspace

\author{Leonidas S.\ Pitsoulis}						%PITSOULIS
\authorspace
\authref  See also \hyperlink{Gautam Appa}{G.~Appa} and \hyperlink{Konstantinos Papalamprou}{K.~Papalamprou}.
\authrefspace

\bsgitem{2014a}{}  \emph{Topics in Matroid Theory}.  SpringerBriefs in Optimization.  Springer, New York, 2014.
\MR{3154793}.  \Zbl{1319.05033}.  
\annot{} Ch.\ 6, ``Signed-graphic matroids'':  Signed graphs; signed-graphic matroids; binary signed-graphic matroids; decomposition.
\codes{\CSG:~\CMatrd}
\itemspace

\author{Leonidas Pitsoulis, Konstantinos Papalamprou, Gautam Appa, \& Bal\'azs Kotnyek}
\authorx{Leonidas Pitsoulis, Konstantinos Papalamprou, Gautam Appa, \& Balazs Kotnyek}
\authorspace
\bsgitem{2009a}{}  On the representability of totally unimodular matrices on bidirected graphs.  \emph{Discrete Math.}\ 309 (2009), no.\ 16, 5024--5042.
\MR{2548904} (2010m:05182).  \Zbl{1182.05120}.					
\annot{} Tour matrices of bidirected graphs are closed under 1-, 2-, and 3-sums.   Possibly, every totally unimodular matrix is a tour matrix.
\par
\codes{\COri:~\CInc(\CGen)}%
\itemspace

\author{Leonidas Pitsoulis \& Eleni-Maria Vretta}
\authorspace
\bsgitem{2015a}{}  Decomposition of quaternary signed-graphic matroids.  Manuscript, 2015.
\arXiv{1510.06727}.  
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\author{Irene Pivotto}						%PIVOTTO
\authorspace
\authref  See also \hyperlink{Rong Chen}{R.~Chen}, \hyperlink{Matt DeVos}{M.~DeVos}, \hyperlink{Daryl Funk}{D.~Funk}, and \hyperlink{Bertrand Guenin}{B.~Guenin}.
\authrefspace

\bsgitem{2006a}{}  \emph{On Excluded Minors for Even Cut Matroids}.  Master's thesis, Univ.\ of Waterloo, 2006.  URL \url{http://hdl.handle.net/10012/2680}
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{2011a}{}  \emph{Even Cycle and Even Cut Matroids}.  Doctoral dissertation, Univ.\ of Waterloo, 2011.  URL \url{http://hdl.handle.net/10012/5956}
\codes{\CSG:~\CMatrd}%
\itemspace

\bsgitem{2013a}{}  Biased graphs and their matroids I, II, III.  \emph{The Matroid Union}, 2013, \url{http://matroidunion.org/?p=161}, \url{http://matroidunion.org/?p=279}, \url{http://matroidunion.org/?p=357}
\codes{\CGG:~\CMatrd:~\CExp}%
\itemspace

\authorx{Erik Plahte}							%PLAHTE
\author{Erik Plahte, Thomas Mestl, \& Stig W.\ Omholt}
\authorspace
\bsgitem{1995a}{}  Feedback loops, stability and multistationarity in dynamical systems.  \emph{J.\ Biol.\ Systems} 3 (1995), no.\ 2, 409--413.
\codes{\Csd:~\CQM:~\CDyn}%
\itemspace

\author{Michael Plantholt} 						%PLANTHOLT
\authorspace
\authref   See \hyperlink{Arthur H.\ Busch}{A.H.~Busch}, \hyperlink{Ajit A.\ Diwan}{A.A.~Diwan}, and \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{Andrey Ploskonosov}						%PLOSKONOSOV
\authorspace
\authref  See \hyperlink{Yurii Burman}{Y.~Burman}.
\itemspace

\author{M.D.\ Plummer}						%PLUMMER
\authorspace
\authref  See \hyperlink{Laszlo Lovasz}{L.~Lov\'asz}.
\itemspace 

\author{Jorn van der Pol}						%POL
\authorspace
\authref  See \hyperlink{Peter Nelson}{P.~Nelson}.
\itemspace

\authorx{Agnieszka Polak}						%POLAK
\author{Agnieszka Polak \& Daniel Simson}
\authorspace
\bsgitem{2013a}{}  Algorithms computing $O(n,\bbZ)$-orbits of $P$-critical edge-bipartite graphs and $P$-critical unit forms using Maple and C\#.  \emph{Algebra Discrete Math.}\ 16 (2013), no.\ 2, 242--286.
\MR{3186088}.  \Zbl{1310.05218}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2013b}{}  On Coxeter spectral classification of $P$-critical edge-bipartite graphs of Euclidean type $\tilde \bbA_n$.  Combinatorics 2012 (Perugia, 2012).  \emph{Electronic Notes in Discrete Math.}\ 40 (2013), 311--316.
\MR{3155275} (volume).  \Zbl{1292.05005} (volume).  
\codes{\CSG}%
\itemspace

\bsgitem{2013c}{}  Algorithmic experiences in Coxeter spectral study of $P$-critical edge-bipartite graphs and posets.  
In:  Nikolaj Bj\"orner \etal, eds., \emph{15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing} (SYNASC 2013, Timisoara, Romania, 2013), pp.\ 375--382.  IEEE, 2013.
\codes{\CSG}%
\itemspace

\bsgitem{2014a}{}  Coxeter spectral classification of almost $TP$-critical one-peak posets using symbolic and numeric computations.  \emph{Linear Algebra Appl.}\ 445 (2014), 223--255.
\MR{3151272}.  \Zbl{1290.16014}.  
\codes{\CSG}%
\itemspace

\author{Oskar E.\ Polansky}						%POLANSKY
\authorspace
\authref  See \hyperlink{Ivan Gutman}{I.~Gutman}.
\itemspace

\author{Svatopluk Poljak}						%POLJAK*
\authorspace
\authref   See also \hyperlink{Yves Crama}{Y.~Crama} and \hyperlink{Bojan Mohar}{B.~Mohar}. 
\authrefspace

\author{Svatopluk Poljak \& Daniel Turz\'{\i}k} 
\authorx{Svatopluk Poljak \& Daniel Turzik} 
\authorspace
\bsgitem{1982a}{}  A polynomial algorithm for constructing a large bipartite subgraph, with an application to a satisfiability problem.
\emph{Canad.\ J.\ Math.}\ 34 (1982), 519--524.  
\MR{0663301} (83j:05048).  \Zbl{471.68041}, (\Zbl{487.68058}).			
\annot{} Main Theorem: For a simple, connected signed graph of order $n$ and size $\#E = m$, the frustration index $l(\Sigma) \leq \frac 12 m - \frac 14(n-1)$.  The proof is algorithmic, by constructing a (relatively) small deletion set.  
Dictionary:  $\Sigma$ is an ``edge-2-colored graph'' $(G, c)$, $E^+$ and $E^-$ are called $E_1$ and $E_2$, a balanced subgraph is ``generalized bipartite'', and $m - l(\Sigma)$ is what is calculated.  
[This gives an upper bound on $D(\Gamma) := \max_\sigma l(\Gamma,\sigma)$ for a connected, simple graph, whereas \xref{Akiyama, Avis, Chv\'atal, and Era}{1981a}{J.\ Akiyama, D.\ Avis, V.\ Chvatal, \& H.\ Era} has a lower bound on $D$.]
\codes{\CSG:~\CFr,~\CAlgor}%
\itemspace

\bsgitem{1986a}{}  A polynomial time heuristic for certain subgraph optimization problems with guaranteed worst case bound.  
\emph{Discrete Math.}\ 58 (1986), 99--104.
\MR{0820844} (87h:68131).  \Zbl{585.05032}. 					
\annot{} Generalizes \xrefd{1982a}{Svatopluk Poljak \& Daniel Turzik}, with application to signed graphs in Cor.\ 3.
\par
\codes{\CSG:~\CFr,~\CAlgor}%
\itemspace

\bsgitem{1987a}{}  On a facet of the balanced subgraph polytope.  \emph{\v{C}asopis P\v{e}st.\ Mat.}\ 112 (1987), 373--380.  
\MR{0921327} (89f:05155).  \Zbl{643.05059}.					
\annot{} The polytope $P_B(\Sigma)$ (the authors write $P_{BL}$) is the convex hull in $\bbR^E$ of characteristic vectors of balanced edge sets.  It generalizes the bipartite subgraph polytope $P_B(\Gamma) = P_B(-\Gamma)$ (see \xref{Barahona, Gr\"otschel, and Mahjoub}{1985a}{Francisco Barahona, Martin Grotschel, \& Ali Ridha Mahjoub}), but is essentially equivalent to it according to Prop.\ 2: The negative-subdivision trick preserves facets of the polytope.  Thm.\ 1 gives new facets, corresponding to certain circulant subgraphs.  (They are certain unions of two Hamilton circles, each having constant sign.)
\codes{\CSG:~\CFr,~\CGeom}%
\itemspace

\bsgitem{1992a}{}  Max-cut in circulant graphs.  \emph{Discrete Math.}\ 108 (1992), 379--392.
\MR{1189859} (93k:05101).  
\annot{} Further development of \xrefd{1987a}{Svatopluk Poljak \& Daniel Turzik} for all-negative $\Sigma$.  The import for general signed graphs is not discussed.  [Developed more in \xref{Kaparis and Letchford}{2018a}{Konstantinos Kaparis \& Adam N.\ Letchford}.]
\codes{\CPar:~\CFr,~\CGeom}%
\itemspace

\author{Svatopluk Poljak \& Zsolt Tuza}
\authorspace
\bsgitem{1995a}{}  Maximum cuts and large bipartite subgraphs.  In: W.~Cook, L.~Lov\'asz, and P.\ Seymour, eds., \emph{Combinatorial Optimization} (DIMACS Special Year, New Brunswick, N.J., 1992--1993), pp.\ 181--244.  DIMACS Ser.\ Discrete Math.\ Theor.\ Computer Sci., Vol.\ 20.  Amer.\ Math.\ Soc., Providence, R.I., 1995. 
\MR{1338615} (97a:90106).  \Zbl{819.00048}.
\annot{} Surveys max-cut and weighted max-cut [that is, max size balanced subgraph and max weight balanced subgraph in all-negative signed graphs].  See esp.\ \S2.9: ``Bipartite subgraph polytope and weakly bipartite graphs''.  [The weakly bipartite classes announced by Gerards suggested that a signed-graph characterization of weakly bipartite graphs is called for.  This is provided by \xref{Guenin}{2001a}{Bertrand Guenin}.] 
\annotctd{} \S1.2, ``Lower bounds, expected size, and heuristics'', surveys results for all-negative signed graphs that are analogous to results in \xref{Akiyama, Avis, Chv\'a\-tal, and Era}{1981a}{J.\ Akiyama, D.\ Avis, V.\ Chvatal, \& H.\ Era} (\qv), etc.  [\Problem.  Generalize any of these results, that are not already generalized, to signed simple graphs and to simply signed graphs.]
\codes{\Cpar:~\CFr,~\Ctg(\CSw):~\CExp,~\CRef}%
\itemspace

\author{Albert Pollatchek}						%POLLATCHEK
\authorspace
\bsgitem{1975a}{}  \emph{Relationships Between Combinatorics and 0-Simple Semigroups.}  Ph.D.\ thesis, Univ.\ of California, Berkeley, 1975
\MR{2940578} (no rev).							
\codes{\Cgg:~\Caut,~\CAlgeb}%
\itemspace

\bsgitem{1976a}{}   Relationships between combinatorics and 0-simple semigroups.  \emph{Semigroup Forum} 12 (1976), no.\ 1, 78--82.
\MR{0486249} (58 \#6017).  \Zbl{322.20029}.					
\annot{} Announcement of \xrefd{1977a}{Albert Pollatchek}.
\codes{\Cgg:~\Caut,~\CAlgeb}%
\itemspace

\bsgitem{1977a}{}   Relationships between combinatorics and 0-simple semigroups.  \emph{J.\ Pure Appl.\ Algebra} 9 (1976/77), no.\ 3, 301--334.
\MR{0453905} (56 \#12158).  \Zbl{421.20029}.					
\codes{\Cgg:~\Caut,~\CAlgeb}%
\itemspace

\author{Alexandr Polyanskii}						%POLYANSKII
\authorspace
\authref  See \hyperlink{Zilin Jiang}{Z.L.~Jiang}.
\itemspace

\author{Y.\ Pomeau} 							%POMEAU
\authorspace
\authref   See \hyperlink{B.\ Derrida}{B.~Derrida}.
\itemspace

\author{Kalpesh M.\ Popat}						%POPAT KM
\authorspace
\authref  See \hyperlink{Samir K.\ Vaidya}{S.K.~Vaidya}.
\itemspace

\author{Dragos Popescu [Drago\c{s}-Radu Popescu]}
\authorx{Dragos Popescu [Dragos-Radu Popescu]}
\authorspace
\authref   See \hyperlink{Dragos Popescu}{D.-R.~Popescu}.
\itemspace

\author{Drago\c{s}-Radu Popescu [Dragos Popescu]}			%POPESCU
\authorx{Drago\c{s}-Radu Popescu}
\authorx{Dragos-Radu Popescu}
\authorx{Dragos Popescu}
\authorspace
\bsgitem{1979a}{}  Proprietati ale grafurilor semnate.  [Properties of signed graphs.]  (In Romanian.  French summary.)  
\emph{Stud.\ Cerc.\ Mat.}\ 31 (1979), 433--452.  
\MR{0560478} (82b:05111).  \Zbl{426.05048}. 
\annot{} A signed $K_n$ is balanced or antibalanced or has a positive and a negative circle of every length $k = 3, \ldots, n$.  For odd $n$, the signed $K_n$ if not balanced has at least $\frac{n-1}{2}$ negative Hamiltonian circles.  For even $n$, $-K_n$ does not maximize the number of negative circles.  A ``circle basis'' is a set of the smallest number of circles whose signs determine all circle signs.  This is proved to have $\binom{n-1}{2}$ members.  Furthermore, there is a basis consisting of $k$-circles for each $k = 3, \ldots, n$.  [A circle basis in this sense is the same as a basis of circles for the binary cycle space.  See \xref{Zaslavsky}{1981b}{Thomas Zaslavsky}, \xref{Topp and Ulatowski}{1987a}{J.\ Topp \& W.\ Ulatowski}.]
\codes{\CSG:~\CFr,~\CCircles}%
\itemspace

\bsgitem{1991a}{}  Cicluri {\^\i}n grafuri semnate.  [Cycles in signed graphs.]  (In Romanian; French summary.)  
\emph{Stud.\ Cerc.\ Mat.}\ 43 (1991), no.\ 3/4, 85--219.  
\MR{1138705} (92j:05114).  \Zbl{751.05060}.					
\annot{} Ch.\ 1: ``$A$-balance'' (p.\ 91).  Let $F$ be a spanning subgraph of $K_n$ and $A$ a signed $K_n$.  The ``product'' of signed graphs is $\Sigma_1 * \Sigma_2$ whose underlying graph is $|\Sigma_1| \cup |\Sigma_2|$, signed as in $\Sigma_i$ for an edge in only one $\Sigma_i$ but with sign $\sigma_1(e)\sigma_2(e)$ if in both.  Let $\cG_F$ denote the group of all signings of $F$; let $\cG_F(A)$ be the group generated by the set of restrictions to $F$ of isomorphs of $A$.  A member of $\cG_F(A)$ is ``$A$-balanced''; other members of $\cG_F$ are $A$-unbalanced.  
We let $\hat \Sigma$ denote the coset of $\Sigma$ and $\approx$ the ``isomorphism'' of cosets induced by graph isomorphism, i.e., cosets are isomorphic if they have isomorphic members.  Let $\dot \Sigma$ be the isomorphism class of $\Sigma$, $\hat{\hat \Sigma}$ the isomorphism class of $\hat \Sigma$, and ${\overset \circ \to \Sigma} := \bigcup \hat{\hat \Sigma}$.  
Now choose a system of representatives of the coset isomorphism classes, $R = \{\Sigma_1, \ldots, \Sigma_l\}$.  Prop.\ 1.4.1:  Each $\dot \Sigma$ intersects exactly one $\hat \Sigma_i$.  Let $R_i = \{\Sigma_{i1}, \ldots, \Sigma_{ia_i}\}$ be a system of representatives of $\hat \Sigma_i / \cong$, arranged so that $\#E^-(\Sigma_{ij})$ is a minimum when $j = 1$.  This minimum value is the ``[line] index of $A$-imbalance'' of each $\Sigma \in {\overset \circ \to \Sigma}_i$ and is denoted by $\delta_A(\Sigma)$.  (\S2.1:  Taking $A$ to be $K_n$ with one vertex star all negative makes this equal the frustration index $l(\Sigma)$.)  
Prop.\ 1.5.1:  $\delta_A(\Sigma)$ is the least number of edges whose sign needs to be changed to make $\Sigma$ $A$-balanced.  Prop.\ 1.5.2.  $\delta_A(\Sigma) = \#E^-(\Sigma)$ iff $\#(E^-(\Sigma) \cap E^-(F,\beta)) \leq \frac{1}{2} \#E^-(F,\beta)$ for every signing $\beta)$ of $F$.  Finally, for each $\Sigma \in \cG_F$ define the ``$\Sigma$-relation'' on coset isomorphism classes $\hat{\hat\Sigma}_i$ to be the relation generated by negating in $\Sigma_1$ all the edges of $E^-(\Sigma)$, extended by isomorphism and transitivity.  This is well defined (Prop.\ 1.6.1) and symmetric (Prop.\ 1.6.2) and is preserved under negation of coset isomorphism classes (Prop.\ 1.6.4, 1.6.5).  Self-negative classes, such that $\hat{\hat\Sigma} \approx -\hat{\hat\Sigma}$, are the subject of Prop.\ 1.6.3.
\annotctd{} Ch.\ 2: ``Signed complete graphs'' (p.\ 106).  \S2.5:  ``$H$-graphs''.  If $H$ is a signed $K_h$, a ``standard $H$-graph'' $\Sigma$ is a signed $K_n$ such that $\Sigma^- \cong H^- \cupdot\ K_{n-h}^c$.  Prop.\ 2.5.3.  Assume certain hypotheses on $n$, $\#X_0$ for $X_0 \subseteq V(\Sigma)$, and a quantity $D^-(H)$ derived from negative degrees.  Then $\#E^- = l(\Sigma) \Rightarrow$ the induced subgraph $G\ind X_0$ is a standard $H$-graph with $\#E^-(\Sigma\ind X_0) = l(\Sigma\ind X_0)$.  The cases $H^- = K_1$, $K_2$, and a $2$-edge path are worked out.  For the former, Prop.\ 2.5.3 reduces to \xref{Soza\'nski's}{1976a}{Tadeusz Sozanski} Thm.\ 3. 
\annotctd{} Ch.\ 3: ``Frustration index'' (p.\ 158).  Some upper bounds.
\annotctd{} Ch.\ 4: ``Evaluations, divisibility properties'' (p.\ 174).  Similar to parts of \xrefd{1996a}{Dragos Popescu} and \xref{Popescu and Tomescu}{1996b}{Dragos-Radu Popescu \& Ioan Tomescu}.
\annotctd{} Ch.\ 5: ``Maximal properties'' (p.\ 198).  \S5.1: ``Minimum number and maximum number of negative stars, resp.\ $2$-stars''.  \S5.2 is a special case of \xref{Popescu and Tomescu}{1996a}{Dragos-Radu Popescu \& Ioan Tomescu}, Thm.\ 2.  \S5.3:  ``On the maximum number of negative cycles in some signed complete graphs''.  Shows that Conjecture 1 is false for even $n \geq 6$.  Some results on the odd case.
\annotctd{} \Conjecture\ 1 (Tomescu).  A signed complete graph of odd order has the most negative circles iff it is antibalanced.  (Partial results are in \S5.3.)  [This example maximizes $l(\Sigma)$.  A somewhat related conjecture is in \xref{Zaslavsky}{1997b}{Thomas Zaslavsky}.]  \Conjecture\ 2.  See \xrefd{1993a}{Dragos Popescu}.  \Conjecture\ 3.  Given $k$ and $m$, there is $n(k,m)$ so that for any $n \geq n(k,m)$, a signed $K_n$ with $m$ negative edges has (a) the most negative $k$-circles iff the negative edges are pairwise nonadjacent; (b) the fewest iff the negative edges form a star.
\codes{\CSG:~\CBal(\CGen),~\CKG,~\CFr,~\CEnum:~\CCircles,~\CPaths}%
\itemspace

\bsgitem{1993a}{}  Problem 17.  \emph{Research Problems} at the Int.\ Conf.\ on Combinatorics (Keszthely, 1993).  Unpublished manuscript.  J\'anos Bolyai Math.\ Soc., Budapest, 1993. 
\annot{} \Conjecture.  An unbalanced signed complete graph has the minimum number of negative circles iff its frustration index equals $1$.  [This has been proved.]
\codes{\CSG:~\CCircles,~\CFr}%
\itemspace

\bsgitem{1996a}{}  Une m\'ethode d'\'enum\'eration des cycles n\'egatifs d'un graphe sign\'e.  \emph{Discrete Math.}\ 150 (1996), 337--345.  
\MR{1392742} (97c:05077).  \Zbl{960.39919}.					
\annot{} The numbers of negative subgraphs, especially circles and paths of length $k$, in an arbitrarily signed $K_n$.  Complicated formulas; divisibility and congruence properties.  Extends part of \xref{Popescu and Tomescu}{1996a}{Dragos-Radu Popescu \& Ioan Tomescu}.
\par
\codes{\CSG:~\CKG,~\CEnum:~\CCircles,~\CPaths}%
\itemspace

\bsgitem{1999a}{}  Balance in systems of finite sets. 
Proc.\ Annual Meeting Fac.\ Math.\ (Bucharest, 1999).  \emph{An.\ Univ.\ Bucure\c{s}ti Mat.\ Inform.}\ 48 (1999), no.\ 2, 29--40.
\MR{1829295} (2002c:05082).  
\annot{} A generalization of signed-graph frustration index.  Let $\cF \subseteq \cP(E)$; let $\delta_{\min}(S|\cF) := \min[ \{ \#(S\oplus F) : F \in \cF\}$ and similarly $\delta_{\max}$.  Application to signed graphs, where $\cF = \cB(\Sigma)$ and $\delta_{\min}(S|\cF) = l(\Sigma|S)$.
\dated{3 Oct 2014}
\codes{\CSG:~\CBal,~\CGen}%
\itemspace

\bsgitem{2001a}{}  An inequality on the maximum number of negative cycles in complete signed graphs.  \emph{Math.\ Rep.\ (Bucur.)} 3(53) (2001), no.\ 1, 53--60.
\MR{1887184} (2002m:\-05193).  \Zbl{1017.05099}.
\annot{} Similar to \xrefd{1999a}{Dragos Popescu}.	
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{2007a}{}  Balance in systems of finite sets with applications.  \emph{J.UCS} 13 (2007), no.\ 11, 1755--1766.
\MR{2390248} (2009c:05245).  
\annot{} Similar to \xrefd{1999a}{Dragos Popescu} but [\cf\ MR] with improved results.
	
\codes{\CSG:~\CFr}%
\itemspace

\author{Drago\c{s}-Radu Popescu \& Ioan Tomescu}
\authorx{Dragos-Radu Popescu \& Ioan Tomescu}
\authorspace

\bsgitem{1996a}{}  Negative cycles in complete signed graphs.  \emph{Discrete Appl.\ Math.}\ 68 (1996), 145--152.  
\MR{1393315} (98f:05098).  \Zbl{960.35935}.					
\annot{} The number $c_p^-$ of negative circles of length $p$ in a signed $K_n$ with $s$ negative edges.  
Thm.\ 1:  For $n$ sufficiently large compared to $p$ and $s$, $c_p^-$ is minimized if $E^-$ is a star (iff, when $s > 3$) and is maximized iff $E^-$ is a matching.   Thm.\ 2:  $c_p^-$ is divisible by $2^{p - 2 - \lfloor \log_2(p-1) \rfloor}$.  
Thm.\ 3:  If $s \sim \lambda n$ and $p \sim \mu n$ and the negative-subgraph degrees are bounded (this is essential), then asymptotically the fraction of negative $p$-circles is $\frac{1}{2}(1 - e^{-4\lambda\mu})$.
[\xref {Kittipassorn \& M\'esz\'aros}{2015a}{Teeradej Kittipassorn \& Gabor Meszaros} performs a detailed study of the number of negative triangles.]
\codes{\CSG:~\CKG:~\CFr,~\CEnum:~\CCircles}%
\itemspace

\bsgitem{1996b}{}  Bonferroni inequalities and negative cycles in large complete signed graphs.  \emph{European J.\ Combin.}\ 17 (1996), 479--483.
\MR{1397155} (97d:05177).  \Zbl{861.05036}.					
\annot{} A much earlier version of \xrefd{1996a}{Dragos-Radu Popescu \& Ioan Tomescu} with delayed publication.  Contains part of \xrefd{1996a}{Dragos-Radu Popescu \& Ioan Tomescu}: a version of Thm.\ 1 and a restricted form of Thm.\ 3.
\par
\codes{\CSG:~\CKG:~\CFr,~\CEnum:~\CCircles}%
\itemspace

\author{L.\ P\'osa}							%POSA
\authorx{L.\ Posa}
\authorspace
\authref  See \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\itemspace

\author{Olaf Post}							%POST O
\authorspace
\authref  See \hyperlink{Carsten Lange}{C.~Lange}.
\itemspace

\author{Luke Postle}							%POSTLE
\authorspace
\authref  See \hyperlink{Zdenek Dvorak}{Z.~Dvo\v{r}\'ak}.
\itemspace

\author{Alexander Postnikov} 						%POSTNIKOV
\authorspace
See also \hyperlink{Federico Ardila}{F.~Ardila}.
\authrefspace

\bsgitem{1997a}{}  Intransitive trees.  \emph{J.\ Combin.\ Theory Ser.\ A} 79 (1997), 360--366.  
\MR{1462563} (98b:05036).  \Zbl{876.05042}.
\annot{} \S4.2 mentions the lift matroid of $\{1\}\vec K_n$, i.e., the integral poise gains of a transitively oriented complete graph, represented by the Linial arrangement.  [\Cf\ \xref{Stanley}{1996a}{Richard P.\ Stanley}.]
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace

\author{Alexander Postnikov \& Richard P.\ Stanley}
\authorspace
\bsgitem{2000a}{}  Deformations of Coxeter hyperplane arrangements.  \emph{J.\ Combin.\ Theory Ser.\ A} 91 (2000), 544--597.
\MR{1780038} (2002g:52032).  \Zbl{962.05004}.						
\annot{} The arrangements are the canonical affine-hyperplane lift representations of certain additive real gain graphs.  Characteristic polynomials of the former, equalling zero-free chromatic polynomials of the latter, are calculated.  And much more.  
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar}
\itemspace

\author{J.\ Poulter}							%POULTER
\authorspace
\authref  See also \hyperlink{A.\ Aromsawa}{A.~Aromsawa}, \hyperlink{J.A.\ Blackman}{J.A.~Blackman}, and \hyperlink{J.R.\ Goncalves}{J.R.~Goncalves}.
\authrefspace

\author{J.\ Poulter \& J.A.\ Blackman}
\authorspace
\bsgitem{2001a}{}  Properties of the $\pm J$ Ising spin glass on the triangular lattice.  \emph{J.\ Phys.\ A} 34 (2001), 7527--7539.
\Zbl{982.82032}.  
\annot{} Triangular lattice graph with a definite proportion of negative edges.  Successor to \xref{Blackman and Poulter}{1991a}{J.A.\ Blackman \& J.\ Poulter}.
\dated{16 Aug 2018} 
\par
\codes{\CPhys:~\Csg:~\CFr}%
\itemspace

\bsgitem{2005a}{}  Exact algorithm for spin-correlation functions of the two-dimensional $\pm J$ Ising spin glass
in the ground state.  \emph{Phys.\ Rev.\ B} 72 (2005), art.\ 104422, 8 pp.
\codes{\CPhys:~\Csg:~\CFr,~\CState}%
\itemspace

\author{Swathy Prabhu [Swathyprabhu Mj]}				%[PRABHU]
\authorspace
\authref  See \hyperlink{Swathyprabhu Mj}{Swathyprabhu}.
\itemspace

\author{Shivaramakrishna Pragada}					%PRAGADA
\authorspace
\authref  See \hyperlink{M.\ Rajesh Kannan}{M.R.~Kannan}.
\itemspace

\authorx{U.M.\ Prajapati}						%PRAJAPATI
\author{U.M.\ Prajapati \& K.K.\ Raval}
\authorspace
\bsgitem{2019a}{}  Wheel related some signed product cordial graphs.  \emph{Int.\ J.\ Sci.\ Res.\ Rev.}\ 8 (2019), no.\ 2, 2934--2940.
\annot{} More examples as in \xref{Baskar Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\par
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\authorx{K.N.\ Prakasha}						%PRAKASHA
\author{K.N.\ Prakasha, P.\ Siva Kota Reddy, \& Ismail Naci Cangul}
\authorspace
\bsgitem{2017a}{}  Partition Laplacian energy of a graph.  \emph{Adv.\ Stud.\ Contemp.\ Math., Kyungshang} 
\Zbl{1386.05114}.  
\codes{\Csgw(\CGen):~\CAdj:~\CEig}%
\itemspace

\authorx{Pranjali}							%PRANJALI
\author{Pranjali [Pranjali Sharma]}
\authorspace
\authref  See also \hyperlink{Mukti Acharya}{M.~Acharya}.
\itemspace

\bsgitem{2023a}{}  Line signed graph of a signed unit graph of commutative rings.  \emph{Commun.\ Combin.\ Optim.}\ 8 (2023), no.\ 2, 313--326.
\MR{4570020}.  
\annot{} Line graph $\Lambda_{BC}(G_\Sigma(R))$ (\xref{Behzad and Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand}) for ring $R$ with unity.  Thm.\ 5 characterizes $R$ with balance.  Thm.\ 14 characterizes $R$ with canonical consistency.
\dated{22 Apr 2023} 
\par
\codes{\CAlgeb:~\CSG:~\CLG:~\CBal}%
\itemspace

\author{Pranjali Sharma \& Mukti Acharya}
\authorspace
\bsgitem{2016a}{}  Balanced signed total graphs of commutative rings.  \emph{Graphs Combin.}\ 32 (2016), no.\ 4, 1585--1597.
\MR{3514985}.  \Zbl{1342.05060}.  
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{Pranjali, Atul Gaur, \& Mukti Acharya}
\authorspace
\bsgitem{2016b}{}  $\cC$-Consistency in signed total graphs of commutative rings.  \emph{Discrete Math.\ Algorithms Appl.}\ 8 (2016), no.\ 3, art.\ 1650041, 14 pp.
\MR{3531893}.  \Zbl{1354.13008}.  
\annot{} $\cC$-Consistency = canonical consistency.
\codes{\CSG:~\CAlgeb:~\CLG(\CGen),~\CVS}%
\itemspace

\author{Viktor K.\ Prasanna}						%PRASANNA
\authorspace
\authref  See \hyperlink{Ajitesh Srivastava}{A.~Srivastava}.
\itemspace

\author{Philips Kokoh Prasetyo}					%PRASETYO
\authorspace
\authref  See \hyperlink{David Lo}{D.~Lo}.
\itemspace

\author{B.\ Prashanth}						%PRASHANTH
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Primo\v{z} Poto\v{c}nik}						%POTOCNIK
\authorx{Primoz Potocnik}
\authorspace
\authref  See also \hyperlink{Tomaz Pisanski}{T.~Pisanski}.
\authrefspace

\author{Primo\v{z} Poto\v{c}nik \& Mateja \v{S}ajna}
\authorx{Primoz Potocnik \& Mateja Sajna}
\authorspace
\bsgitem{2007a}{}  Self-complementary two-graphs and almost self-complementary double covers.  \emph{European J.\ Combin.}\ 28 (2007), 1561--1574.
\MR{2339485} (2008g:05177).  \Zbl{1123.05079}.  
\annot{} $\Gamma$ is ``almost self-complementary'' if $\Gamma \cong K_n\setm M \setm \Gamma$ where $M$ is a perfect matching in $K_{2n} \setm \Gamma$.  Such graphs are double coverings $(\Delta,\sigma)\tilde{}$ that are $\cong (\Delta^c,\sigma^c)\tilde{}$ for some $\sigma^c$.	
Dictionary:  ``$\bbZ_2$-voltage assignment'' = signature.
\dated{22 Aug 2013} 
\codes{\CTG:~\CCov,~\CAut}%
\itemspace

\bsgitem{2009a}{}  Brick assignments and homogeneously almost self-complementary graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 99 (2009), 185--201.
\MR{2467825} (2009i:05107).  \Zbl{1200.05182}.  
\codes{\Csg:~\CCov,~\CAut}%
\itemspace

\author{Annie K.\ Powell}						%POWELL
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\author{Julia Preusse}						%PREUSSE
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\authorx{K.O.\ Price}							%PRICE
\author{K.O.\ Price, E.\ Harburg \& T.M.\ Newcomb}
\authorspace
\bsgitem{1966a}{}  Psychological balance in situations of negative interpersonal attitudes.  \emph{J.\ Personality Social Psychol.}\ 3 (1966), 265--270.
	
\codes{\CPsS}%
\itemspace

\author{Noah Prince}							%PRINCE N
\authorspace
\authref  See \hyperlink{Henry Liu}{H.~Liu}.
\itemspace

\author{Geert Prins}							%PRINS*
\authorspace
\authref   See \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{Ion Prisecaru}						%PRISECARU
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\author{G.S.\ Shanmuga Priya}						%PRIYA
\authorspace
\authref  See \hyperlink{M.\ Venkata Anusha}{M.V.~Anusha}.
\itemspace

\author{Sharon Pronchik}						%PRONCHIK
\authorspace
\authref   See \hyperlink{Lori Fern}{L.~Fern}.
\itemspace

\author{James Propp}						%PROPP
\authorspace
\bsgitem{2001a}{}  A reciprocity theorem for domino tilings.  \emph{Electronic J.\  Combin.}\ 8 (2001), no.\ 1, Res.\ paper 18, 5 pp.
\MR{1855859} (2003e:05032).  \Zbl{982.05012}.  \arXiv{math/0104011}.			
\codes{\CSG:~\CAppl}%
\itemspace

\authorx{Anton Proskurnikov}						%PROSKURNIKOV
\author{Anton V.\ Proskurnikov and Ming Cao}
\authorspace
\bsgitem{2017a}{}  Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 2003--2008.  IEEE, 2017.
\codes{\CSG}%
\itemspace

\author{Anton Proskurnikov, Alexey Matveev, \&  Ming Cao}
\authorspace
\bsgitem{2014a}{}  Consensus and polarization in Altafini's model with bidirectional time-varying network topologies.  In:  \emph{53rd IEEE Conference on Decision and Control} (Los Angeles, 2014), pp.\ 2112--2117.  IEEE, 2014.
\codes{\CSG:~\CLap:~\CEig,~\CDyn}%
\itemspace

\author{Anton V.\ Proskurnikov \& Roberto Tempo}
\authorspace
\bsgitem{2018a}{}  A tutorial on modeling and analysis of dynamic social networks. Part II.  \emph{Annual Rev.\ Control} 45 (2018), 166--190.
\MR{3880766}.  
\annot{} \S6, ``Disagreement via negative influence'':  Expounds \xref{Altafini's}{2012a}{Claudio Altafini}, \xrefd{2013a}{Claudio Altafini} linear model of signed (weighted) (di)graph dynamics.  \S6.1, ``Balance theory'' for signed graphs including Altafini's Laplacian.  \S6.2, ``Altafini's model of opinion formation'':  Static and dynamic, balanced and unbalanced.
\dated{24 Dec 2018} 
\par
\codes{\CPsS:~\CSD,~\CWG:~\CBal,~\CDyn:~\CExp}%
\itemspace

\author{Anton V.\ Proskurnikov, Roberto Tempo, Ming Cao, \& Noah E. Friedkin}
\authorspace
\bsgitem{2017a}{}  Opinion evolution in time-varying social influence networks with prejudiced agents.  \emph{IFAC-PapersOnLine}\ 50 (2017), no.\ 1, 11896--11901.
\arXiv{1704.06900}.  
\codes{\CSD:~\CKG:~\CBal(\CGen),~\CDyn}%
\itemspace

\author{Andrzey Proskurowski}						%PROSKUROWSKI
\authorspace
\authref   See \hyperlink{Arthur M.\ Farley}{A.M.~Farley}.
\itemspace

\author{Alexandre Proutiere}						%PROUTIERE
\authorspace
\authref  See \hyperlink{Guodong Shi}{G.-D.~Shi}.
\itemspace

\author{J.\ Scott Provan} 						%PROVAN
\authorspace
\bsgitem{1983a}{}  Determinacy in linear systems and networks.  \emph{SIAM J.\ Algebraic Discrete Methods} 4 (1983), 262--278.  
\MR{0699780} (84g:90061).  \Zbl{558.93018}.
\codes{\CSol,~\CGN}%
\itemspace

\bsgitem{1987a}{}  Substitutes and complements in constrained linear models.  \emph{SIAM J.\ Algebraic Discrete Methods} 8 (1987), 585--603.  
\MR{0918061} (89c:90072).  \Zbl{645.90049}.
\annot{} \S4:  ``Determinacy in a class of network models.''  [Fig.\ 1 and Thm.\ 4.7 hint at a possible digraph version of the signed-graph or gain-graph frame matroid.]
\codes{\Csg?,~\Cgg:~\Cmatrd(\Cbases?):~\Cgen}%
\itemspace

\author{Pavel Prozorov}						%PROZOROV
\authorspace
\authref  See \hyperlink{Danila Cherkashin}{D.~Cherkashin}.
\itemspace

\authorx{Teresa M.\ Przytycka}						%PRZYTYCKA
\author{Teresa M.\ Przytycka \& J\'ozef H.\ Przytycki}
\authorx{Teresa M.\ Przytycka \& Jozef H.\ Przytycki}
\authorspace
\bsgitem{1988a}{}  Invariants of chromatic graphs.  Tech.\ Rep.\ No.\ 88-22, University of British Columbia, Vancouver, B.C., 1988.
\annot{} Generalizing concepts from \xref{Kauffman}{1989a}{Louis H.\ Kauffman}.  [\Cf\ \xref{Traldi}{1989a}{Lorenzo Traldi} and \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}.]
	
\codes{\CSGc:~\CGen:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{1993a}{}  Subexponentially computable truncations of Jones-type polynomials.  
In: Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 63--108.  Contemp.\ Math., Vol.\ 147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.
\MR{1224697} (95c:57016).  \Zbl{812.57010}.					
\annot{} A ``chromatic graph'' is a graph with edges weighted from the set $Z \times \{d,l\}$, $Z$ being [apparently] an arbitrary set of ``colors''.  A ``dichromatic graph'' has $Z = \{+,-\}$.  
Such graphs have general dichromatic polynomials [see \xref{Przytycka and Przytycki}{1988a}{Teresa M.\ Przytycka \& Jozef H.\ Przytycki}, \xref{Traldi}{1989a}{Lorenzo Traldi}, and \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}], as [partially] anticipated by \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn}.  
I will not attempt to summarize this paper.  
	
\par
\codes{\CSGc:~\CInvar,~\CKnot,~\CRef}%
\itemspace

\author{J\'ozef H.\ Przytycki}						%PRZYTYCKI
\authorx{Jozef H.\ Przytycki}
\authorspace
\authref   See \hyperlink{Kunio Murasugi}{K.~Murasugi} and \hyperlink{Teresa M.\ Przytycka}{T.M.~Przytycka}.
\itemspace

\author{Vlastimil Ptak} 						%PTAK
\authorspace
\authref   See \hyperlink{Miroslav Fiedler}{M.~Fiedler}.
\itemspace

\authorx{Charles J.\ Puccia}						%PUCCIA
\author{Charles J.\ Puccia \& Richard Levins}
\authorspace
\bsgitem{1986a}{}  \emph{Qualitative Modeling of Complex Systems: An Introduction to Loop Analysis and Time Averaging}.  
Harvard Univ.\ Press, Cambridge, Mass., 1986. 
\par
\codes{\CSD:~\CQM:~\CSta:~\CCycles}
\itemspace

\authorx{Pakize Simin Pulat}						%PULAT
\author{P.\ Simin Pulat}
\authorspace
\bsgitem{1989a}{}  A decomposition algorithm to determine the maximum flow in a generalized network.  \emph{Computers Oper.\ Res.}\ 10 (1989), no.\ 2, 161--172.
\MR{985143} (90a:90203).  \Zbl{674.90023}.  
\annot{} Decomposing an $\bbR_{>0}$-gain graph.
\dated{8 Jan 2016} 
\codes{\Cgg:~\CAlgor}%
\itemspace

\author{William R.\ Pulleyblank}						%PULLEYBLANK
\authorspace
\authref   See \hyperlink{Jean-Marie Bourjolly}{J.-M.~Bourjolly} and \hyperlink{Martin Grotschel}{M.~Gr\"otschel}.
\itemspace

\authorx{G.N.\ Purohit}						%PUROHIT
\author{G.N.\ Purohit \& Ritu Rani Meherwal}
\authorspace
\bsgitem{2012a}{}  Signed graph partitioning by spectral rounding.  \emph{Int.\ J.\ Contemp.\ Math.\ Sci.}\ 7 (2012), no.\ 43, 2117--2124.
\MR{2980861} (no rev).  \Zbl{1255.05150}.  
\par
\codes{\CSG:~\CWG:~\CEig,~\CAlgor}%
\itemspace

\author{L.\ Pushpalatha}						%PUSHPALATHA
\authorspace
\authref  See \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\author{L.\ Pyber}							%PYBER
\authorspace
\authref   See \hyperlink{Laszlo Lovasz}{L.~Lov\'asz}.
\itemspace

\author{Sung-Soo Pyo}						%PYO SS
\authorspace
\authref  See \hyperlink{Jin-Woo Park}{J.-W.~Park}.
\itemspace


\label{LETTER Q} \hypertarget{LETTER Q}{}	% LETTER  Q


\author{Hao Qi}							%QI H
\authorspace
\bsgitem{2019a}{}  Critical permutation sets for generalized signed graph coloring.  In:  \emph{Taipei International Workshop on Combinatorics and Graph Theory} (Taipei, 2019).  Academia Sinica, Taipei, 2019.  1 p.
\annot{} Abstract.  Fix $\Gamma$ and $S \subseteq \fS_k$.  $\Gamma$ is ``$S$-colorable'' if every $S$-permutation gain function $\phi$ makes $(\Gamma,\phi)$ $[k]$-colorable.  ``Critical'' means $\forall$ $S' \supset S$, $\exists$ $\Gamma$ that is $S$- but not $S'$-colorable.  \Cf\ \xref{Jin, Wong, and Zhu}{20xxa}{Ligang Jin, Tsai-Lien Wong, \& Xuding Zhu}.
\dated{29 Jul 2019} 
\codes{\CGG,~\CSG:~\CCol}%
\itemspace

\author{Hao Qi, Tsai-Lien Wong, \& Xuding Zhu}
\authorspace
\bsgitem{2019a}{}  Chromatic number and orientations of graphs and signed graphs.  \emph{Taiwanese J.\ Math.}\ 23 (2019), no.\ 4, 767--776.
\MR{3982060}.  \Zbl{1417.05078}.  
\codes{\CSG:~\CCol}%
\itemspace

\author{Jian Qi}							%QI JA
\authorspace
\authref  See \hyperlink{Shang Wang Tan}{S.W.~Tan}.
\itemspace

\authorx{Xingqin Qi}							%QI XQ
\author{Xingqin Qi, Huimin Song, Jianliang Wu, Edgar Fuller, Rong Luo, \& Cun-Quan Zhang}
\authorspace
\bsgitem{2017a}{}  Eb\&D: A new clustering approach for signed social networks based on both edge-betweenness centrality and density of subgraphs.  \emph{Physica A} 482 (2017), 147--157.
\MR{3651415} (no rev).  
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Jianguo Qian}						%QIAN JG
\authorspace
\authref  See \hyperlink{Xiangyu Ren}{X.-Y.~Ren}, \hyperlink{Wei Wang [mr21]}{W.~Wang}, and \hyperlink{Wei Wang [mr36]}{W.~Wang}.
\itemspace

\authorx{Yi Qian}							%QIAN Y
\author{Yi Qian \& Sibel Adal\i}
\authorspace
\bsgitem{2013a}{}  Extended structural balance theory for modeling trust in social networks.  In:  Jordi Castell\`a-Roca \etal, eds., \emph{2013 Eleventh Annual Conference on Privacy, Security and Trust} (PST2013, Tarragona, Catalonia, 2013), pp.\ 283--290.  IEEE, 2013.
\codes{\CSG:~\CBal:~\CGeom,~\CDyn}%
\itemspace

\author{Tianyong Qiang}						%QIANG TY
\authorspace
\authref  See \hyperlink{Bao Jiao}{B.~Jiao}.
\itemspace

\author{Hongxun Qin}						%QIN HX
\authorspace
\authref  See also \hyperlink{Joseph E.\ Bonin}{J.E.~Bonin}, \hyperlink{Peter Brooksbank}{P.~Brooksbank}, \hyperlink{Thomas A.\ Dowling}{T.~Dowling}, and \hyperlink{Daniel C.\ Slilaty}{D.C.~Slilaty}.
\authrefspace

\bsgitem{2004a}{}  Complete principal truncations of Dowling lattices.  \emph{Adv.\ Appl.\ Math.}\ 32 (2004), no.\ 1-2, 364--379.
\MR{2037636} (2005e:06003).  \Zbl{1041.05019}.						
\annot{} These matroids are determined by their Tutte polynomials, except that only the order of the group can be determined.
\codes{\Cgg:~\CMatrd:~\CInc}%
\itemspace

\author{Hongxun Qin, Daniel C.\ Slilaty, \& Xiangqian Zhou}
\authorspace
\bsgitem{2009a}{}  The regular excluded minors for signed-graphic matroids.  \emph{Combin.\ Prob.\ Computing} 18 (2009), 953--978.
\MR{2550378} (2010m:05062).  \Zbl{1231.05063}.  
\annot{} The complete list of 31 forbidden minors that are regular matroids.
\dated{10 Sept 2010}
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\author{Qingfeng Qin}						%QIN QF
\authorspace
\authref  See \hyperlink{Wenqiang Duan}{W.-Q.~Duan}.
\itemspace

\author{Li Qiu}							%QIU L
\authorspace
\authref  See \hyperlink{Wei Chen}{W.~Chen}.
\itemspace

\author{Lihong Qiu}							%QIU LH
\authorspace
\authref  See \xref{Wang, Qiu, Qian, and Wang}{2020a}{Wei Wang, Lihong Qiu, Jianguo Qian, \& Wei Wang}.
\itemspace

\author{Wen-Yuan Qiu}						%QIU WY
\authorspace
\authref  See \hyperlink{Guang Hu}{G.~Hu}.
\itemspace

\author{Hui Qu}							%QU H
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.-H.~Yu}.
\itemspace

\authorx{Eduardo Queiroga}						%QUEIROGA
\author{Eduardo Queiroga, Anand Subramanian, Rosa Figueiredo, \& Yuri Frota}
\authorspace
\bsgitem{2021a}{}  Integer programming formulations and efficient local search for relaxed correlation clustering.  \emph{J.\ Global Optim.}\ 81 (2021), 919--966.
\MR{4337982}.  \Zbl{1481.90278}.  
\annot{} 
Dictionary: ``balance'' = clusterability.
\codes{\CSD(\Csg),~\CWG:~\CClu:~\CAlgor}%
\itemspace

\author{Rachel Quinlan}						%QUINLAN
\authorspace
\authref  See \hyperlink{Cian O'Brien}{C.~O'Brien}.
\itemspace

\author{Louis V.\ Quintas} 						%QUINTAS
\authorspace
\authref  See \hyperlink{Michael L.\ Gargano}{M.~Gargano}.
\itemspace

\author{James P.\ Quirk} 						%QUIRK
\authorspace
\authref  See also \hyperlink{Lowell Bassett}{L.~Bassett} and \hyperlink{John S.\ Maybee}{J.S.~Maybee}.
\authrefspace

\bsgitem{1974a}{}  A class of generalized Metzlerian matrices.  In:  George Horwich and Paul A.\ Samuelson, eds., \emph{Trade, Stability, and Macroeconomics:  Essays in Honor of Lloyd A.\ Metzler}, pp.\ 203--220.  Academic Press, New York, 1974.
\par
\codes{\CQM:~\CSta:~\Csd}%
\itemspace

\bsgitem{1981a}{}  Qualitative stability of matrices and economic theory: a survey article.  
In:  Harvey J.\ Greenberg and John S.\ Maybee, eds., \emph{Computer-Assisted Analysis and Model Simplification} (Proc.\ Sympos., Boulder, Col., 1980), pp.\ 113--164.  Discussion, pp.\ 193--199.  Academic Press, New York, 1981.  
\MR{0617930} (82g:00016) (book).  \Zbl{495.93001}) (book).
\annot{} Comments by W.M.\ Gorman (pp.\ 175--189) and Eli Hellerman (pp.\ 191--192).  Discussion:  see pp.\ 193--196.
\codes{\CQM:~\CSta:~\Csd,~\Cbal:~\CExp}%
\itemspace

\author{James Quirk \& Richard Ruppert}
\authorspace
\bsgitem{1965a}{}  Qualitative economics and the stability of equilibrium.  \emph{Rev.\ Economic Stud.}\ 32 (1965), 311--326.
\codes{\CQM:~\CSta:~\Csd}%
\itemspace


\label{LETTER R} \hypertarget{LETTER R}{}	% LETTER  R


\author{Aya Rabie}							%RABIE
\authorspace
\authref  See \hyperlink{Shokry Nada}{S.~Nada}.
\itemspace

\author{[N.\ Jafari Rad]}						%[RAD]
\authorspace
\authref  See \hyperlink{N.\ Jafari Rad}{N.\ Jafari Rad} (under ``J'').
\itemspace

\author{Nicole Radde}						%RADDE
\authorspace
\bsgitem{2010a}{}  Fixed point characterization of biological networks with complex graph topology.  \emph{Bioinformatics} 26 (2010), no.\ 22, 2874--2880.
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2011a}{}  The role of feedback mechanisms in biological network models - a tutorial.  \emph{Asian J.\ Control} 13 (2011), no.\ 5, 597--610.
\MR{2860960} (no rev).  \Zbl{1303.93087}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Nicole Radde, Nadav S.\ Bar, \& Murad Banaji}
\authorspace
\bsgitem{2010a}{}  Graphical methods for analysing feedback in biological networks -- a survey.  \emph{Int.\ J.\ Systems Sci.}\ 41 (2010), no.\ 1, 35--46.
\MR{2599706} (no rev).  \Zbl{1291.92060}.					
\par
\codes{\CSD,~\CBiol:~\CDyn:~\CExp}%
\itemspace

\authorx{Filippo Radicchi}						%RADICCHI
\author{Filippo Radicchi, Daniele Vilone, \& Hildegard Meyer-Ort\-manns}
\authorspace
\bsgitem{2007a}{}  Universality class of triad dynamics on a triangular lattice.  \emph{Phys.\ Rev.\ E} 75 (2007), 021118.
\codes{\CSG:~\CBal}%
\itemspace

\author{Filippo Radicchi, Daniele Vilone, Sooeyon Yoon, \& Hildegard Meyer-Ort\-manns}
\authorspace
\bsgitem{2007a}{}  Social balance as a satisfiability problem of computer science.  \emph{Phys.\ Rev.\ E (3)} 75 (2007), no.\ 2, 026106, 17 pp.
\MR{2354025} (2008g:91190).  
\annot{} \xref{Antal, Krapivsky, and Redner}{2005a}{T.\ Antal, P.L.\ Krapivsky, \& S.\ Redner} is generalized to $k$-cycle dynamics.  
\dated{20 Jun 2011}
\codes{\CSG:~\CBal:~\CAlgor}%
\itemspace

\author{Marko Radovanovi\'c}						%RADOVANOVIC
\authorx{Marko Radovanovic}
\authorspace
\authref  See \hyperlink{Pierre Aboulker}{P.~Aboulker}.
\itemspace

\author{Ileana Rodica R\u{a}dulescu}					%RADULESCU
\authorx{Ileana Rodica Radulescu}
\authorspace
\authref  See \hyperlink{Eva Kaslik}{E.~Kaslik}.
\itemspace

\author{Fahimeh Rahimi}						%RAHIMI
\authorspace
\bsgitem{2016a}{}  \emph{Covering Graphs and Equiangular Tight Frames}.  Master's thesis, Univ.\ of Waterloo, 2016.
\annot{} \S0.1:  Coverings of permutation gain graphs.  \S0.27, ``Two-graphs''.  \Cf\ \xref{Coutinho--Godsil--Shirazi--Zhan}{2016a}{G.\ Coutinho, C.\ Godsil, H.\ Shirazi, \& H.\ Zhan}, \xref{Godsil}{1996a}{C.D.\ Godsil}.
\dated{7 Nov 2020} 
\codes{\Cgg:~\CCov:~\CExp}\codesctd{\CTG:~\CCov:~\CExp}%
\itemspace

\author{Mourad Rahmani}						%RAHMANI
\authorspace
\bsgitem{2014a}{}  Some results on Whitney numbers of Dowling lattices.  \emph{Arab J.\ Math.\ Sci.}\ 20 (2014), no.\ 1, 11--27.
\MR{3148044}.  \Zbl{1377.11032}.  \arXiv{1212.0954}.				
\annot{} \Cf\ \xref{Mez\H{o}}{2010a}{Istvan Mezo}, \xref{Cheon and Jung}{2012a}{Gi-Sang Cheon \& Ji-Hwan Jung}.  Many formulas and identities for numbers and polynomials associated with Dowling lattices $Q_n(\fG)$ and the $r$-Whitney and $r$-Dowling numbers and polynomials of Mez\H{o} and Cheon--Jung.
\dated{28 May 2018} 
\codes{\Cgg:~\Cmatrd:~\CInvar}%
\itemspace

\author{Ashutosh Rai}						%RAI
\authorspace
\authref  See \hyperlink{Geevarghese Philip}{G.~Philip}.
\itemspace

\author{W.M.\ Raike} 						%RAIKE
\authorspace
\authref  See \hyperlink{A.\ Charnes}{A.~Charnes}.
\itemspace

\author{Justin Raimondi}						%RAIMONDI
\authorspace
\authref  See \hyperlink{Joel Foisy}{J.~Foisy}.
\itemspace

\author{M.A.\ Rajan}							%RAJAN
\authorspace
\authref  See \hyperlink{P.\ Balamuralidhar}{P.~Balamuralidhar} and \hyperlink{Hemant Kumar Rath}{H.K.~Rath}.
\itemspace

\author{K.R.\ Rajanna}						%RAJANNA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Rajeevan P}							%RAJEEVAN
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\author{R.\ Rajendra}						%RAJENDRA R
\authorspace
\authref  See also \hyperlink{P.\ Siva Kota Reddy}{P.S.K.\ Reddy}.
\authrefspace

\author{R.\ Rajendra, P.\ Siva Kota Reddy, \& V.M.\ Siddalingaswamy}
\authorspace
\bsgitem{2018a}{}  On some signed graphs of finite groups.  \emph{South East Asian J.\ Math.\ Math.\ Sci.}\ 14 (2018), no.\ 3, 57--62.
\MR{3964934}.  \Zbl{1428.05127}.  
\annot{} Very elementary.
\dated{19 Jan 2020} 
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{Supriya Rajendra}						%RAJENDRA S
\authorspace
\authref  See \hyperlink{Reshma R}{Reshma R}.
\itemspace

\authorx{M.R.\ Rajesh Kanna}						%RAJESH KANNA
\author{M.R.\ Rajesh Kanna, R.\ Jagadeesh, \& Mohammad Reza Farahani}
\authorspace
\bsgitem{2016a}{}  Minimum covering Seidel energy of a graph.  \emph{J.\ Indonesian Math.\ Soc.}\ 22 (2016), no.\ 1, 71--82.
\MR{3503081}.  \Zbl{1463.05350}.  
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{M.R.\ Rajesh Kanna, R.\ Jagadeesh, \& B.K.\ Kempegowda}
\authorspace
\bsgitem{2016a}{}  Minimum dominating Seidel energy of a graph.  \emph{Int.\ J.\ Sci.\ Eng.\ Res.}\  7 (2016), no.\ 5, 10--14.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{M.R.\ Rajesh Kanna, R.\ Pradeep Kumar, \& Mohammad Reza Farahani}
\authorspace
\bsgitem{2016a}{}  Milovanovi\'c bounds for Seidel energy of a graph.  \emph{Adv.\ Theor.\ Appl.\ Math.}\ 10 (2016), no.\ 1, 37--44.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{M.R.\ Rajesh Kanna, R.\ Pradeep Kumar, \& R.\ Jagadeesh}
\authorspace
\bsgitem{2015a}{}  Minimum covering color energy of a graph.  \emph{Int.\ J.\ Math.\ Anal.}\ 9 (2015), no.\ 8, 351--364.
\annot{} From $\Gamma$ define $\Sigma$ as in \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.  Fix $C$ a minimum vertex cover of $\Gamma$; let $I_C$ have $1$ in $(i,i)$ for $v_i\in C$, otherwise 0.  Minimum covering color energy $\cE^C$ := energy of $A(\Sigma)+I_C$.
	
\dated{22 Dec 2018} 
\codes{\Csg:~\CAdj:~\CEig}%
\itemspace

\author{Ramakrishnan K O}						%RAMAKRISHNAN
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\author{Harishchandra S.\ Ramane}					%RAMANE
\authorspace
\authref  See also \hyperlink{Prabhakar Ramrao Hampiholi}{P.R.~Hampiholi}.
\authrefspace

\author{Harishchandra S.\ Ramane \& Mahadevappa M.\ Gundloor}
\authorspace
\bsgitem{2017a}{}  More equienergetic signed graphs.  \emph{Math.\ Interdisciplinary Res.}\ 2 (2017), 169--179.
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Harishchandra S.\ Ramane, Mahadevappa M.\ Gundloor, \& Sunilkumar M.\ Hosamani}
\authorspace
\bsgitem{2016a}{}  Seidel equienergetic graphs.  \emph{Bull.\ Math.\ Sci.\ Appl.}\ 16 (2016), 62--69.
\par
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Harishchandra S.\ Ramane, Ivan Gutman, \& Mahadevappa M.\ Gundloor}
\authorspace
\bsgitem{2015a}{}  Seidel energy of iterated line graphs of regular graphs.  \emph{Kragujevac J.\ Math.}\ 39 (2015), no.\ 1, 7--12.
\MR{3361336} (no rev).  \Zbl{1464.05249}.  
\par
\codes{\Csg:~\CLG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\authorx{Nacim Ramdani}						%RAMDANI
\author{Nacim Ramdani, Nacim Meslem, \& Yves Candau}
\authorspace
\bsgitem{2010a}{}  Computing reachable sets for uncertain nonlinear monotone systems.  \emph{Nonlinear Analysis: Hybrid Systems} 4 (2010), 263--278.
\MR{2607907} (2011b:93054).  \Zbl{1201.93019}.  
\codes{\CSD:~\CQM}%
\itemspace

\author{Fahimeh Ramezani}						%RAMEZANI
\authorspace
\authref  See \hyperlink{Giovanni Iacono}{G.~Iacono} and \hyperlink{Nicola Soranzo}{N.~Soranzo}.
\itemspace

\author{Farzaneh Ramezani}						%RAMEZANI
\authorspace
\authref  See also \hyperlink{Y.\ Bagheri}{Y.~Bagheri}.
\authrefspace

\bsgitem{2016a}{}  Cospectral lifts of graphs.  Manuscript, 2016.
\arXiv{1601.02354}.  
\annot{} Cospectrality of covering graphs of (apparently) abelian gain graphs.
\dated{27 Jul 2022} 
\codes{\Cgg:~\CCov}%
\itemspace

\bsgitem{2018a}{}  Constructing signed strongly regular graphs via star complement technique.  \emph{Math.\ Sci.}\ 12 (2018), no.\ 3, 157--161.
\MR{3861443}.  \Zbl{1417.05125}.  
\par
\codes{\CSG:~\CAdj,~\CEig}%
\itemspace

\bsgitem{2019a}{}  Coloring problem of signed interval graphs.  \emph{Trans.\ Combin.}\ 8 (2019), no.\ 4, 1--9.
\MR{4023079}.  \Zbl{1474.05170}.  \arXiv{1612.03280}.				
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{2020a}{}  On the signed graphs with two distinct eigenvalues.  \emph{Utilitas Math.}\ 114 (2020), 33--48.
\MR{4230326}.  \Zbl{1453.05063}.  \arXiv{1511.03511}.				
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Some regular signed graphs with only two distinct eigenvalues.  \emph{ Linear Multilinear Algebra} 70 (2022), no.\ 3, 517--530.
\MR{4388842}.  \Zbl{1484.05091}.  \arXiv{1909.06817}.  
\codes{\CSG:~\CAdj,~\CEig}%
\itemspace

\bsgitem{20xxa}{}  Some non-sign-symmetric signed graphs with symmetric spectrum.  Submitted.  \emph{}\ 
\arXiv{1909.06821}.  
\codes{\CSG:~\CAdj,~\CEig}%
\itemspace

\author{Farzaneh Ramezani, Peter Rowlinson, \& Zoran Stani\'c}
\authorspace
\bsgitem{2020a}{}  On eigenvalue multiplicity in signed graphs.  \emph{Discrete Math.}\ 343 (2020), no.\ 10, art.\ 111982, 8 pp.
\MR{4107756}.  \Zbl{1445.05062}.  \arXiv{1911.01113}.  
\annot{} Eigenvalue $\lambda \neq 0, \pm1$ of multiplicity $m$ satisfies $n \leq \binom{n-m+2}{3}$.
\dated{30 May 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Signed graphs with at most three eigenvalues.  \emph{Czechoslovak Math.\ J.} 72(147) (2022), no.\ 1, 59--77. 
\MR{4389106}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022b}{}  More on signed graphs with at most three eigenvalues.  \emph{Discuss.\ Math.\ Graph Theory} 42 (2022), no.\ 4, 1313--1331. 
\MR{4445854}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Farzaneh Ramezani \& Zoran Stani\'c}
\authorspace
\bsgitem{2021a}{}  An upper bound for the Laplacian index of a signed graph.  \emph{Discrete Math.\ Lett.}\ 5 (2021), 24--28.
\MR{4227397}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Some upper bounds for the net Laplacian index of a signed graph.  \emph{Bull.\ Iranian Math.\ Soc.}\ 48 (2022), no.\ 1, 243--250.
\MR{4391959}.  \Zbl{1483.05072}.  
\par
\codes{\CSG:~\CLap(\CGen):~\CEig}%
\itemspace

\author{Jos\'e L.\ Ram{\'\i}rez}						%RAMIREZ JL
\authorx{Jose L.\ Ramirez}
\authorspace
\authref  See \hyperlink{Miguel A.\ Mendez}{M.A.~M\'endez}.
\itemspace

\author{A.J.\ Ramirez-Pastor [Antonio Jos\'e Ramirez Pastor]}			%RAMIREZ-PASTOR
\authorx{A.J.\ Ram{\'\i}rez-Pastor}
\authorx{A.J.\ Ramirez-Pastor}
\authorspace
\authref  See also \hyperlink{F.\ Roma}{F.~Rom\'a}.
\authrefspace

\author{A.J.\ Ram{\'\i}rez-Pastor, F.\ Nieto, S.\ Contreras, \& E.E.\ Vogel}
\authorx{A.J.\ Ramirez-Pastor, F.\ Nieto, S.\ Contreras, \& E.E.\ Vogel}
\authorspace
\bsgitem{2000a}{}  Site order parameters for $\pm J$ Ising lattices.  \emph{Physica A} 283 (2000), 94--99.
\par
\codes{\CSG:~\CFr,~\CPhys}%
\itemspace

\author{A.J.\ Ram{\'\i}rez-Pastor, F.\ Nieto, \& E.E.\ Vogel}
\authorx{A.J.\ Ramirez-Pastor, F.\ Nieto, \& E.E.\ Vogel}
\authorspace
\bsgitem{1997a}{}  Ising lattices with $\pm J$ second-nearest-neighbor interactions.  \emph{Phys.\ Rev.\ B} 55 (1997), no.\ 21, 14323--14329.
\annot{} Randomly signed square lattice with half positive and half negative edges plus randomly signed second-neighbor edges, added according to various schemes and calculated for random examples.  Compares properties to 3-dimensional cubic and planar lattices, in particular to the pure square lattice in \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas} (\qv\ for dictionary).
	
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CFr,~\CSw}%
\itemspace

\author{A.J.\ Ramirez-Pastor, F.\ Rom\'a, F.\ Nieto, \& E.E.\ Vogel}
\authorx{A.J.\ Ramirez-Pastor, F.\ Roma, F.\ Nieto, \& E.E.\ Vogel}
\authorspace
\bsgitem{2004a}{}  Effect of the ground-state structure on order parameters in $\pm J$ Ising lattices.  \emph{Physica A} 336 (2004), 454--460.
\codes{\CSG:~\CState(\Cfr),~\CSw,~\CPhys}%
\itemspace

\author{R.\ Rammal} 							%RAMMAL
\authorspace
\authref  See \hyperlink{Francisco Barahona}{F.~Barahona} and \hyperlink{I.\ Bieche}{I.~Bieche}.
\itemspace

\author{K.\ Ranganathan}						%RANGANATHAN
\authorspace
\authref  See \hyperlink{R.\ Balakrishnan}{R.~Balakrishnan}.
\itemspace

\author{R.\ Rangarajan}						%RANGARAJAN
\authorspace
\authref  See also \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\authrefspace

\author{R.\ Rangarajan \& P.\ Siva Kota Reddy}
\authorspace
\bsgitem{2008a}{}  Notions of balance in symmetric $n$-sigraphs.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 11 (2008), no.\ 2, 145--151.
\MR{2482598} (2010h:05143).  \Zbl{1205.05102}.					
\annot{} $S_n$ is a symmetric $n$-signed graph.  Further definitions as in the notes to 
\xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
\S2, ``Balance in an $n$-sigraph $S_n = (G,\sigma)$.''  Prop.\ 1 (generalizing \xref{Harary}{1953a}{Frank Harary} for signed graphs): $S_n$ is balanced iff for each pair $u,v\in V$, every $uv$-path has the same gain.  [The simple proof of $\implies$, which depends on the fact that the gain group has exponent 2, is the best I have seen.  The proof of $\Leftarrow$ is incorrect.]  Prop.\ 4:  $\Sigma_{S_n}$ is balanced iff $V = V_1 \cupdot V_2$ such that an edge has identity gain iff it lies within $V_1$ or $V_2$.  Good proof via $\min$ as defined in the cited notes.
\S3, ``Clustering in an $n$-sigraph $S_n = (G,\sigma)$.''  $S_n$ is ``clusterable'' if $V$ has a partition $\pi$ such that an edge has identity gain iff it lies within a part of $\pi$.  Prop.\ 5 generalizes \xref{Davis}{1967a}{James A.\ Davis} to $n$-signed graphs.  
\S3.1:  ``Local balance (Local $i$-balance) in an $n$-sigraph $S_n = (G,\sigma)$.''  Prop.\ 6 generalizes \xref{Harary}{1955a}{Frank Harary}) on local balance [with a good proof].  Prop.\ 8:  A complete $S_n$ is balanced iff every triangle on one vertex is balanced.  Prop.\ 9 [incorrect]:  The same for imbalance.  Prop.\ 10 gives the number of balanced $S_n = (K_k,\sigma)$ [incorrect; the correct value is $2^{\lceil k/2 \rceil (n-1)}$].  
[The results are equally true, \emph{mutatis mutandis}, without assuming symmetry.]
[Minor typos require correction.]  
\dated{9 July 2009}
\codes{\CSG(\CGen),~\Cgg:~\CBal}%
\itemspace

\bsgitem{2008b}{}  Switching invariant 2-path sigraphs.  \emph{mySCIENCE} III (2008), no.\ 1, 20--25.
\par
\codes{\CSG:~\CSw}%
\itemspace

\bsgitem{2008c}{}  Identity and non-identity graphs on $n$-sigraphs.  \emph{Int.\ J.\ Math.\ Sci.\ Engg.\ Appl.}\ 2 (2008), no.\ 3, 111--117.
\Zbl{1210.05058}.  
\codes{\CSG(\CGen)}%
\itemspace

\bsgitem{2009a}{}  Notions of balance and consistency on symmetric $n$-marked graphs.  \emph{Bull.\ Pure Appl.\ Math.}\ 3 (2009), no.\ 1, 1--8.
\MR{2537685} (2010i:05156).  \Zbl{1200.05097}.					
	
\par
\codes{\CVS(\CGen),~\CSG(\CGen),~\Cgg:~\CBal}%
\itemspace

\bsgitem{2010a}{}  The edge $C_4$ signed graph of a signed graph.  \emph{Southeast Asian Bull.\ Math.}\ 34 (2010), 1066--1082.
\MR{2746741} (2011k:05100).  \Zbl{1240.05141}.					
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
The edge $C_4$ signed graph $E_4(\Sigma) := (V',E',\sigma_S)$ where $V' := E$ and $E' := \{ef : \exists\ C_4 \ni e,f \text{ in } |\Sigma|\}$.
Prop.\ 2.1:  $E_4(\Sigma)$ is balanced.  Cor.\ 2.5:  $E_4(\Sigma) = E_4(-\Sigma)$.  Prop.\ 2.3:  $\Sigma \simeq E_4(\Sigma)$ iff $\Sigma$ is a balanced signing of $C_n$, $n \geq 5$.  Prop.\ 3.1:  $\Sigma'$ is an $E_4(\Sigma)$ iff it is balanced and $|\Sigma'|$ is an $E_4(\Gamma)$.
\dated{2 Aug 2009, 20 Dec 2010}
\codes{\CSG:~\CBal,~\CSw,~\CLG(\CGen)}%
\itemspace

\author{R.\ Rangarajan, P.\ Siva Kota Reddy, \& N.D.\ Soner}
\authorspace
\bsgitem{2009a}{}  Switching equivalence in symmetric $n$-sigraphs.  II.  \emph{J.\ Orissa Math.\ Soc.}\ 28 (2009), no.\ 1--2, 1--12. 
\MR{2664129} (2011k:05099).  \Zbl{1244.05109}.					
\annot{} Continuation of \xref{Rangarajan, Reddy, and Subramanya}{2009a}{R.\ Rangarajan, P.\ Siva Kota Reddy, \& M.S.\ Subramanya} and \xref{Reddy and Prashanth}{2009a}{P.\ Siva Kota Reddy \& B.\ Prashanth}.  
Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  $\Phi$ is a symmetric $n$-signed graph.  
Prop.\ 4:  $\Phi$ is the $({\leq}\, m)$-distance graph $D_m(\Phi')$ of some $\Phi'$ iff it is balanced and $\|\Phi\|$ is a $({\leq}\, m)$-distance graph.  [Sufficiency is incorrect.]  
Solved [possibly incorrectly]:  $\Phi^c$ or $\Lambda_S(\Phi^c) \simeq D_m(\Lambda_S(\Phi))$;  $\Lambda_S(\Phi)$ or $\Lambda_S^2(\Phi) \simeq D_m(\Phi^{[c]})^{[c]}$ (except $\Lambda_S^2(\Phi) \simeq D_m(\Phi^c)^c$).
[The results are equally true without requiring symmetry.]
\dated{3 Aug 2009}
\par
\codes{\CSG(\CGen),~\Cgg:~\CSw,~\CLG}%
\itemspace

\bsgitem{2012a}{}  $m^\text{th}$ Power symmetric $n$-sigraphs.  \emph{Italian J.\ Pure Appl.\ Math.}\ No.\ 29 (2012), 87--92.
\MR{3009596}.  \Zbl{1328.05074}.  
\codes{\CSG(\CGen)}%
\itemspace

\author{R.\ Rangarajan, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}
\authorspace
\bsgitem{2009a}{}  Switching equivalence in symmetric $n$-sigraphs.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 18 (2009), no.\ 1, 79--85.
\\ \MR{2479750} (2011a:05141).  \Zbl{1183.05033}.					
\annot{} Continuation of \xref{Reddy, Vijay, and Lokesha}{2009a}{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha}, \xrefd{2010a}{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha}.  
Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
Prop.\ 4 characterizes $C_E(\Phi)$.  
Solved:  $\Lambda_S(\Phi) \simeq \Phi$; $\Phi \simeq \Lambda_S(\Phi)$; $\Lambda_S(\Phi) \simeq C_E(\Phi)$; $J(\Phi) \simeq C_E(\Phi)$.  
[The results remain true without assuming symmetry.]
\dated{2 Aug 2009}
\codes{\CSG(\CGen),~\Cgg:~\CSw}%
\itemspace

\author{R.\ Rangarajan, M.S.\ Subramanya, \& P.\ Siva Kota Reddy}
\authorspace
\bsgitem{2010a}{}  The $H$-line signed graph of a signed graph.  \emph{Int.\ J.\ Math.\ Combin.}\ 2010 (2010), no.\ 2, 37--43.
\Zbl{1216.05052}.  
\annot{} $H$ is a connected graph of order $\geq 3$.  $HL(\Sigma) \subseteq \Lambda_S(\Sigma)$ (defined at \xref{Sampathkumar, Reddy, and Subramanya}{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}); $ef \in E(\Lambda_S(\Sigma))$ is in $HL(\Sigma)$ iff $e,f$ are in a copy of $H$ in $|\Sigma|$.  
$\Sigma'$ is an $HL(\Sigma)$ iff it is balanced and $|\Sigma'|$ is an $H$-line graph.  
Solved:  $HL(\Sigma) \simeq \Sigma$ for $H = C_k, P_k$, $K_rL(\Sigma) \simeq \Lambda_S(\Sigma)$.  
Connections with graphs derived from matrices.
\dated{7 Jan 2011} 
\codes{\CSG:~\CLG(\CGen),~\CBal,~\CAdj}%
\itemspace

\bsgitem{2012a}{}  Neighborhood signed graphs.  \emph{Southeast Asian Bull.\ Math.}\ 36 (2012), no.\ 3, 389--397.
\MR{3005090} (no rev).  \Zbl{1274.05208}.						
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
The neighborhood signed graph or 2-path graph $P_2(\Sigma)$ is $(V,E_2,\sigma^c)$ where $E_2 := \{vw : \exists\ \text{$vw$-path of length } 2 \}$.
Thm.\ 5:  $P_2(\Sigma)$ is balanced and the signature can be any balanced signature (by appropriate choice of $\sigma$).  
Solved: $\Sigma$, $P_2(\Sigma) \simeq \Sigma$; $P_2(\Sigma) \simeq \Sigma^c$; $P_2(\Sigma) \simeq \Lambda_S(\Sigma)$.  
For connected $\Sigma$: $P_2^r(\Sigma) \simeq \Lambda_S(\Sigma)$; $P_2(\Sigma) \simeq J_S(\Sigma)$.  
Also, $P_2^r(\Sigma) \simeq \Lambda_S^s(\Sigma)$ when $|\Sigma|$ is unicyclic with circle length $l$ and $r,s < l/2$. 
\dated{2 Aug 2009}
\codes{\CSG:~\CBal,~\CSw,~\CLG(\CGen)}%
\annotctd{} \S5, ``$(-1,0,1)$-Matrices and neighborhood signed graphs'':  Given a $(-1,0,1)$-matrix $A$ with columns $a_1,\ldots,a_n$.  Let $V_A := [n]$, $E_A := \{ij : (\exists\ k)\ a_{ki}a_{kj} \neq 0\}$, and $\sigma_A(ij) := \mu_i\mu_j$ where $\mu_i :=$ product of nonzero entries in $a_i$.  Thm.\ 20:  This signed graph of $A(\Sigma)$ is $P_2(\Sigma)$.
\dated{10 Apr, 2 Aug 2009}
\codes{\CSG:~\CAdj:~\CBal}%
\itemspace

\authorx{Athira P Ranjith}						%RANJITH
\author{Athira P Ranjith \& Joseph Varghese Kureethara}
\authorspace
\bsgitem{2020a}{}  Sum signed graphs - I.  In:  Samayan Narayanamoorthy, ed., \emph{Advances In Applicable Mathematics-- ICAAM2020} (Coimbatore, India, 2020).  AIP Conf.\ Proc., Vol.\ 2261, art.\ 030047, 4 pp.
\annot{} ``Sum signed graph'': $\sigma_f(uv) = \sgn(n+\frac12-f(u)-f(v))$ for a fixed bijection $f: [n] \to V$.  Thms.:  $\exists$ negative edge [if $n>1$].  \emph{rna} number (\cf\ \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara}) $\rna(\Sigma) := \min_{f: \sigma_f=\sigma} \#E^-$ = 1 for path and star, 2 for circle, $\lfloor n^2/4 \rfloor$ for $K_n$.
\dated{13 Dec 2020} 
\par
\codes{\CLab:~\CSG}%
\itemspace


\author{A.R.\ Rao}							%RAO AR
\authorspace
\authref  See \xref{Acharya, Joshi, Rao, and Rao}{2003a}{B.D.\ Acharya, S.\ Joshi, A.R.\ Rao, \& S.B.\ Rao}.
\itemspace

\author{Angeline Rao}						%RAO A
\authorspace
\authref  See \hyperlink{Vinciane Chen}{V.~Chen}.
\itemspace

\author{Anita Kumari Rao}						%RAO AK
\authorspace
\authref  See \hyperlink{Deepa Sinha \& Anita Kumari Rao}{D.~Sinha}.
\itemspace

\author{M.R.\ Rao} 							%RAO MR 
\authorspace
\authref  See \hyperlink{Yvo M.I.\ Dirickx}{Y.M.I.~Dirickx}.
\itemspace

\author{S.B.\ Rao} 							%RAO SB
\authorspace
\authref  See also \hyperlink{B.\ Devadas Acharya}{B.D.~Acharya}, \hyperlink{Prabir Das}{P.~Das}, and \hyperlink{G.R.\ Vijayakumar}{[G.R.]~Vijaya~Kumar}. 
\authrefspace

\bsgitem{1984a}{} Characterizations of harmonious marked graphs and consistent nets.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 9 (1984), 97--112.
\MR{0959057} (89h:05048).  \Zbl{625.05049}. 
\annot{} A complicated solution, with a polynomial-time algorithm, to the problem of characterizing consistency in vertex-signed graphs (\cf\ \xref{Beineke and Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}).  Thm.\ 4.1 points out that graphs with signed vertices and edges can be easily converted to graphs with signed vertices only; thus harmony in graphs with signed vertices and edges is characterized as well.  
[This paper was independent of and approximately simultaneous with \xref{B.D.\ Acharya}{1983b}{B.\ Devadas Acharya}, \xrefd{1984a}{B.\ Devadas Acharya}.]  
[See \xref{Joglekar, Shah, and Diwan}{2010a}{Manas Joglekar, Nisarg Shah, \& Ajit A.\ Diwan} for the last word.] 
\codes{\CSG,~\CVS:~\CBal,~\CAlgor}%
\itemspace

\author{S.B.\ Rao, B.D.\ Acharya, T.\ Singh, \& Mukti Acharya}
\authorspace
\bsgitem{2005a}{}  Graceful complete signed graphs.  
In:  S.\ Arumugam, B.D.\ Acharya, and S.B.\ Rao, eds., \emph{Graphs, Combinatorics, Algorithms and Applications} (Proc.\ Nat.\ Conf., Anand Nagar, Krishnankul, India, 2004), pp.\ 123--124.  Narosa Publishing House, New Delhi, 2005.
\annot{} Extended abstract without proofs.  ``Graceful'' means $(1,1)$-graceful, $r=1$, as at \xref{M.\ Acharya and Singh}{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  
Thm.\ 1:  $(K_n,\sigma)$ is graceful iff $n\leq3$, $n=4$ and $\#E^-\neq3$, or $n=5$ and $\#E^-\neq5$ is odd and neither $\Sigma^+$ nor $\Sigma^-$ is $K_{1,3}$.  The proof involves a recursive labelling procedure.	
\dated{21 July 2010} 
\codes{\CSGc:~\CLab}%
\itemspace

\author{S.B.\ Rao, N.M.\ Singhi, \& K.S.\ Vijayan} 
\authorspace
\bsgitem{1981a}{}  The minimal forbidden subgraphs for generalized line graphs.  In:  S.B.\ Rao, ed., \emph{Combinatorics and Graph Theory} (Proc.\ Sympos., Calcutta, 1980), pp.\ 459--472.  \LNM 885.  Springer-Verlag, Berlin, 1981.  
\MR{0655644} (83i:05062).  \Zbl{494.05053}.					
\annot{} These are the minimal forbidden induced subgraphs for an all-negative signed simple graph to be the reduced line graph of a signed graph.
\par
\codes{\Csg:~\CLG,~\Cpar}%
\itemspace

\author{Vasant Rao}							%RAO V
\authorspace
\authref  See \hyperlink{Madhav Desai}{M.~Desai}.
\itemspace

\author{Anatol Rapoport}						%RAPOPORT
\authorspace
\bsgitem{1963a}{}  Mathematical models of social interaction.  In: R.D. Luce, R.R. Bush, and E. Galanter, eds., \emph{Handbook of Mathematical Psychology}, Vol.\ II, pp.\ 493--579.  Wiley, New York, 1963.
\annot{} \S3.3, ``The theory of structural balance''.  Nontechnical exposition.
\dated{12 Nov 2018} 
\codes{\CPsS:~\CSG:~\CBal:~\CExp}%
\itemspace

\author{A.M.\ Rappoport}						%RAPPOPORT
\authorspace
\authref  See \hyperlink{Ya.R.\ Grinberg}{Ya.R.~Grinberg}.
\itemspace

\authorx{Thomas Raschle}						%RASCHLE
\author{Thomas Raschle \& Klaus Simon}
\authorspace
\bsgitem{1995a}{}  Recognition of graphs with threshold dimension two.  
In: \emph{Proceedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing} (Las Vegas, 1995), pp.\ 650--661.  
\annot{} Expounded by \xref{Mahadev and Peled}{1995a}{N.V.R.\ Mahadev \& U.N.\ Peled}, \S8.5 (\qv).
\par
\codes{\Cpar:~\Cori,~\CAlgor}%
\itemspace

\author{Andr\'e Raspaud}						%RASPAUD
\authorx{Andre Raspaud}
\authref  See also \hyperlink{Edita Macajova}{E.~M\'a\v{c}ajov\'a}, \hyperlink{Amanda Montejano}{A.~Montejano}, and \hyperlink{Jaroslav Nesetril}{J.~Ne\v{s}et\v{r}il}.
\authrefspace

\author{Andre Raspaud \& Xuding Zhu}
\authorspace
\bsgitem{2011a}{}  Circular flow on signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 101 (2011), 464--479.
\MR{2832812} (2012j:05191).  \Zbl{05987722}.					
\annot{} Thm.\ 1:  $\Sigma$ has a nowhere-zero integral and circular [i.e., real] 4-flow if it is edge 4-connected.  It has a nowhere-zero circular $r$-flow with $r<4$ if it is edge 6-connected.  
A signed cut $D$ [\cf\ \xref{Chen and Wang}{2009a}{Beifang Chen \& Jue Wang}] is described by a signed subset $X=X^+\cupdot X^-$ of $V$.  Lemma 3:  $\Sigma$ has a circular $r$-flow iff it has an orientation such that $1/(r-1) \leq \#\partial^+(X)/\#\partial^-(X) \leq r-1$ for every $X$.  Here $\partial^\epsilon(X)$ is the set of ends in $X$, of $e \in D$, that have a certain sign.
\dated{23 March 2010} 
\par
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\authorx{Hemant Kumar Rath}						%RATH
\author{Hemant Kumar Rath, M.\ Rajan, \& P.\ Balamuralidhar}
\authorspace
\bsgitem{2011a}{}  Monotonic signed graph approach for cross-layer congestion control in wireless ad-hoc networks.  In:  \emph{2011 IEEE GLOBECOM Workshops} (GC Wkshps, Bangalore, 2011), pp.\ 309--314.  IEEE, 2011.
\codes{\CSD:~\CAppl}%
\itemspace

\author{H.\ Donald Ratliff} 						%RATLIFF
\authorspace
\authref   See \hyperlink{Jean-Claude Picard}{J.-Cl.~Picard}.
\itemspace

\author{Alberto Ravagnani}						%RAVAGNANI
\authorspace
\bsgitem{2022a}{\dag}  Whitney numbers of combinatorial geometries and higher-weight Dowling lattices.  \emph{SIAM J.\ Appl.\ Algebra Geom.}\ 6 (2022), no.\ 2, 156--189.
\MR{4406906}.  
\arXiv{1909.10249}.  
\annot{} The first serious computational methods and formulas for characteristic polynomials of higher-weight Dowling lattices (\cf\ \xref{Bonin}{1993a}{Joseph E.\ Bonin}).  Many results, such as Thm.\ 5.1:  For fixed weight $k$ and size $q$, the Whitney number $w_i$ for higher ranks $n$ are known in terms of their values for $n<ki$.  Also, some explicit formulas for $k=3$.  Etc.  [\Problem.  These lattices might be thought of as hypergraphic analogs of frame matroids of complete gain graphs.  If that interpretation can be implemented, the polynomials might be chromatic polynomials.]
[Related: \xref{Koga, Maruta, and Shiromoto}{2018a}{Yoshitaka Koga, Tatsuya Maruta, \& Keisuke Shiromoto}.]
\dated{22 Jul 2022} 
\codes{\CMatrd:~\CInvar,~\Cgg(\CGen)}%
\itemspace

\author{Dieter Rautenbach}						%RAUTENBACH
\authorspace
\authref  See also \hyperlink{Carlos V.G.C.\ Lima}{C.V.G.C.~Lima}.
\authrefspace

\author{Dieter Rautenbach \& Bruce Reed}
\authorspace
\bsgitem{2001a}{} The Erd\"os--P\'osa property for odd cycles in highly connected graphs.  Paul Erd\"os and His Mathematics (Budapest, 1999).  \emph{Combinatorica} 21 (2001), no.\ 2, 267--278.
\MR{1832451} (2002i:05073).  \Zbl{981.05066}.
\annot{} The smallest sufficient connectivity in \xref{Thomassen}{2001a}{Carsten Thomassen} is about $976k$.  [For more, see \xref{Hochst\"attler, Nickel, and Peis}{2006a}{Winfried Hochstattler, Robert Nickel, \& Britta Peis}.]
\par
\codes{\Cpar:~\CFr:~\CCircles}%
\itemspace

\author{K.K.\ Raval}							%RAVAL
\authorspace
\authref  See \hyperlink{U.M.\ Prajapati}{U.M.~Prajapati}.
\itemspace

\author{Bertram H.\ Raven} 						%RAVEN
\authorspace
\authref   See \hyperlink{Barry E.\ Collins}{B.E.~Collins}.
\itemspace

\author{E.V.\ Ravve}							%RAVVE
\authorspace
\authref  See \hyperlink{E.\ Fischer}{E.~Fischer}.
\itemspace

\authorx{D.K.\ Ray-Chaudhuri}						%RAY-CHAUDHURI
\author{D.K.\ Ray-Chaudhuri, N.M.\ Singhi, \& G.R.\ Vijayakumar}
\authorspace
\bsgitem{1992a}{}  Signed graphs having least eigenvalue around $-2$.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 17 (1992), 148--165.
\MR{1216974} (94g:05056).  
\codes{\CSG:~\CEig,~\CGeom:~\CExp}%
\itemspace

\author{Abigail Raz}							%RAZ
\authorspace
\authref  See \hyperlink{John Brown}{J.~Brown} and \hyperlink{Devlin Mallory, Abigail Raz, Christino Tamon, \& Thomas Zaslavsky}{D.~Mallory}.
\itemspace

\author{Igor Razgon}						%RAZGON
\authorspace
\authref  See \hyperlink{Gregory Gutin}{G.~Gutin}.
\itemspace

\author{Margaret A.\ Readdy}						%READDY
\authorspace
\authref  See also \hyperlink{Richard Ehrenborg}{R.~Ehrenborg}.
\authrefspace

\bsgitem{2001a}{}  The Yuri Manin ring and its $\cB_n$-analogue.  \emph{Adv.\ Appl.\ Math.}\ 26 (2001), 154--167.
\MR{1808445} (2001k:13034).  \Zbl{989.13016}.						
\annot{} P.\ 164:  Lattice of signed compositions (``ordered signed partitions''), from \xref{Ehrenborg and Readdy}{1999a}{Richard Ehrenborg \& Margaret A.\ Readdy}, \S6.  Pp.\ 164--165:  Signed permutahedron [equivalent to acyclotope of $\pm K_n\full$].
\codes{\CS}\codesctd{\Csg:~\Ckg:~\CGeom}%
\itemspace

\author{S.\ Redner}							%REDNER
\authorspace
\authref  See \hyperlink{T.\ Antal}{T.~Antal}.
\itemspace

\author{A.\ Sashi Kanth Reddy}						%REDDY ASK
\authorspace
\authref  See \xref{Reddy, Vaidya, and Reddy}{2011a}{P.\ Siva Kota Reddy, Gurunath Rao Vaidya, \& A.\ Sashi Kanth Reddy}.
\itemspace

\author{P.\ Siva Kota Reddy}						%REDDY PSK
\authorspace
\authref  See also \hyperlink{V.\ Lokesha}{V.~Lokesha}, \hyperlink{K.N.\ Prakasha}{K.N.~Prakasha}, \hyperlink{R.\ Rajendra}{R.~Rajendra}, \hyperlink{R.\ Rangarajan}{R.~Rangarajan}, \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}, and \hyperlink{M.S.\ Subramanya}{M.S.~Subramanya}.
\authrefspace

\bsgitem{2010a}{}  $t$-path sigraphs.  \emph{Tamsui Oxford J.\ Math.\ Sci.}\ 26 (2010), no.\ 4, 433--441.
\MR{2840769} (2012g:05096).  \Zbl{1236.05097}.  
\codes{\CSG}%
\itemspace

\bsgitem{2013a}{}  Smarandache directionally $n$-signed graphs --- A survey.  \emph{Int.\ J.\ Math.\ Combin.}\ 2013 (2013), no.\ 2, 34--43.
\Zbl{1288.05123}.  
\annot{} [Smarandache is irrelevant.]  \Cf\ \xref{Sampathkumar, Reddy, and Subramanya}{2010b}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.
\codes{\CSG:~\CGen}%
\itemspace

\bsgitem{20xxa}{}  Switching invariant $t$-path sigraphs.  Manuscript.  
\annot{} In the $t$-path signed graph $(\Sigma)_t$, $u, v$ are adjacent when joined by a path of length $t$, with signature $\sigma^c$ (see \xref{Sampathkumar, Reddy, and Subramanya}{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}).  
(The signature differs from that of \xref{Gill and Patwardhan}{1986a}{M.K.\ Gill \& G.A.\ Patwardhan} and \xref{M.\ Acharya}{1988a}{Mukti Acharya}.)
Solved: $\Sigma \simeq (\Sigma)_2,\ (\Sigma)_3$.  
\dated{10 Apr 2009}
\codes{\CSG:~\CSw,~\CLG(\CGen)}%
\itemspace

\bsgitem{20xxb}{}  A note on characterization of jump signed graphs.  Manuscript.  
\codes{\CSG:~\CLG}%
\itemspace

\author{P.\ Siva Kota Reddy, M.C.\ Geetha, \& K.R.\ Rajanna}
\authorspace
\bsgitem{2011a}{}  Switching equivalence in symmetric $n$-sigraphs--IV.  \emph{Scientia Magna} 7 (2011), no.\ 3, 34--38.
\annot{} See definitions at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  The antipodal graph $A(\Phi)$ has $V(A):=V$, $E(A):=\{uv: d(u,v)={\max}\}$, $\sigma_A(uv)=\mu_\phi(u)\mu_\phi(v)$ where $\mu_\phi=$ canonical vertex labelling.  Solved:  $A(\Phi)\simeq\Phi$ [trivial], $A(\Phi)\simeq\Phi^c$, etc.\ [elementary].  [Cors.\ 3.2, 3.3 are wrongly stated.]
\dated{13 Jul 2013} 
\codes{\CSG(\CGen):~\CSw}%
\itemspace

\bsgitem{2012a}{}  Switching equivalence in symmetric $n$-sigraphs.  V.  \emph{Int.\ J.\ Math.\ Combin.}\ 2012 (2012), no.\ 3, 58--63.
\codes{\CSG(\CGen):~\CSw}%
\itemspace

\author{P.\ Siva Kota Reddy, P.S.\ Hemavathi, \& R.\ Rajendra}
\authorspace
\bsgitem{2014a}{}  Radial signed digraphs.  \emph{Math.\ Aeterna} 4 (2014), no.\ 3, 191--196.
\MR{3214737} (no rev).  
\annot{} Radial digraph of $(D,\sigma)$ with certain balanced signs.
\dated{22 Jan 2020} 
\codes{\CSD,~\Csg:~\CBal,~\CSw,~\CVS}%
\itemspace

\author{P.\ Siva Kota Reddy, V.\ Lokesha, \& Gurunath Rao Vaidya}
\authorspace
\bsgitem{2010a}{}  A note on Smarandachely consistent symmetric $n$-marked graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2010 (2010), no.\ 3, 41--44.
\Zbl{1238.05120}.  
\annot{} [The name Smarandache is used for no reason.]  Symmetric $n$-marking means $\mu: V \to H_n := \{\text{symmetric sign $n$-vectors}\}$.  Consistent: product around every circle is $(+,\ldots,+)$.  Complement: replace $\mu$ by $t\mu$ where $t \in H_n$.  Trivial results.
\dated{13 Jan 2023} 
\codes{\CVS(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy, V.\ Lokesha, B.\ Prashanth, \& S.\ Vijay}
\authorspace
\bsgitem{2011a}{}  Some properties on super line signed graph $\cL_3(S)$.  \emph{J.\ Analysis Comput.}\ 7 (2011), no.\ 1, 45--48.
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy, V.\ Lokesha, \& Gurunath Rao Vaidya}
\authorspace
\bsgitem{2010a}{}  The line $n$-sigraph of a symmetric $n$-sigraph.  II.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 13 (2010), no.\ 3, 305--312.
\MR{3890295 }.  \Zbl{1223.05109}.  						
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.	
\par
\codes{\CGG(\CGen):~\CLG}%
\itemspace

\bsgitem{2010b}{}  The line $n$-sigraph of a symmetric $n$-sigraph.  III.  \emph{Int.\ J.\ Open Problems Computer Sci.\ Math.}\ 3 (2010), no.\ 5, 172--178.
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.	
\par
\codes{\CGG(\CGen):~\CLG}%
\itemspace

\bsgitem{2011a}{}  Switching equivalence in symmetric $n$-sigraphs.  III.  \emph{Int.\ J.\ Math.\ Sci.\ Engg.\ Appl.}\ 5 (2011), no.\ 1, 95--101.
\MR{2791536} (2012a:05141).  
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.	
\par
\codes{\CGG(\CGen):~\CSw}%
\itemspace

\author{P.\ Siva Kota Reddy \& U.K.\ Misra}
\authorspace
\bsgitem{2012a}{}  The common minimal equitable dominating signed graphs.  \emph{Notes Number Theory Discrete Math.}\ 18 (2012), no.\ 4, 40--46.
\codes{\CSG:~\CDom}%
\itemspace

\bsgitem{2013a}{}  The equitable associate signed graphs.  \emph{Bull.\ Int.\ Math.\ Virtual Inst.}\ 3 (2013), no.\ 1, 15--20.
\codes{\CSG}%
\itemspace

\bsgitem{2013b}{}  Directionally $n$-signed graphs-III: The notion of symmetric balance.  \emph{Trans.\ Combin.}\ 2 (2013), no.\ 4, 53--62.
\MR{3150452} (no rev).  \Zbl{1301.05160}.  
\annot{} Directional $n$-signing: see \xref{Sampathkumar--Reddy--Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Symmetric balance:  the gain of every circle is symmetric.  Main result:  Thm.\ 4.1:  $\Phi$ is symmetically balanced iff every circle has an even number of unsymmetric edge gains.  [False.  Let $\phi(v_iv_j) = (a_1,\ldots,a_n)$, $b_i:=a_ia_{n+1-i})=+$, and $\phi'(v_iv_j):=(b_1,\ldots,b_{\lceil n/2 \rceil})$.  Note that $\phi'$ is a nondirectional $n$-signing.  Then $\phi(C)$ is symmetrically balanced iff $\phi'$ is a balanced gain graph.  This contradicts Thm.\ 4.1.]
\dated{3 Feb 2014} 
\codes{\CGG(\CGen):~\CSG(\CGen):~\CBal(\CGen)}
\itemspace

\bsgitem{2013c}{}  Graphoidal signed graphs.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 23 (2013), no.\ 3, 451--460.
\MR{3113160}.  
\codes{\CSG}
\itemspace

\bsgitem{2013d}{}  Restricted super line signed graph $\cR\cL_r(S)$.  \emph{Notes Number Theory Discrete Math.}\ 19 (2013), no.\ 4, 86--92.
\annot{} \Cf\ \xref{Reddy and Vijay}{2012a}{P.\ Siva Kota Reddy \& S.\ Vijay}.  $\cR\cL_r(\Sigma)$ has vertex set $V(\cR\cL_r) = \cP_r(E)$, edge set $E(\cR\cL_r) = \{ PQ \in \cP_r(E) : \exists! \text{ adjacent } p\in P,\ q \in Q\}$, and signs $\sigma'(PQ) = \sigma(P)\sigma(Q)$.  Solved:  $\cR\cL_r(\Sigma)\simeq\cL_r(\Sigma)$.  Prop.\ 10, characterizing signed graphs of the form $\cR\cL_r(\Sigma)$, has a wrong proof and appears to be false.
\dated{30 Apr 2022} 
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\bsgitem{2014a}{}  Edge neighborhood signed graphs.  \emph{ Appl.\ Math.\ Sci.\ (Ruse)} 8 (2014), no.\ 29--32, 1473--1482.
\MR{3200146} (no rev).  
\codes{\CSG}%
\itemspace

\author{P.\ Siva Kota Reddy, U.K.\ Misra, \& P. S.\ Hemavathi}
\authorspace
\bsgitem{2014a}{}  Note on neighborhood signed graphs and jump signed graphs.  \emph{Adv.\ Studies Contemp.\ Math., Kyungshang} 24, no.\ 2, 217--226 (2014).
\codes{\CSG}%
\itemspace

\author{P.\ Siva Kota Reddy, U.K.\ Misra, \& P.N.\ Samanta}
\authorspace
\bsgitem{2013a}{}  The minimal equitable dominating signed graphs.  \emph{Scientia Magna} 9 (2013), no.\ 4, 64--70.
\annot{} The intersection graph of minimal equitable dominating sets $P$ of $|\Sigma|$, with $\sgn(PP')=\mu_\sigma(P)\mu_\sigma(P')$.  [Results are trivial or (Prop.\ 3.1) wrong.]  Dictionary: ``complement'' of $\Sigma$ is $(|\Sigma|^c,\mu_\sigma')$ where $\mu_\sigma'(uv)=\mu_\sigma(u)\mu_\sigma(v)$.
\dated{4 Oct 2019} 
\codes{\CSG,~\CBal:~\CDom}%
\itemspace

\author{P.\ Siva Kota Reddy, K.M.\ Nagaraja, \& M.C.\ Geetha}
\authorspace
\bsgitem{2012a}{}  The line $n$-sigraph of a symmetric $n$-sigraph.  IV.  \emph{Int.\ J.\ Math.\ Combin.}\ 2012 (2012), no.\ 1, 106--112.
Zbl{1272.05069}.  
\codes{\CSG(\CGen):~\CLG}%
\itemspace

\bsgitem{2014a}{}  The line $n$-sigraph of a symmetric $n$-sigraph.  V.  \emph{Kyungpook Math.\ J.}\ 54 (2014), no.\ 1, 95--101.
\MR{3190412}.  \Zbl{1295.05118}.  
\codes{\CSG(\CGen):~\CLG}%
\itemspace

\author{P.\ Siva Kota Reddy, K.M.\ Nagaraja, \& V.M.\ Siddalingaswamy}
\authorspace
\bsgitem{2015a}{}  Edge $C_k$ signed graphs.  \emph{Int.\ J.\ Pure Appl.\ Math.}\ 98 (2014), no.\ 2, 231--238.
\annot{} $E_k(\Sigma)$ has $V_k=E(\Gamma)$, $ef \in E_k$ iff $e,f$ are adjacent or opposite in a $C_k$, $\sigma_k(ef)=\sigma(e)\sigma(f)$.  Generalizes $k=3,4$ \xref{Subramanya--Reddy}{2009a}{M.S.\ Subramanya \& P.\ Siva Kota Reddy}, \xref{Rangarajan--Reddy}{2010a}{R.\ Rangarajan \& P.\ Siva Kota Reddy}.  Thm.\ 7:  $\Sigma \sim E_k(\Sigma)$ iff $\Sigma=$ balanced $C_n$, $n\geq5$.  Minor graph equations solved.
\dated{2 Jul 2019} 
\par
\codes{\CSG:~\CLG(\CGen):~\CBal,~\CSw}%
\itemspace

\author{P.\ Siva Kota Reddy \& Kavita S.\ Permi}
\authorspace
\bsgitem{2014a}{}  Signed graph equations:  $N(\Sigma) \sim CMD$; $CMD(\Sigma) \sim MD(\Sigma)$; $MD(\Sigma) \sim L(\overline\Sigma)$.  \emph{Bull.\ Int.\ Math.\ Virtual Institute}\ 4 (2014), 27--35.
\annot{} $N$ = neighborhood signed graph (\xref{Rangarajan, Subramanya, and Reddy}{2012a}{R.\ Rangarajan, M.S.\ Subramanya, \& P.\ Siva Kota Reddy}), $(C)MD$ = (common) minimal dominating signed graph (\xref{Reddy and Prashanth}{2012a}{P.\ Siva Kota Reddy \& B.\ Prashanth}, \xrefd{2013a}{P.\ Siva Kota Reddy \& B.\ Prashanth}).
\dated{9 Apr 2020} 
\codes{\CSG}\codesctd{\CSG:~\CDom}%
\itemspace

\author{P.\ Siva Kota Reddy, Kavita S.\ Permi, \& K.R.\ Rajanna}
\authorspace
\bsgitem{2012a}{}  Combinatorial aspects of a measure of rank correlation due to Kendall and its relation to complete signed digraphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2012 (2012), no.\ 1, 74--77.
\codes{\CSD:~\CAppl}%
\itemspace

\author{P.\ Siva Kota Reddy \& B.\ Prashanth}
\authorspace
\bsgitem{2009a}{}  Switching equivalence in symmetric $n$-sigraphs.  I.  \emph{Adv.\ Appl.\ Discrete Math.}\ 4 (2009), no.\ 1, 25--32.
\MR{2555623} (2010k:05122).  \Zbl{1176.05034}.					
\annot{} Continuation of \xref{Rangarajan, Reddy, and  Subramanya}{2009a}{R.\ Rangarajan, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Solved for an $n$-signed graph $\Phi$:  $\Lambda_S(\Phi) \simeq \Phi^c$;  $\Lambda_S^k(\Phi) \simeq \Phi^c$.  [The results remain true without assuming symmetry.]
\codes{\CSG(\CGen):~\CSw,~\CLG}%
\itemspace

\bsgitem{2012a}{}  The common minimal dominating signed graph.  \emph{Trans.\ Combin.}\ 1 (2012), no.\ 3, 39--46.
\codes{\CSG}%
\itemspace

\bsgitem{2012b}{}  $\cS$-Antipodal signed graphs.  \emph{Tamsui Oxford J.\ Inform.\ Math.\ Sci.}\ 28 (2012), no.\ 2, 165--174.
\codes{\CSG}%
\itemspace

\bsgitem{2013a}{}  A note on minimal dominating signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2013 (2013), no.\ 4, 96--102.
\codes{\CSG:~\CDom}%
\itemspace

\author{P.\ Siva Kota Reddy, B.\ Prashanth, \& T.R.\ Vasanth Kumar}
\authorspace
\bsgitem{2011a}{}  Antipodal signed digraphs.  
\emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 21 (2011), no.\ 4, 355--360.
\MR{2884999} (2012k:05172).  \Zbl{1250.05054}.  
\codes{\CSD}%
\itemspace

\author{P.\ Siva Kota Reddy, B.\ Prashanth, \& V.\ Lokesha}
\authorspace
\bsgitem{2010a}{}  Smarandachely $t$-path step signed graphs.  \emph{Scientia Magna} 6 (2010), no.\ 3, 89--92.
\annot{} [The name Smarandache is used for no reason.]
\codes{\CSG}%
\itemspace

\author{P.\ Siva Kota Reddy, B.\ Prashanth, \& Kavita S.\ Permi}
\authorspace
\bsgitem{2011a}{}  A note on antipodal signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2011 (2011), no.\ 1, 107--112.
\MR{2829740} (2012d:05173).  \Zbl{1230.05146}.					
\annot{} The antipodal graph $A$ of $|\Sigma|$ with $\sigma_A(uv)=\mu_\sigma(u)\mu_\sigma(v)$, hence balanced.  Solved:  $\Sigma \sim (A,\sigma_A)$, $-\Sigma \sim (A,\sigma_A)$.
[The name Smarandache is used for no reason.]
\dated{3 Sept 2019} 
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\bsgitem{2011b}{}  A note on switching in symmetric $n$-sigraphs.  \emph{Notes Number Theory Discrete Math.}\ 17 (2011), no.\ 3, 22--25. 
\Zbl{1250.05055}.  
\annot{} Switching multiple signs $\sigma(e) \in \{+,-\}^k$ by signs $\mu(v) \in \{+,-\}$.  [Equivalent to restricted switching, i.e., $\mu(v) \in \{\pm(+,\ldots,+)\}$.]  Characterized by cutset negation.
\dated{7 Jan 2011} 
\codes{\CSG(\CGen):~\CSw}%
\itemspace

\author{P.\ Siva Kota Reddy, B.\ Prashanth, \& M.\ Ruby Salestina}
\authorspace
\bsgitem{2010a}{}  Smarandachely antipodal signed digraphs.  \emph{Scientia Magna} 6 (2010), no.\ 3, 84--88.
\annot{} [The name Smarandache is used for no reason.]
\codes{\CSD}%
\itemspace

\author{P.\ Siva Kota Reddy, K.R.\ Rajanna, \& Kavita S.\ Permi}
\authorspace
\bsgitem{2013a}{}  The common minimal common neighborhood dominating signed graphs.  \emph{Trans.\ Combin.}\ 2 (2013), no.\ 1, 1--8.
\MR{3150480}.  \Zbl{1320.05057}.  
\codes{\CSG}%
\itemspace

\author{P.\ Siva Kota Reddy, R.\ Rangarajan, \& M.S.\ Subramanya}
\authorspace
\bsgitem{2011a}{}  Switching invariant neighborhood signed graphs.   \emph{Proc.\ Jangjeon Math.\ Soc.}\ 14 (2011), no.\ 2, 249--258.
\MR{2829740} (2012d:05173).  \Zbl{1238.05121}.					
\par
\codes{\CSG,~\CVS:~\CLG}%
\itemspace

\author{P.\ Siva Kota Reddy, R.\ Rajendra, \& M.C.\ Geetha}
\authorspace
\bsgitem{2016a}{}  Boundary $n$-signed graphs.  \emph{Int.\ J.\ Math.\ Sci.\ Engg.\ Appl.}\ 10 (2016), no.\ II, 161--168.
\codes{\CSG(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy, E.\ Sampathkumar, \& M.S.\ Subramanya}
\authorspace
\bsgitem{2010a}{}  Common-edge signed graph of a signed graph.  \emph{J.\ Indonesian Math.\ Soc.}\ 16 (2010), no.\ 2, 105--113.
\MR{2752773} (no rev).  \Zbl{1236.05098}.						
\codes{\CSG}%
\itemspace

\author{P.\ Siva Kota Reddy, K.\ Shivashankara, \& K.V.\ Madhusudhan}
\authorspace
\bsgitem{2010a}{}  Negation switching equivalence in signed graphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2010 (2010), no.\ 3, 85--90.
\Zbl{1238.05122}.  
\annot{} Solved: $-\Sigma, \ \Lambda_{\times}^k(\Sigma) \simeq \Lambda_{\times}^2(\Sigma)$, based on existing solutions for unsigned isomorphism.  (See \xref{M.\ Acharya}{2009a}{Mukti Acharya} for $\Lambda_{\times}$.)
\dated{6 Feb 2011} 
\par
\codes{\CSG:~\CLG,~\CSw}%
\itemspace

\author{P.\ Siva Kota Reddy \& M.S.\ Subramanya}
\authorspace

\bsgitem{2007a}{}  A characterization of symmetric 3-sigraphs whose line symmetric 3-sigraphs are switching equivalent.  \emph{J.\ Appl.\ Math.\ Anal.\ Appl.}\ 3 (2007), no.\ 1, 23--31.
\MR{2479512} (2009m:05080).  
\codes{\CSG(\CGen):~\CSw,~\CLG}%
\itemspace

\bsgitem{2008a}{}  Jump symmetric 3-sigraphs.  \emph{Int.\ J.\ Phys.\ Sci.}\ 20 (2008), no.\ 2, 431--434.
\par
\codes{\CSG(\CGen):~\CLG}%
\itemspace

\bsgitem{2009a}{}  Signed graph equation $L^ \Lap(S) \sim \bar S$.  \emph{Int.\ J.\ Math.\ Combin.}\ 4 (2009), 84--88 (2010).
\MR{2598675} (no rev).  \Zbl{1238.05123}.						
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Solved:  $\Sigma^c \simeq \Lambda_{\times}^2(\Sigma)$; $\Lambda_{\times}^k(\Sigma) \simeq \Sigma^c$.  
[$\Lambda_{\times}$ as in \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]
[Continued in \xref{Reddy, Vijay, and Lokesha}{2009a}{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha}, \xrefd{2010a}{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha}].
\dated{3 Aug 2009}
\codes{\CSG:~\CBal,~\CSw,~\CLG}%
\itemspace

\bsgitem{2009b}{}  Note on path signed graphs.  \emph{Notes Number Theory Discrete Math.}\ 15 (2009), no.\ 4, 1--6.
\annot{} $V(P_k(\Sigma)) := \{\text{paths}\}$, $PP' \in E(P_k(\Sigma))$ iff $P \cup P'$ is a path of order $k+1$ or a $C_k$, $\sigma(PP') = \sigma(P)\sigma(P')$.  This is balanced.  Solved:  $\Sigma \simeq P_3(\Sigma), P_4(\Sigma)$.
\dated{7 Jan 2011} 
\codes{\CSG:~\CLG(\CGen),~\CBal}%
\itemspace

\author{P.\ Siva Kota Reddy, M.S.\ Subramanya, \& R.\ Rajendra}
\authorspace
\bsgitem{2007a}{}  Symmetric 3-sigraphs.  \emph{J.\ Intelligent Systems Res.}\ 1 (2007), no.\ 2, 155--159.
\par
\codes{\CSG(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy, Gurunath Rao Vaidya, \& A.\ Sashi Kanth Reddy}
\authorspace
\bsgitem{2011a}{}  Neighborhood symmetric $n$-sigraphs.  \emph{Scientia Magna} 7 (2011), no.\ 2, 99--105.
\codes{\CSG(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy \& S.\ Vijay}
\authorspace
\bsgitem{2010a}{}  Total minimal dominating signed graph.  \emph{Int.\ J.\ Math.\ Combin.}\ 3 (2010), 11--16.
\Zbl{1238.05124}.  
\annot{} The intersection graph $M_t$ of all total minimal dominating sets of $|\Sigma|$ is signed to be balanced using the canonical vertex signature of $\Sigma$.  Such signed graphs are characterized.  $M_t \simeq \Sigma, -\Sigma$ are solved, based on existing solutions for unsigned isomorphism.
\dated{6 Feb 2011} 
\par
\codes{\CSG:~\CDom}%
\itemspace

\bsgitem{2011a}{}  A note on characterization of jump signed graphs.  \emph{Bull.\ Pure Appl.\ Math.}\ 5 (2011), no.\ 2, 276--277.
\MR{3027214} (no rev).  
\codes{\CSG:~\CLG}%
\itemspace

\bsgitem{2012a}{}  The super line signed graph $\cL_r(S)$ of a signed graph.  \emph{Southeast Asian Bull.\ Math.}\ 36 (2012), no.\ 6, 875--882.
\MR{3057818}.  
\annot{} $V(\cL_r(\Sigma)) := \cP_r(E)$ with edge $PQ_{e,f} \in E(\cL_r(\Sigma))$ for each adjacent $e \in P,\ f\in Q$ and $\sigma_\cL(PQ_{e,f}) = \sigma(P)\sigma(Q)$.  This is balanced.  Solved:  $\Sigma, \Lambda_{\times}(\Sigma) \simeq \cL_2(\Sigma)$, $\Sigma \cong \cL_2(\Sigma)$, \etal  [\Cf \xref{Reddy and Misra}{2013d}{P.\ Siva Kota Reddy \& U.K.\ Misra}.]
\dated{7 Jan 2011} 
\codes{\CSG:~\CLG(\CGen),~\CBal}%
\itemspace

\author{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha}
\authorspace
\bsgitem{2009a}{}  $n^{\text th}$ power signed graphs.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 12 (2009), no.\ 3, 307--313.
\MR{2582796} (2011e:05109).  \Zbl{1213.05121}.					
\annot{} Definitions and notation as in \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
\annotctd{$D_m$} The ``$m$th power signed graph'' $\Sigma^m$ [I will say ``$\leq m$-distance signed graph'' $D_m(\Sigma)$] is the graph of distance $\leq m$ in $|\Sigma|$ with signature $\sigma^c$.  
Prop.\ 5:  $\Sigma$ has the form $D_m(\Sigma')$ iff it is balanced and $|\Sigma|$ is a $({\leq}\, m)$-distance graph.  [Sufficiency is incorrect.]
Solved [possibly incorrectly]:  $\Sigma^c$ or $\Lambda_{\times}(\Sigma^c) \simeq D_m(\Lambda_{\times}(\Sigma))$;  $\Lambda_{\times}(\Sigma)^c \simeq D_m(\Sigma^c)$;  $\Lambda_{\times}^2(\Sigma) \simeq D_m(\Sigma),\ D_m(\Sigma)^c$, $D_m(\Sigma^c)$.  
[$\Lambda_{\times}$ as in \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]
\dated{12 Apr 2009}
\codes{\CSG:~\CBal,~\CSw,~\CLG}%
\itemspace

\bsgitem{2010a}{}  The $n^{th}$ power signed graphs.  II.  \emph{Int.\ J.\ Math.\ Combin.}\ 1 (2010), 74--79.
\MR{2662418} (no rev).  \Zbl{1207.05074}.					
\annot{} Continuing \xrefd{2009a}{P.\ Siva Kota Reddy, S.\ Vijay, \& V.\ Lokesha} with: $\Lambda_{\times}(\Sigma) \simeq D_m(\Sigma^{[c]})$; $\Lambda_{\times}(\Sigma)^c \simeq D_m(\Sigma)$.
\dated{10 Apr 2009}
\codes{\CSG:~\CBal,~\CSw,~\CLG}%
\itemspace

\author{P.\ Siva Kota Reddy, S.\ Vijay, \& Kavita S.\ Permi}
\authorspace
\bsgitem{2013a}{}  Note on equitable signed graphs.  \emph{Math.\ Aeterna} 3 (2013), no.\ 3, 169--177.
\MR{3056111} (no rev).  
\codes{\CSG}
\itemspace

\author{P.\ Siva Kota Reddy, S.\ Vijay, \& Kavita S Permi}
\authorspace
\bsgitem{2016a}{}  Radial signed graphs.  \emph{Math.\ Combin.\ Book Ser.}\ 4 (2016), 128--134.
\annot{} Radial graph of $\Sigma$ with certain balanced signs.
\dated{22 Jan 2020} 
\codes{\CSG:~\CBal,~\CSw,~\CVS}%
\itemspace

\author{P.\ Siva Kota Reddy, S.\ Vijay, \& B.\ Prashanth}
\authorspace
\bsgitem{2009a}{}  The edge $C_4$ $n$-sigraph of a symmetric $n$-sigraph.  \emph{Int.\ J.\ Math.\ Sci.\ Engg.\ Appl.}\ 3 (2009), no.\ 2, 21--27.
\codes{\CSG(\CGen):~\CLG(\CGen)}%
\itemspace

\author{P.\ Siva Kota Reddy, S.\ Vijay, \& H.C.\ Savithri}
\authorspace
\bsgitem{2010a}{}  A note on path signed digraphs.  \emph{Int.\ J.\ Math.\ Combin.}\ 2010 (2010), vol.\ 1, 42--46.
\MR{2662415} (no rev).  \Zbl{1203.05065}.						
\codes{\CSD:~\CLG(\CGen)}%
\itemspace

\author{Bruce Reed}							%REED B
\authorspace
\authref  See also \hyperlink{Claude Berge}{C.~Berge}, \hyperlink{Samuel Fiorini}{S.~Fiorini}, \hyperlink{Jim Geelen}{J.~Geelen}, \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi}, and \hyperlink{Dieter Rautenbach}{D.~Rautenbach}.
\authrefspace

\bsgitem{1999a}{}  Mangoes and blueberries.  \emph{ Combinatorica} 19 (1999), no.\ 2, 267--296.
\MR{1723044} (2000i:05160).  \Zbl{0928.05059}.  
\annot{} Given $k$, $\exists\ f(k)$ such that $\Gamma$ has a 2-packing of $2k$ odd circles, or $l_0(-\Gamma) \leq f(k)$.  [Simpler proof in \xref{Kawarabayashi and Reed}{2010a}{Ken-Ichi Kawarabayashi \& Bruce Reed}.]
\dated{29 Dec 2020} 
\codes{\Cpar:~\Cfr}%
\itemspace

\author{Bruce Reed, Kaleigh Smith, \& Adrian Vetta}
\authorspace
\bsgitem{2004a}{}  Finding odd cycle transversals.  \emph{Oper.\ Res.\ Lett.}\ 32 (2004), 299--301.
\MR{2057781} (no rev).  \Zbl{1052.05061}.  
\annot{} Algorithm, superseded by faster \xref{Kawarabayashi and Reed}{2010a}{Ken-Ichi Kawarabayashi \& Bruce Reed}.
\dated{29 Dec 2020} 
\codes{\Csg:~\Cpar:~\Cfr:~\CAlgor}%
\itemspace

\author{P.\ Reed} 							%REED P
\authorspace
\authref   See \hyperlink{A.J.\ Bray}{A.J.~Bray}.
\itemspace

\author{David Rees}							%REES
\authorspace
\bsgitem{1940a}{}  On semi-groups.  \emph{Proc.\ Cambridge Philos.\ Soc.}\ 36 (1940), no.\ 4, 387--400.
\MR{0002893} (2, 127g).  \Zbl{28.00401}	.					
\annot{} A fundamental paper introducing bipartite gain graphs and their switching automorphisms (all not recognized as such) into the foundations of semigroup theory.  
\Cf\ \xref{Graham}{1968a}{R.L.\ Graham}, \xref{Houghton}{1977a}{C.H.\ Houghton}, \xref{Pollatchek}{1977a}{Albert Pollatchek}, \xref{Rhodes and Steinberg}{2009a}{John Rhodes \& Benjamin Steinberg}.
\dated{21 Aug 2019} 
\par
\codes{\Cgg:~\Caut,~\Csw:~\CAlgeb}%
\itemspace

\author{Nathan Reff}							%REFF
\authorspace
\authref   See also \hyperlink{Francesco Belardo}{F.~Belardo}, \hyperlink{Luke Duttweiler}{L.~Duttweiler} and \hyperlink{Ouail Kitouni}{O.~Kitouni}.
\authrefspace

\bsgitem{2011a}{}  New bounds for the Laplacian spectral radius of a signed graph.  Manuscript, 2011.  
\arXiv{1103.4629}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2012a}{}  Spectral properties of complex unit gain graphs.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 9, 3165--3176.
\MR{2900705}.  \Zbl{1241.05085}.  \arXiv{1110.4554}.				
\annot{$\bbT$} Complex unit gain graphs have gain group $\bbT := \{z \in \bbC: |z|=1\}$.  Bounds on largest and smallest eigenvalues of $A(\Phi)$ (\S3, ``Eigenvalues of the adjacency matrix'') and $\Lap(\Phi)$ (\S4, ``Eigenvalues of the Laplacian matrix'').  Most (except Thm.\ 4.9, where the edge gains affect the bounds) generalize known bounds for graphs, the signless Laplacian $\Lap(-\Gamma)$, or signed graphs.  Some generalizations are not obvious.  
Lemmas 3.1, 4.1: The spectrum of $A$ or $\Lap$ depends only on the switching class.  Lemmas 3.2, 4.2:  If $\Phi$ is balanced, the spectra are the same as those of $\|\Phi\|$.  [\Problem.  Generalize \xref{B.D.\ Acharya}{1980a}{B.\ Devadas Acharya} by proving the converses.]  
Thm.\ 5.1:  Exact eigenvalues for circle graphs.  Thm.\ 5.4:  Lemma 3.1 of \xref{Hou, Li, and Pan}{2003a}{Yaoping Hou, Jiongsheng Li, \& Yongliang Pan} generalized to complex unit gain graphs.
\dated{30 Oct 2011, rev 20 Jan 2017} 
\codes{\CGG:~\CEig,~\CInc}%
\itemspace

\bsgitem{2012b}{}  \emph{Gain Graphs, Group-Oriented Hypergraphs, and Matrices}.  Doctoral dissertation, Binghamton University, 2012.
\MR{3078448} (no rev).  
\par
\codes{\CSG,~\CGG,~\CGH:~\CAdj,~\CLap,~\CEig,~\CInc}%
\itemspace

\bsgitem{2014a}{}  Spectral properties of oriented hypergraphs.  \emph{Electronic J.\ Linear Algebra} 27 (2014), no.\ 1, Art.\ 24, 373--391.
\MR{3218780}.  \Zbl{1327.05220}.  \arXiv{1506.05054}.				
\codes{\CSH:~\COri:~\CEig}%
\itemspace

\bsgitem{2016a}{}  Intersection graphs of oriented hypergraphs and their matrices.  \emph{ Australasian J.\ Combin.}\ 65 (2016), 108--123. 
\MR{3490209}.  \Zbl{1344.05103}.  \arXiv{1509.02132}.				
\codes{\CSH:~\COri}%
\itemspace

\bsgitem{2016b}{}  Oriented gain graphs, line graphs and eigenvalues.  \emph{Linear Algebra Appl.}\ 506 (2016), 316--328.
\MR{3530682}.  \Zbl{1346.05103}.  \arXiv{1506.05059}.				
\par
\codes{\CGG:~\COri:~\CEig,~\CLG}%
\itemspace

\author{Nathan Reff \& Lucas J.\ Rusnak}
\authorspace
\bsgitem{2012a}{}  An oriented hypergraphic approach to algebraic graph theory.  \emph{Linear Algebra Appl.}\ 437 (2012), no.\ 9, 2262--2270.
\MR{2954488}.  \Zbl{1247.05164}.  \arXiv{1506.05051}.				
\annot{} Walks, the adjacency matrix and its powers, and the incidence matrix and the Laplacian matrix $\Lap$ of an oriented hypergraph have the same relationships as with graphs.  
\dated{19 Oct 2012} 
\par
\codes{\CSH:~\COri:~\CInc,~\CAdj,~\CEig}%
\itemspace

\author{Nathan Reff \& Howard Skogman}
\authorspace
\bsgitem{2017a}{}  A connection between Hadamard matrices, oriented hypergraphs and signed graphs.  \emph{Linear Algebra Appl.}\ 529 (2017), 115--125.
\MR{3659795}.  \Zbl{1365.05184}.  
\codes{\CSG,~\CSH,~\CSw}%
\itemspace


\author{Damien Regnault}						%REGNAULT
\authorspace
\authref  See \hyperlink{Mathilde Noual}{M.~Noual}.
\itemspace

\author{F.\ Regonati}							%REGONATI
\authorspace
\authref  See \hyperlink{E.\ Damiani}{E.~Damiani}.
\itemspace

\author{Guus Regts}							%REGTS
\authorspace
\authref  See \hyperlink{Andrew Goodall}{A.~Goodall}.
\itemspace

\authorx{J\"org Reichardt}						%REICHARDT
\authorx{Jorg Reichardt}
\author{J\"org Reichardt \& Stefan Bornholdt}
\authorx{Jorg Reichardt \& Stefan Bornholdt}
\authorspace
\bsgitem{2006a}{}  Statistical mechanics of community detection.  \emph{Phys.\ Rev.\ E}\ 74 (2006), 016110, 14 pp.
\MR{2276596} (2007h:82089).  
\codes{\Csg:~\Ckg:~\CPhys,~\CClu}%
\itemspace

\author{Philip F.\ Reichmeider}						%REICHMEIDER
\authorspace
\bsgitem{1984a}{}  \emph{The Equivalence of Some Combinatorial Matching Theorems}.  Polygonal Publ.\ House, Washington, N.J., 1984.
\MR{0781348} (86j:05001).  \Zbl{562.05020}.						
\annot{} Thm.\ 7.6, p.\ 107, attributed to \xref{Hoffman}{1960a}{Alan J.\ Hoffman} (who credits \xref{Heller and Tompkins}{1956a}{I.\ Heller \& C.B.\ Tompkins}): in effect the incidence matrix of a balanced signed graph is totally unimodular.  K\"onig's and Hall's theorems are corollaries, per Hoffman.
\dated{8 Nov 2015} 
\codes{\Csg:~\Cinc,~\Cbal:~\CExp}%
\itemspace

\author{Talmage James Reid}						%REID TJ
\authorspace
\authref  See also \hyperlink{Torina Lewis}{T.~Lewis}.
\authrefspace

\author{Talmage James Reid \& Lee Inmon Virden}
\authorspace
\bsgitem{1997a}{}  On rounded five-element lines of matroids.  \emph{DiscreteMath.}\ 163 (1997), 119--127.
\MR{1428563} (97m:05068).  \Zbl{873.05032}.  
\annot{} The Dowling geometry $Q_3(\bbF_3^\times)$ is one of two crucial matroids.
\dated{9 Apr 2016} 
\codes{\CMatrd:~\CStr:~\Cgg}%
\itemspace

\author{Michael Reilly}						%REILLY
\authorspace
\authref  See \hyperlink{Oscar Coppola}{O.~Coppola}.
\itemspace

\author{Gerhard Reinelt} 						%REINELT
\authorspace
\authref   See \hyperlink{Francisco Barahona}{F.~Barahona}, \hyperlink{C.\ De Simone}{C.~De~Simone}, \hyperlink{Martin Grotschel}{M.~Gr\"otschel}, and \hyperlink{Frauke Liers}{F.~Liers}.
\itemspace

\author{Victor Reiner} 						%REINER
\authorspace
\authref   See also \hyperlink{Paul H.\ Edelman}{P.H.~Edelman}.
\authrefspace

\bsgitem{1993a}{}  Signed posets.  \emph{J.\ Combin.\ Theory Ser.\ A} 62 (1993), 324--360. 
\MR{1207741} (94d:06011).  \Zbl{773.06008}. 
\annot{} They are equivalent to acyclic bidirected graphs.  
\par
\codes{\CS,~\Csg:~\COri:~\CStr,~\Cgeom}%
\itemspace

\bsgitem{1997a}{}  Non-crossing partitions for classical reflection groups.  \emph{Discrete Math.}\ 177 (1997), 195--222.
\MR{1483446} (99f:06005).  \Zbl{892.06001}.  
\annot{} ``Noncrossing partitions [actually, partial partitions] of type $B$'' are elements of the Dowling lattice $Q_n(\{+,-\})$ that, regarded as sign-symmetric partitions of $\pm[n]$, are noncrossing when drawn on the circular arrangement $[1,2,\ldots,n,-1,-2,\ldots,-n]$.
	
\dated{28 Jan 2015} 
\codes{\Csg:~\CMatrd}%
\itemspace

\author{Victor Reiner \& Dennis Tseng}
\authorspace
\bsgitem{2014a}{}  Critical groups of covering, voltage and signed graphs.  \emph{Discrete Math.}\ 318 (2014), 10--40.
\MR{3141623}.  \Zbl{1281.05072}.  \arXiv{1301.2977}.				
\codes{\CGG,~\CSG:~\CCov}%
\itemspace

\author{Timothy C.\ Reluga}						%RELUGA
\authorspace
\authref  See \hyperlink{Assieh Saadatpour}{A.~Saadatpour}.
\itemspace

\authorx{Jeffrey B.\ Remmel}						%REMMEL*
\author{J.B.\ Remmel \& Michelle L.\ Wachs}
\authorspace
\bsgitem{2004a}{}  Rook theory, generalized Stirling numbers and $(p; q)$-analogues.  \emph{Electronic J.\ Combin.}\ 11 (2004), no.\ 1, art.\ \#R84, 48 pp.
\MR{2114188} (2005k:05007).  \Zbl{1065.05018}.  
\annot{} \S5, ``Permutation statistics, colored partitions, and restricted growth functions'':  Connection with Dowling lattices $Q_n(\fG)$.  E.g., their $S_{n,k}^{1,j}(1,1)$ $= W_{n-k+1}(Q_n(\bbZ_j))$.
\dated{28 May 2018} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Remna K P}							%REMNA
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\author{\'Elisabeth Remy}						%REMY
\authorx{Elisabeth Remy}
\authorspace
\authref  See also \hyperlink{Gilles Didier}{G.~Didier} and \hyperlink{Aurelien Naldi}{A.~Naldi}.
\authrefspace

\author{\'Elisabeth Remy \& Paul Ruet}
\authorx{Elisabeth Remy \& Paul Ruet}
\authorspace
\bsgitem{2007a}{}  On differentiation and homeostatic behaviours of Boolean dynamical systems. 
In:  Corrado Priami, ed., \emph{Transactions on Computational Systems Biology VIII}, pp.\ 92--101.
Lect.\ Notes in Bioinformatics.  \LNCS 4780.  Springer, Berlin, 2007. 
\MR{2303765}.  \Zbl{1141.92330}.						
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2008a}{}  [From elementary signed circuits to the dynamics of Boolean regulatory networks.]  Manuscript, 2008.
\annot{} \xrefd{2008b}{Elisabeth Remy \& Paul Ruet} under a previous title; often cited as such.
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2008b}{}  From minimal signed circuits to the dynamics of Boolean regulatory networks.  \emph{Bioinformatics} 24 (2008), i220--i226.
\codes{\CSD:~\CDyn}%
\itemspace

\author{\'Elisabeth Remy, Paul Ruet, \& Denis Thieffry}
\authorx{Elisabeth Remy, Paul Ruet, \& Denis Thieffry}
\authorspace
\bsgitem{2006a}{}  Positive or negative regulatory circuit inference from multilevel dynamics. 
In:  Christian Commault \etal, eds., \emph{Positive Systems} (Proceedings of the second multidisciplinary international symposium on positive systems: Theory and applications, POSTA 06, Grenoble, 2006), pp.\ 263--270. 
Lect.\ Notes in Control and Inform.\ Sci., Vol.\ 341.  Springer, Berlin, 2006. 
\MR{2250264} (2007e:92032).  \Zbl{1132.93305}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2008a}{}  Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. 
\emph{Adv.\ Appl.\ Math.}\ 41 (2008), no.\ 3, 335--350.
\MR{2449595} (2010j:37018).  \Zbl{1169.05333}.					
\codes{\CSD:~\CDyn}%
\itemspace

\author{Han Ren}							%REN H
\authorspace
\authref  See \hyperlink{Fayun Cao}{F.-Y.~Cao}.
\itemspace

\author{Ling-Zhi Ren}						%REN LZ
\authorspace
\authref  See \hyperlink{Yue Liu}{Y.~Liu}.
\itemspace

\author{Qing Jun Ren}						%REN QJ
\authorspace
\authref  See also \hyperlink{Hong Shan Du}{H.S.~Du}.
\authrefspace

\bsgitem{2001a}{}  A note on the quasi-Laplacian spectra of graphs.  (In Chinese.)  \emph{J.\ Nanjing Normal Univ.\ Nat.\ Sci.\ Ed.}\ 24 (2001), no.\ 2, 23--25.
\MR{1849157} (no rev).  \Zbl{984.05059}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Xiangyu Ren}						%REN XY
\author{Xiangyu Ren \& Xiaowang Li}
\authorspace
\bsgitem{2023a}{}  Note on a zero net-regular signed graph.  \emph{Discrete Appli.\ Math.}\ 327 (2023), 28--32.
\annot{} Thm.\ 3.1:  Connected $\Sigma$ has $d^\pm(v) \equiv 0$ iff $\exists$ alternating Eulerian tour.
\dated{28 Dec 2022} 
\codes{\CSG}%
\itemspace

\author{Xiangyu Ren \& Jianguo Qian}
\authorspace
\bsgitem{2019a}{}  Flow polynomials of a signed graph.  \emph{Electronic J.\ Combin.}\ 26 (2019), no.\ 3, art.\ 3.37, 16 pp.
\MR{4014607}.  \Zbl{1419.05095}.  \arXiv{1805.07878}.  
\annot{} Builds on \xref{DeVos, Rollov\'a, and \v{S}\'amal}{2019a}{Matt DeVos, Edita Rollova, \& Robert Samal}.
	
\codes{\CSG:~\CFlows:~\CInvar}%
\itemspace

\author{Xiangyu Ren, Jianguo Qian, Sumin Huang, \& Junxia Zhang}
\authorspace
\bsgitem{2022a}{}  The odd-valued chromatic polynomial of a signed graph.  \emph{Discrete Math.}\ 345 (2022), no.\ 11, art.\ 113042, 10 pp.
\MR{4440102}.  
\codes{\CSG:~\CHom,~\CCol:~\CInvar}%
\itemspace

\author{Raghunathan Rengaswamy}					%RENGASWAMY
\authorspace
\authref  See \hyperlink{Mani Bhushan}{M.~Bhushan} and \hyperlink{Mano Ram Maurya}{M.R.~Maurya}.
\itemspace

\authorx{Reshma R}							%RESHMA
\author{Reshma R, Gayathri H, \& Supriya Rajendra}	
\authorspace
\bsgitem{2021a}{}  On some parameters of parity signed graphs.  \emph{Turkish J.\ Computer Math.\ Educ.}\ 12 (2021), no.\ 13, 1992--1998.
\annot{} \Cf\ \xref{Acharya--Kureethara--Zaslavsky}{2021a}{Mukti Acharya, Joseph Varghese Kureethara, \& Thomas Zaslavsky}.  Additional examples, with partial proofs.
\dated{20 Jun 2021} 
\codes{\CLab:~\CSG}%
\itemspace

\authorx{Enrique Reyes}						%REYES
\author{Enrique Reyes, Christos Tatakis, \& Apostolos Thoma}
\authorspace
\bsgitem{2012a}{}  Minimal generators of toric ideals of graphs.  \emph{Adv.\ Appl.\ Math.}\ 48 (2012), 64--78.
\MR{2845507}.  \Zbl{1266.14041}.  
\annot{} Cor.\ 3.3 (restated):  A closed walk in $\Gamma$ is primitive iff its graph is a cactus tree in which each cutpoint is in exactly 2 blocks, the outer blocks are odd circles, and the inner blocks are even circles or isthmi.  [Generalization:  A closed walk in $\Sigma$ is primitive iff its graph is a cactus tree in which each cutpoint is in exactly 2 blocks, the outer blocks are negative circles, and the inner blocks are positive circles or isthmi.  \Problem: Prove it.]  
\annot{} Thm.\ 4.13:  Characterizes closed walks that give a minimal binomial of the ideal.  
\annot{} Thm.\ 4.16 (restated):  A closed walk in $\Gamma$ is fundamental iff it is a minimal walk of an even circle $C$ with at most one chord, which creates two odd circles (i.e., $\Gamma{:}V(C)$ contains no other even circle, equivalently no other frame circuit), or of a handcuff circuit, of $\bfF(-\Gamma)$.  [Generalization:  A closed walk in $\Sigma$ is fundamental iff it is a minimal walk of an positive circle $C$ with at most one chord, which creates two negative circles (i.e., $\Sigma{:}V(C)$ contains no other positive circle, equivalently no other frame circuit), or of a handcuff circuit, of $\bfF(\Sigma)$.  \Problem: Prove it.]
\dated{22 Oct 2020} 
\codes{\Cecyc:~\CAlgeb,~\Cincid}%
\itemspace

\author{Josephine Reynes}						%REYNES
\authorspace
\authref  See \hyperlink{Will Grilliette}{W.~Grilliette} and \hyperlink{Lucas Rusnak}{L.J.~Rusnak}.
\itemspace

\author{Rezvaneh Rezapour}						%REZAPOUR
\authorspace
\authref  See \hyperlink{Ly Dinh}{L.~Dinh}.
\itemspace

\author{Brendon Rhoades}						%RHOADES
\authorspace
\authref  See \hyperlink{Drew Armstrong \& Brendon Rhoades}{D.~Armstrong} and \hyperlink{Emily Leven}{E.~Leven}.
\itemspace

\author{John Rhodes}						%RHODES
\authorspace
\authref  See also \hyperlink{Stuart Margolis}{S.~Margolis}.
\authrefspace

\author{John Rhodes \& Benjamin Steinberg}
\authorspace
\bsgitem{2009a}{}  \emph{The $\mathfrak{q}$-Theory of Finite Semigroups}.  Springer Monographs in Math.  Springer, Boston, 2009.
\MR{2472427} (2010h:20132).  \Zbl{1186.20043}.  
\annot{} \S4.13.1, ``Topology of labeled graphs''.  \S4.13.2.  \Cf\ \xref{Rees}{1940a}{David Rees}, \xref{Graham}{1968a}{R.L.\ Graham}, \xref{Houghton}{1977a}{C.H.\ Houghton}, \xref{Pollatchek}{1977a}{Albert Pollatchek}.
\par
\codes{\Cgg:~\Csw,~\Caut:~\CAlgeb:~\CExp}%
\itemspace

\author{Ricardo Riaza}						%RIAZA
\authorspace
\bsgitem{2015a}{}  Some qualitative problems in network dynamics.  Manuscript, 2015.
\arXiv{1501.01904}.  
\annot{} Preliminary version of \xrefd{2017a}{Ricardo Riaza}.  \S4.5, ``Signed graphs''.
\codes{\CSG:~\CDyn}%
\itemspace

\bsgitem{2017a}{}  Structure and stability of the equilibrium set in potential-driven flow networks.  \emph{J.\ Math.\ Anal.\ Appl.}\ 454 (2017), no.\ 1, 335--360.
\MR{3649856}.  \Zbl{1373.37196}.  
\annot{} \S5.1, ``Signed graphs''.
\codes{\CSG:~\CDyn}%
\itemspace

\author{M.\ Riazi}							%RIAZI
\authorspace
\authref  See \hyperlink{A.\ Kargaran}{A.~Kargaran}.
\itemspace

\author{Federico Ricci-Tersenghi}					%RICCI-TERSENGHI
\authorspace
\authref  See \hyperlink{Alexander K.\ Hartmann}{A.K.~Hartmann}.
\itemspace

\author{Adrien Richard}						%RICHARD
\authorspace
\authref  See also \hyperlink{Julio Aracena}{J.~Aracena}, \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, and \hyperlink{Loic Pauleve}{L.~Paulev\'e}.
\authrefspace

\bsgitem{2009a}{}  Positive circuits and maximal number of fixed points in discrete dynamical systems.  
\emph{Discrete Appl.\ Math.}\ 157 (2009), no.\ 15, 3281--3288.
\MR{2554797} (2010m:\-05132).  \Zbl{1193.05085}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2010a}{}  Negative circuits and sustained oscillations in asynchronous automata networks.  \emph{Adv.\ Appl.\ Math.}\ 44 (2010), 378--392.
\MR{2600786} (2011a:92007).  \Zbl{1201.37117}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2011a}{}  Local negative circuits and fixed points in non-expansive Boolean networks.  \emph{Discrete Appl.\ Math.}\ 159 (2011), no.\ 11, 1085--1093.
\MR{2794310} (2012d:94121).  \Zbl{1253.94082}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2015a}{}  Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks.  \emph{Theor.\ Computer Sci.}\ 583 (2015), 1--26.
\MR{3337943}.  \Zbl{1310.94244}.  
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2018a}{}  Fixed points and connections between positive and negative cycles in Boolean networks.  \emph{Discrete Appl.\ Math.}\ 243 (2018), 1--10.
\MR{3804734}.  \Zbl{}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Adrien Richard \& Jean-Paul Comet}
\authorspace
\bsgitem{2007a}{}  Necessary conditions for multistationarity in discrete dynamical systems.  
\emph{Discrete Appl.\ Math.}\ 155 (2007), no.\ 18, 2403--2413.
\MR{2365052} (2008m:37032).  \Zbl{1125.37062}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{2011a}{}  Stable periodicity and negative circuits in differential systems.  \emph{J.\ Math.\ Biol.}\ 63 (2011), no.\ 3, 593--600.
\MR{2824982} (2012h:34096).  \Zbl{1311.34096}.  
Erratum.  \emph{Ibid.}\ 70 (2015), no.\ 4, 957--958.
\MR{3306622}.  \Zbl{1311.34097}.  
\par
\codes{\CSD:~\CDyn}%
\itemspace

\author{Adrien Richard, Jean-Paul Comet, \& Gilles Bernot}
\authorspace
\bsgitem{2005a}{}  R.\ Thomas' modeling of biological regulatory networks: Introduction of singular states in the qualitative dynamics.  \emph{Fundamenta Inform.}\ 65 (2005), 373--392.
\MR{2348098} (no rev).  \Zbl{1092.92001}.  
\annot{} ``Regulatory network'' = signed digraph.
	
\codes{\CSD:~\CGen:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2005b}{}  Graph-based modeling of biological regulatory networks: Introduction of singular states.  In:  Vincent Danos \etal., eds., \emph{Computational methods in systems biology} (Int.\ Conf.\ CMBS 2004, Paris), pp.\ 58--72.  \LNCS 3082.  Springer, Berlin, 2005.
\MR{2278705}.  \Zbl{1088.93506}.  
\codes{\CSD:~\CGen:~\CDyn,~\CBiol}%
\itemspace

\author{Adrien Richard \& Paul Ruet}
\authorspace
\bsgitem{2013a}{}  From kernels in directed graphs to fixed points and negative cycles in Boolean networks.  \emph{Discrete Appl.\ Math.}\ 161 (2013), no.\ 7-8, 1106--1117.
\MR{3030595}.  \Zbl{1315.05068}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Jean Richelle}						%RICHELLE
\authorspace
\authref  See also \hyperlink{Denis Thieffry}{D.~Thieffry} and \hyperlink{Rene Thomas}{R.~Thomas}.
\authrefspace

\bsgitem{1979a}{}  Comparative analysis of negative loops by continuous, boolean and stochastic approaches.  
In:  Ren\'e Thomas, ed., \emph{Kinetic Logic:  A Boolean Approach to the Analysis of Complex Regulatory Systems} (Proc.\ EMBO Course, Brussels, 1977), Ch.\ XIV, pp.\ 281--325.  Lect.\ Notes in Biomath., Vol.\ 29.  Springer-Verlag, Berlin, 1979. 
\annot{} The dynamics of a negative cycle differ under different models: continuous (differential equations), boolean (2-valued states), and stochastic, with or without the time delays found in biology.
\dated{4 Aug 2018} 
\codes{\Csd:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{1980a}{}  Analyses bool\'eenne et continue de syst\`emes comportant des boucles de r\'etro\-action.  I.\ Analyse continue des boucles de r\'etroaction simples.  (In French.)  \emph{Acad.\ Roy.\ Belg.\ Bull.\ Cl.\ Sci.}\ (5) 66 (1980), no.\ 11, 890--912.
\MR{0637745} (84h:92020a).  \Zbl{493.92003}.					
\annot{} A positive signed digraph cycle allows a stationary state while a negative cycle allows multiple stationary states.  
Part II: \xref{Leclercq and Thomas}{1981a}{J.\ Leclercq \& R.\ Thomas}.
\dated{10 Jul 2022} 
\codes{\Csd:~\CDyn:~\CBiol}%
\itemspace

\author{Daniel J.\ Richman}						%RICHMAN
\authorspace
\authref   See \hyperlink{John S.\ Maybee}{J.S.~Maybee}.
\itemspace

\author{R.J.\ Riddell}							%RIDDELL
\authorspace
\bsgitem{1951a}{}  \emph{Contributions to the Theory of Condensation}.  Dissertation, University of Michigan, Ann Arbor, 1951.
\annot{} Includes the number of labelled simple $1$-trees of order $n$, i.e., bases of bicircular matroid $\bfF(K_n,\eset)$.  [Sequence A057500 in N.J.A.\ Sloane, \emph{The On-Line Encyclopedia of Integer Sequences}, URL \url{http://oeis.org/A057500}.               
[\Cf\ \xref{Neudauer, Meyers, and Stevens}{2001a}{Nancy Ann Neudauer, Andrew M.\ Meyers, \& Brett Stevens}.]
\par
\codes{\Cbic:~\CInvar(\Cbases)}%
\itemspace

\author{Heiko Rieger}						%RIEGER
\authorspace
\authref  See also \hyperlink{M.J.\ Alava}{M.J.~Alava}, \hyperlink{Barbara Coluzzi}{B.~Coluzzi}, \hyperlink{Alexander K.\ Hartmann}{A.K.~Hartmann}, \hyperlink{N.\ Kawashima}{N.~Kawashima}, and \hyperlink{J.D.\ Noh, H.\ Rieger, M.\ Enderle, \& K.\ Knorr}{J.D.~Noh}.
\authrefspace

\bsgitem{1998a}{}  Frustrated systems: Ground state properties via combinatorial optimization.  In:  J\'anos Kert\'esz and Imre Kondor, eds., \emph{Advances in Computer Simulation} (Budapest, 1996), pp.\ 122--158.  Lect.\ Notes in Phys., Vol.\ 501.  Springer-Verlag, Berlin, 1998.
\Zbl{898.65089}.  \arXiv{cond-mat/9705010}.  
\par
\codes{\CPhys,~\Csg:~\CState(\Cfr),~\CAlgor:~\CExp}%
\itemspace

\author{Robert G.\ Rieper}						%RIEPER
\authorspace
\authref   See \hyperlink{Jianer Chen}{J.~Chen}.
\itemspace

\author{M.J.\ Rigby}							%RIGBY
\authorspace
\authref   See \hyperlink{A.C.\ Day, R.B.\ Mallion, \& M.J.\ Rigby}{A.C.~Day}.
\itemspace

\authorx{Simone Righi}						%RIGHI
\author{Simone Righi \& K\'aroly Tak\'acs}
\authorx{Simone Righi \& Karoly Takacs}
\authorspace
\bsgitem{2013a}{}  Signed networks, triadic interactions and the evolution of cooperation.  In:  Alex Graudenzi \etal, eds., \emph{Proceedings Wivace 2013 - Italian Workshop on Artificial Life and Evolutionary Computation} (Wivace 2013, Milan, 2013), pp.\ 104--107.  Electronic Proc.\ Theor.\ Computer Sci., Vol.\ 130, 2013.
\arXiv{1309.7698}.  
\annot{}  Extended abstract.  See \xrefd{2014b}{Simone Righi \& Karoly Takacs}.
\codes{\CSG:~\CDyn,~\CPsS}%
\itemspace

\bsgitem{2014a}{}  Triadic balance and closure as drivers of the evolution of cooperation.  In:  F.J.\ Miguel Quesada \etal, eds., \emph{Advances in Computational Social Science and Social Simulation} (Barcelona, 2014), 4 pp.  Social Simulation Conference, Bellaterra, Catalu\~na, 2014.
URL \url{http://ddd.uab.cat/record/125597}
\annot{} Extended abstract.  See \xrefd{2014b}{Simone Righi \& Karoly Takacs}.
\codes{\CSG:~\CDyn,~\CPsS}%
\itemspace

\bsgitem{2014b}{}  Emotional strategies as catalysts for cooperation in signed networks.  \emph{Adv.\ Complex Systems} 17 (2014), no.\ 2, art.\ 1450011, 23 pp.  
\MR{3224041}.  \arXiv{1401.1996}.  
\annot{} Random pairs of adjacent nodes (= persons) of $\Sigma$ play Prisoner's Dilemma.  Each node is a Cooperator, Defector, or Conditional player who cooperates iff the edge is positive.  $\Sigma$ and (in some papers) the node strategies evolve depending on the outcome of each round, which is either a single play (dyadic) or a round robin in a triangle (triadic).  Some conditions evolve into universal cooperation, some into universal defection; conditions for each outcome are explored in these articles.
\dated{8 Jan 2016} 
\codes{\CSG:~\CDyn,~\CPsS}%
\itemspace

\bsgitem{2014c}{}  Parallel versus sequential update and the evolution of cooperation with the assistance of emotional strategies.  Manuscript, 2014.
\arXiv{1401.4672}.  
\annot{} See \xrefd{2014b}{Simone Righi \& Karoly Takacs}.
\codes{\CSG:~\CDyn,~\CPsS}%
\itemspace

\bsgitem{2014d}{}  Degree variance and emotional strategies catalyze cooperation in dynamic signed networks.  In:   Flaminio Squazzoni \etal, eds., \emph{Proceedings 28th European Conference on Modelling and Simulation} (ECMS 2014, Brescia), 7 pp.  European Council for Modelling and Simulation (ECMS), 2014.
\arXiv{1404.6097}.  
\annot{} See \xrefd{2014b}{Simone Righi \& Karoly Takacs}.
\codes{\CSG:~\CDyn,~\CPsS}%
\itemspace

\author{[Arnout van de Rijt]}						%RIJT
\authorspace
\authref   See \hyperlink{Arnout van de Rijt}{A.~van~de~Rijt} (under `V').
\itemspace

\author{James E.\ Riley}						%RILEY
\authorspace
\bsgitem{1969a}{}  An application of graph theory to social psychology.  In: G.\ Chartrand and S.F.\ Kapoor, eds., \emph{The Many Facets of Graph Theory} (Proc., Kalamazoo, 1968), pp.\ 275--280.  \LNM 110.  Springer, 1969.
\MR{0252266} (40 \#5487).  \Zbl{198.52802}.  					
\codes{\CPsS:~\CSG:~\CExp}%
\itemspace

\author{Chong S.\ Rim} 						%RIM
\authorspace
\authref   See \hyperlink{Hyeong-ah Choi}{H.~Choi}.
\itemspace

\author{Giovanni Rinaldi}						%RINALDI
\authorspace
\authref  See \hyperlink{C.\ De Simone}{C.~De~Simone}, \hyperlink{Frauke Liers}{F.~Liers}, and \hyperlink{J.\ Lukic}{J.~Lukic}.
\itemspace

\author{R.D.\ Ringeisen} 						%RINGEISEN
\authorspace
\authref   See also \hyperlink{Marc J.\ Lipman}{M.J.~Lipman}.
\authrefspace

\bsgitem{1974a}{} Isolation, a game on a graph.  \emph{Math.\ Mag.}\ 47 (May, 1974), no.\ 3, 132--138.  
\MR{0373642} (51 \#9842).  
\annot{}  Mathematical examination of a game based on switching vertices until a player creates an isolated vertex and wins.
\dated{26 May 2017}
\par
\codes{\Ctg:~\CSw}%
\itemspace

\author{Gerhard Ringel}						%RINGEL*
\authorspace
\authref   See also \hyperlink{Nora Hartsfield}{N.~Hartsfield} and \hyperlink{Mark Jungerman}{M.~Jungerman}.
 \authrefspace

\bsgitem{1974a}{}  \emph{Map Color Theorem}.  Grundl.\ math.\ Wiss., B.\ 209.  Springer-Verlag, Berlin, 1974.  
\MR{0349461} (50 \#1955).  \Zbl{287.05102}.
\annot{} ``Cascades'' (\S 8.3): see \xref{Youngs}{1968a}{J.W.T.\ Youngs}.  
\annotctd{} [Nonorientable embedding of $K_n$ means an orientation embedding (\cf\ \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}) of some signed $K_n$.  \Question\ (minor).  What signs?  For $K_{16}$, $\Sigma^- = K_{8,8}$ with $C_8$ in each side of the bipartition (found with \hyperlink{Shengxiang Lyu}{S.-X.~Lyu}).]  \dated{8 Oct 2016}
\codes{\Csg:~\COri:~\CAppl}%
\authrefspace

\bsgitem{1974b}{}  \emph{Teorema o Raskraske Kart}.  Transl.\ V.B.\ Alekseev.  Ed.\ G.P.\ Gavrilov.  ``Mir'', Moskva, 1977.  
\MR{0465923} (57 \#5809).  \Zbl{439.05019}.
\annot{} Russian translation of \xrefd{1974a}{Gerhard Ringel}. 
\codes{\Csg:~\COri:~\CAppl}%
\itemspace

\bsgitem{1977a}{}  The combinatorial map color theorem.  \emph{J.\ Graph Theory} 1 (1977), 141--155.  
\MR{0444509} (56 \#2860).  \Zbl{386.05030}.
\annot{} Signed rotation systems for graphs.  Thm.\ 12: Signed rotation systems describe all cellular embeddings of a graph; an embedding is orientable iff its signature is balanced.  \Cf\ \xref{Stahl}{1978a}{Saul Stahl}.  Dictionary:  ``Triple'' = graph with signed rotation system.  ``Orientable'' triple = balanced signature.  ``Oriented'' = all positive.
\codes{\CSG:~\CTop,~\CSw}%
\itemspace

\author{Oliver Riordan}						%RIORDAN
\authorspace
\authref  See \hyperlink{Bela Bollobas}{B.~Bollob\'as}.
\itemspace

\author{S.\ Risau-Gusman}						%RISAU-GUSMAN
\authorspace
\authref  See \hyperlink{F.\ Roma}{F.~Rom\'a}.
\itemspace

\author{F.\ Ritort}							%RITORT
\authorspace
\authref  See \hyperlink{Enzo Marinari}{E.~Marinari}.
\itemspace

\author{Vincent Rivasseau}						%RIVASSEAU
\authorspace
\authref  See \hyperlink{Thomas Krajewski}{T.~Krajewski}.
\itemspace

\author{Nicolas Rivier}						%RIVIER
\authorspace
\bsgitem{1990a}{}  Gauge theory and geometry of condensed matter.  In:  J.F.\ Sadoc, ed., \emph{Geometry in Condensed Matter Physics}, Ch.\ I, pp.\ 1--88.  World Scientific, Singapore, 1990.
\MR{1090337} (no rev).  
\annot{} Engaging if physics-intensive exposition of continuous and discrete gauge transformations via fiber bundles.  E.g., mixed Ising models (= signed graphs $\Sigma$) with discrete $\pm1$ or continuous $S^3$ spins.  ``Odd ring'' defects (p.\ 21) treated via all-negative graph $-\Gamma$; ``odd lines'' (pp.\ 22 ff.)\ = negative paths.  $\bbZ_2$ is implicit in sign of gauge-invariant physical configuration (\S3.3), explicit in magnetization (p.\ 38).  \S4, ``Gauge invariance in discrete space''.  
\S4.1, ``Discrete gauge invariance in spin glasses'':  Ising spin glass = general $\Sigma$.  $\bbZ_2$ gauge transformation (switching) ``cannot be meaningfully generalized to \dots\ XY or Heisenberg \dots\ spins [\Question.  Is that true?] because [signs $\pm1$] are real numbers''.  
\S4.2.1, ``Potential valleys in configuration space'':  ``The configuration space is a direct product of \dots\ odd [= negative?]\ line[s].''  ``Tunnelling'' between valleys (regions closer to ground state potential).  
\S4.2.2, ``Elasticity of random networks'':  Fiber bundle = covering graph of permutation gain graph with $\fS_N$ action.  Fig.\ 6: Signed graph on torus, cut along negative edges to become planar all-positive.  
\S5.2, ``Theory of surfaces'': \P\P1--3 must be read.
\S7, ``Conclusions'':  Ising spin glass (i.e., $\Sigma$), p.\ 82.  
\dated{7 Aug 2018} 
\codes{\CSG,~\Cgg:~\CVS:~\CPhys,~\CBal,~\CFr:~\CExp}%
\itemspace

\author{Romeo Rizzi}						%RIZZI
\authorspace
\bsgitem{2001a}{}  On 4-connected graphs without even cycle decompositions.  \emph{Discrete Math.}\ 234 (2001), 181--186.
\MR{1826832} (2001m:05156).  \Zbl{983.05067}.  
\annot{} On decomposing $-\Gamma$ into positive circles.  \Cf\ \xref{C.-Q.~Zhang}{1994a}{Cun-Quan Zhang}, \xref{Markstr\"om}{2012a}{Klas Markstrom}, \xref{M\'a\v{c}ajov\'a and Maz\'ak}{2013a}{Edita Macajova \& Jan Mazak}, \xref{Liu, Cui, and Wang}{2022a}{Wenzhong Liu, Qing Cui, \& Yunzhuo Wang}, etal
\dated{13 Aug 2013} 
\codes{\CPar:~\CStr:~\CCircles}%
\itemspace

\author{Mar{\'\i}a Robbiano}						%ROBBIANO
\authorx{Maria Robbiano}
\authorspace
\authref  See also \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~de~Abreu} and \hyperlink{Ivan Gutman}{I.~Gutman}.
\authrefspace

\author{Mar\'{\i}a Robbiano, Katherine Tapia Morales, \& Bernardo San Mart\'{\i}n}
\authorx{Maria Robbiano, Katherine Tapia Morales, \& Bernardo San Martin}
\authorspace
\bsgitem{2016a}{}  Extremal graphs with bounded vertex bipartiteness number.  \emph{Linear Algebra Appl.}\ 493 (2016), 28--36.
\MR{3452724}.  \Zbl{1329.05198}.  
\annot{} They find simple $\Gamma$ with ``vertex bipartiteness number'' $l_0(-\Gamma)\leq k$ that maximize  spectral radii of $A(-\Gamma), \Lap(-\Gamma)$.  [\Problem:  Generalize to frustration number of $\Sigma$.]  
[\Cf\ \xref{Liu and Wang}{20xxa}{Jia-Bao Liu \& Shaohui Wang}.]
\dated{8 May 2017} 
\codes{\Csg:~\CPar:~\CFr,~\CEig}%
\itemspace

\author{Jakayla R.\ Robbins}						%ROBBINS J
\authorspace
\bsgitem{2003a}{}  \emph{On Orientations of the Free Spikes}.  Doctoral dissertation, University of Kentucky, 2003.
\MR{2704379} (no rev).  
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2007a}{}  Enumerating orientations of the free spikes.  \emph{European J.\ Combin.}\  28 (2007), 868--875.
\MR{2300767} (2007k:05047).  \Zbl{1112.05022}.					
\annot{} The number of orientations of the free spike matroid $\bfL(2C_n,\eset)$ is $2^{n-1}D_n$, $D_n :=$ Dedekind number.
\dated{29 Sept 2011} 
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2008a}{}  Representable orientations of the free spikes.  \emph{Discrete Math.}\ 308 (2008), no.\ 22, 5174--5183.
\MR{2450452} (2009h:05052).  \Zbl{1157.05020}.					
\annot{} In general, not all orientations of  the free spike matroid $\bfL(2C_n,\eset)$ have a real vector representation.  Also, bounds on the number of representable orientations.
\dated{29 Sept 2011} 
\codes{\Cgg:~\CMatrd:~\CGeom,~\CInvar}%
\itemspace

\author{Jakayla Robbins, Daniel Slilaty, \& Xiangqian Zhou}
\authorspace
\bsgitem{2016a}{}  Clones in 3-connected frame matroids.  \emph{Discrete Math.}\ 339 (2016), 1329--1334.
\MR{3442541}.  \Zbl{1329.05060}.  
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Fred S.\ Roberts} 						%ROBERTS F
\authorspace
\authref   See also \hyperlink{Thomas A.\ Brown}{T.A.~Brown} and \hyperlink{Robert Z.\ Norman}{R.Z.~Norman}.
\authrefspace

\bsgitem{1974a}{} Structural characterizations of stability of signed digraphs under pulse processes.  
In: Ruth A.\ Bari and Frank Harary, eds., \emph{Graphs and Combinatorics} (Proc.\ Capital Conf., Geo.\ Washington Univ., 1973), pp.\ 330--338.  \LNM 406.
Springer-Verlag, Berlin, 1974.  
\MR{0360342} (50 \#12792).  \Zbl{302.05107}.
\codes{\CSDw}%
\itemspace

\bsgitem{1976a}{}  \emph{Discrete Mathematical Models, With Applications to Social, Biological, and Environmental Problems}.  
Prentice-Hall, Englewood Cliffs, N.J., 1976.  
\Zbl{363.90002}.  
\annot{} \S3.1:  ``Signed graphs and the theory of structural balance.''  Many topics are developed in the exercises.  Exercise 4.2.7 (from \xref{Phillips}{1967a}{J.L.\ Phillips}).
\par
\codes{\CSG,~\CSD:~\CBal,~\CAlgor,~\CAdj,~\CClu,~\CFr,~\CPsS:~\CExp,~\CExr}%
\annotctd{} Ch.\ 4: ``Weighted digraphs and pulse processes.''  Signed digraphs here are treated as unit-weighted digraphs.  Note esp.:  \S4.3: ``The signed or weighted digraph as a tool for modelling complex systems.''  Conclusions about models are drawn from very simple properties of their signed digraphs.  \S4.4: ``Pulse processes.''  \S4.5: ``Stability in pulse processes.''  Stability is connected to eigenvalues of $A(\Sigma)$.
\par
\codes{\CSDw,~\CSD,~\CWD:~\CBal,~\CEig,~\CPsS:~\CExp,~\CExr,~\CRef}%
\itemspace

\bsgitem{1978a}{} \emph{Graph Theory and Its Applications to Problems of Society}.  
CBMS-NSF Regional Conf.\ Ser.\ in Appl.\ Math., 29.  Soc.\ Indust.\ Appl.\ Math., Philadelphia, 1978.  
\MR{0508050} (80g:90036).  \Zbl{452.05001}.
\annot{} Ch.\ 9:  ``Balance theory and social inequalities.''  Ch.\ 10: ``Pulse processes and their applications.''  Ch.\ 11:  ``Qualitative matrices.''
\par
\codes{\CSG,~\CSD,~\CSDw:~\CBal,~\CPsS,~\CQM:~\CExp,~\CRef}%
\itemspace

\bsgitem{1979a}{}  Graph theory and the social sciences.  
In: Robin J.\ Wilson and Lowell W.\ Beineke, eds., \emph{Applications of Graph Theory}, Ch.\ 9, pp.\ 255--291.  Academic Press, London, 1979.  
\MR{0567125} (81h:05050) (book).  \Zbl{444.92018}.
\annot{} \S2: ``Balance and clusterability.''  Basics in brief.  \S7: ``Signed and weighted digraphs as decision-making models.''  Cursory.
\par
\codes{\CSG,~\CPsS,~\CSD,~\CSDw:~\CBal,~\CClu,~\CKG:~\CExp,~\CRef}%
\itemspace

\bsgitem{1986a}{}  \emph{Diskretnye matematicheskie modeli s prilozheniyami k sotsialnym, biologicheskim i ekologicheskim zadacham}.  
Transl.\ A.M.~Rappoport and S.I.~Travkin.  Ed.\ and preface by A.I.~Te\u\i man.  Teoriya i Metody Sistemnogo Analiza [Theory and Methods of Systems Analysis].  ``Nauka'', Moscow, 1986.  
\MR{0889897} (88e:00020).  \Zbl{662.90002}.
\annot{} Russian edition of \xrefd{1976a}{Fred S.\ Roberts}.
\par
\codes{\CSG,~\CSD:~\CBal,~\CAlgor,~\CAdj,~\CClu,~\CFr,~\CPsS:~\CExp,~\CExr}%
\par
\codes{\CSDw,~\CSD,~\CWD:~\CBal,~\CEig,~\CPsS:~\CExp,~\CExr,~\CRef}%
\itemspace

\bsgitem{1989a}{}  Applications of combinatorics and graph theory to the biological and social sciences:  Seven fundamental ideas.  
In: Fred Roberts, ed., \emph{Applications of Combinatorics and Graph Theory to the Biological and Social Sciences},  pp.\ 1--37.  IMA Vols.\ Math.\ Appl., Vol.\ 17.  Springer-Verlag, New York, 1989.  
\MR{1009370} (91c:92001).  
\annot{} \S4: ``Qualitative stability.''  A fine, concise basic survey.
\par
\codes{\CQM:~\CSD:~\CExp,~\CRef}%
\annotctd{} \S5:  ``Balanced signed graphs.''  Another concise basic survey, and two open problems (p.\ 20). 
\codes{\CSG:~\CBal:~\CExp,~\CRef}%
\itemspace

\bsgitem{1995a}{}  On the problem of consistent marking of a graph.  \emph{Linear Algebra Appl.}\ 217 (1995), 255--263.  
\MR{1322554} (95k:05157).  \Zbl{830.05059}.
\annot{} Several characterizations of consistent vertex signatures of a graph.  
$\Gamma$ is ``markable'' iff it has a consistent vertex signature that is not all $+$.  Thm.:  $3$-connected $\Gamma$ is markable iff it is bipartite.  
Thm.:  A classification of markable $2$-connected graphs with girth $\leq 5$.  
[See \xref{Hoede}{1992a}{Cornelis Hoede}.]
\dated{27 Apr 2009}
\codes{\CVS:~\CBal}%
\itemspace

\bsgitem{1999a}{}  On balanced signed graphs and consistent marked graphs.  
Ordinal and Symbolic Data Analysis (OSDA '98, University of Massachusetts, Amherst, Mass., 1998).  \emph{Electronic Notes Discrete Math.}\ 2 (1999), 12 pp.
\MR{1990307} (2004e:\-05088).  \Zbl{971.68115}.					
\annot{} A survey of balance in signed graphs and consistency in vertex-signed graphs and their supposed applications in social psychology and elsewhere.  Results from \xref{Xu}{1998a}{Shaoji Xu} and \xref{Roberts and Xu}{2003a}{Fred S.\ Roberts \& Shaoji Xu}.  
\S4: ``Connections among balance, consistency, and other graph-theoretical notions''.  Lists some special and general equivalences, esp., with bipartiteness, or with all circle lengths divisible by $4$.  
\S5: ``Coherent paths''.  Characterizations of consistency or balance from \xref{Beineke and Harary}{1978b}{Lowell W.\ Beineke \& Frank Harary}, \xref{Roberts and Xu}{2003a}{Fred S.\ Roberts \& Shaoji Xu}, \xref{Acharya}{1983a}{B.\ Devadas Acharya}, \xref{Rao}{1984a}{S.B.\ Rao}.
\S6: ``Fundamental cycles and cycle bases''.  \xref{Hoede's}{1992a}{Cornelis Hoede} characterization of consistency, a variant, and one from \xref{Roberts and Xu}{2003a}{Fred S.\ Roberts \& Shaoji Xu} in terms of a circle basis.
\S7: ``Markable graphs''.  $\Gamma$ is ``markable'' iff it has a consistent vertex signature that is not all $+$.  Thm.\ (Roberts ).  $3$-connected $\Gamma$ is markable iff it is bipartite.  See \xref{Roberts}{1995a}{Fred S.\ Roberts} and \xref{S.\ Xu}{1998a}{Shaoji Xu}.
\S8: ``Open questions''.
\dated{27 Apr 2009}
\par
\codes{\CSG,~\CVS:~\CBal,~\CPsS,~\CAppl:~\CExp}%
\itemspace

\bsgitem{2001a}{}  Discrete mathematics.  In:  Neil J.\ Smelser and Paul B.\ Baltes, eds., \emph{International Encyclopedia of the Social \& Behavioral Sciences}, pp.\ 3743--3746.  Pergamon (Elsevier), 2001.
\annot{} \S5: ``Signed and marked graphs''.
\codes{\CSG,~\CVS,~\CPsS:~\CExp}
\itemspace

\author{Fred S.\ Roberts \& Thomas A.\ Brown} 
\authorspace
\bsgitem{1975a}{} Signed digraphs and the energy crisis.
\emph{Amer.\ Math.\ Monthly} 82 (June--July, 1975), no.\ 6, 577--594.  
\MR{0368957} (51 \#5195).  \Zbl{357.90070}.
\codes{\CSD,~\CSDw}%
\itemspace

\bsgitem{1977a}{} [Reply to \xref{Waterhouse}{1977a}{William C.\ Waterhouse}].  \emph{Amer.\ Math.\ Monthly} 84 (1977), 27.
\itemspace

\author{Fred S.\ Roberts \& Shaoji Xu}
\authorspace
\bsgitem{2003a}{}  Characterizations of consistent marked graphs.  1998 Conf.\ Ordinal and Symbolic Data Analysis (OSDA '98) (Amherst, Mass.).  \emph{Discrete Appl.\ Math.}\ 127 (2003), 357--371.
\MR{1984094} (2004b:05097).  \Zbl{1026.05054}.					
\annot{} Several characterizations of consistent vertex-signed graphs, and algorithms to determine consistency, are surveyed or proved.  
Thm.:  A vertex-signed graph is consistent iff every circle in some circle basis is positive and every two 3-connected vertices have the same sign.	
\dated{26 Apr 2009}
\codes{\CSG,~\CVS:~\CBal,~\CAlgor}
\itemspace

\author{Edmund Robertson}						%ROBERTSON E
\authorspace
\authref  See \hyperlink{Peter Brooksbank}{P.~Brooksbank}.
\itemspace

\authorx{Neil Robertson}						%ROBERTSON N
\author{Neil Robertson, P.D.\ Seymour, \& Robin Thomas}
\authorspace
\authref  See also \hyperlink{William McCuaig}{W.~McCuaig} and \hyperlink{John Maharry}{J.~Maharry}.
\authrefspace

\bsgitem{1999a}{\dag}  Permanents, Pfaffian orientations, and even directed circuits.  \emph{Ann.\ of Math.}\ (2) 150 (1999), no.\ 3, 929--975.
\MR{1740989} (2001b:15013).  \Zbl{947.05066}.
\annot{} Question 1.  Does a given digraph $D$ have an even cycle?  Question 2.  Can a given digraph $D$ be signed so that every cycle is negative?  (These problems are easily seen to be equivalent.)  The main theorem (the ``Even Dicycle Thm.'') is a structural characterization of digraphs that have a signing in which every cycle is negative.  (These were previously characterized by forbidden minors in \xref{Seymour and Thomassen}{1987a}{Paul Seymour \& Carsten Thomassen}.)
\annotctd{} The main theorem is proved also in \xref{McCuaig}{2004a}{William McCuaig}.  See the joint announcement, \xref{McCuaig, Robertson, Seymour, and Thomas}{1997a}{William McCuaig, Neil Robertson, P.D.\ Seymour, \& Robin Thomas}.
\par
\codes{\CSD:~\Cpar:~\CStr:~\CCycles}
\itemspace

\authorx{Garry Robins}						%ROBINS
\author{Garry Robins \& Yoshi Kashima}
\authorspace
\bsgitem{2008a}{}  Social psychology and social networks: Individuals and social systems.  \emph{Asian J.\ Social Psychology} 11 (2008), 1--12.
\annot{} Pp.\ 9--10: a critical review of signed-graph balance theory in social psychology.  [Also see that of {R.C.~Roistacher}{1974a}{Richard C.\ Roistacher}.]
\dated{21 Aug 2014} 
\codes{\CPsS:~\CSG}%
\itemspace

\author{Ellen Robinson}						%ROBINSON E
\authorspace
\authref  See \hyperlink{Lucas Rusnak}{L.~Rusnak} and \hyperlink{Gina Chen}{G.~Chen}.
\itemspace

\author{Herbert A.\ Robinson}						%ROBINSON HA
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{Robert W.\ Robinson} 						%ROBINSON RW
\authorspace
\authref  See also \xref{Harary, Palmer, Robinson, and Schwenk}{1977a}{Frank Harary, Edgar M.\ Palmer, Robert W.\ Robinson, \& Allen J.\ Schwenk} and \xref{Harary and Robinson}{1977a}{Frank Harary \& Robert W.\ Robinson}.
\authrefspace

\bsgitem{1981a}{}  Counting graphs with a duality property.  
In:  H.N.V.~Temperley, ed., \emph{Combinatorics}  (Proc.\ Eighth British Combinatorial Conf., Swansea, 1981), pp.\ 156--186.  London Math.\ Soc.\ Lect.\ Note Ser., 52.  Cambridge Univ.\ Press, Cambridge, England, 1981. 
\MR{0633654} (83c:05071).  \Zbl{462.05035}.						
\annot{} The ``bilayered digraphs'' of \S7 are identical to simply signed, loop-free digraphs (where multiple arcs are allowed if they differ in sign or direction).  Thm.\ 1:  Their number $b_n =$ number of self-complementary digraphs of order $2n$.  Cor.\ 1:  Equality holds if the vertices are signed and $k$-colored.  In \S8, Cor.\ 2 concerns vertex-signed and $2$-colored digraphs; Cor.\ 3 concerns vertex-signed tournaments.  Assorted remarks on previous signed enumerations, mainly from \xref{Harary, Palmer, Robinson, and Schwenk}{1977a}{Frank Harary, Edgar M.\ Palmer, Robert W.\ Robinson, \& Allen J.\ Schwenk}, are scattered about the article.
\codes{\CSD,~\CVS,~\CSG:~\CEnum}%
\itemspace

\author{Paul Rochet}						%ROCHET
\authorspace
\authref  See \hyperlink{Pierre-Louis Giscard}{P.-L.~Giscard}.
\itemspace

\authorx{G.J.\ Rodgers}						%RODGERS
\author{G.J.\ Rodgers \& A.J.\ Bray}
\authorspace
\bsgitem{1988a}{}  Density of states of a sparse random matrix.  \emph{Phys.\ Rev.\ B} 37 (1988), no.\ 7, 3557--3562.
\MR{0932406} (89d:82054).  
\annot{} Physical quantities via spectrum of random $A(\Sigma)$ with $n\to\infty$ assuming expected degree $d_\Sigma(v) \sim pn$ where $0<p<1$.
\dated{29 Dec 2012} 
\par
\codes{\CPhys:~\Csg:~\CRand:~\CEig}%
\itemspace

\author{G.J.\ Rodgers \& C.\ De Dominicis}
\authorspace
\bsgitem{1990a}{}  Density of states of sparse random matrices.  \emph{J.\ Phys.\ A} 23 (1990), 1567--1573.
\MR{1048785} (91a:82043).  
\annot{} \S 3, ``Solutions'':  Random $A(\Sigma)$ where $\Sigma$ is sparse.  
\dated{29 Dec 2012} 
\codes{\CPhys:~\Csg:~\CRand:~\CEig}%
\itemspace

\author{Y.\ Roditty}							%RODITTY
\authorspace
\authref   See \hyperlink{I.\ Krasikov}{I.~Krasikov}.
\itemspace

\author{Vojt\v{e}ch R\"odl}						%RODL
\authorspace
\authref  See \hyperlink{Richard A.\ Duke}{R.A.~Duke}.
\itemspace

\author{Jose Antonio Rodriguez}					%RODRIGUEZ
\authorspace
\authref  See \hyperlink{F.T.\ Boesch}{R.T.~Boesch}.
\itemspace

\author{Elisabeth Rodr{\'\i}guez-Heck}					%RODRIGUEZ-HECK
\authorx{Elisabeth Rodriguez-Heck}
\authorspace
\bsgitem{2018a}{}  \emph{Linear and Quadratic Reformulations of Nonlinear Optimization Problems in Binary Variables}.  Doctoral thesis, Li\'ege Universit\'e, 2018.
\par
\codes{\CSH,~\CSG:~\CBal,~\CGeom}%
\itemspace

\bsgitem{2019a}{}  Linear and quadratic reformulations of nonlinear optimization problems in binary variables.  \emph{4OR} 17 (2019), no.\ 2, 221--222.
\annot{} Abstract of \xrefd{2018a}{Elisabeth Rodriguez-Heck}.
\codes{\CSH,~\CSG:~\CBal,~\CGeom}%
\itemspace

\author{Juan A.\ Rodr{\'\i}guez-Vel\'azquez}				%RODRIGUEZ-VELAZQUEZ
\authorx{Juan A.\ Rodriguez-Velazquez}
\authorspace
\authref  See \hyperlink{Ernesto Estrada}{E.~Estrada}.
\itemspace

\author{Vladimir Rogojin}						%ROGOJIN
\authorspace
\authref  See \hyperlink{Artiom Alhazov}{A.~Alhazov}.
\itemspace

\author{Richard C.\ Roistacher}						%ROISTACHER
\authorspace
\bsgitem{1974a}{}  A review of mathematical methods in sociometry.  \emph{Sociological Methods Res.}\ 3 (1974), no.\ 2, 123--171.
\annot{} Pp.\ 20--22: a critical review of the use of signed graphs in social psychology.  
[Also see that of \xref{Robins and Kashima}{2008a}{Garry Robins \& Yoshi Kashima}.]
\dated{21 Aug 2014} 
\codes{\CPsS:~\CSG}%
\itemspace

\author{Oscar Rojo}							%ROJO
\authorspace
\authref  See also \hyperlink{Ivan Gutman}{I.~Gutman} and \hyperlink{Germain Pasten}{G.~Past\'en}.
\authrefspace

\bsgitem{2009a}{}  Spectra of copies of a generalized Bethe tree attached to any graph.  \emph{Linear Algebra Appl.}\ 431 (2009), 863--882.
\MR{2535558} (2011c:05209).  \Zbl{1168.05328}.  
\par
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2011a}{}  Line graph eigenvalues and line energy of caterpillars.  \emph{Linear Algebra Appl.}\ 435 (2011), 2077--2086.
\MR{2810648} (2012e:05243).  \Zbl{1222.05177}.					
\par
\codes{\Cpar:~\CLG:~\CAdj:~\CEig}%
\itemspace

\author{Oscar Rojo \& Ra\'ul D.\ Jim\'enez}
\authorx{Oscar Rojo \& Raul D.\ Jimenez}
\authorspace
\bsgitem{2011a}{}  Line graph of combinations of generalized Bethe trees:  Eigenvalues and energy.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2402--2419.
\MR{2811125} (2012f:05185).  \Zbl{1222.05178}.					
\codes{\Cpar:~\CAdj:~\CEig,~\CLG}
\itemspace

\author{Oscar Rojo \& Luis Medina}
\authorspace
\bsgitem{2010a}{}  Spectral characterization of some weighted rooted graphs with cliques.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 7, 1388--1409.
\MR{2680266} (2011h:05163).  \Zbl{1194.05095}.					
\codes{\Cpar:~\CAdj:~\CEig}%
\itemspace

\author{Edita Rollov\'a}						%ROLLOVA
\authorx{Edita Rollova}
\authorspace
\authref  See also \hyperlink{Matt DeVos}{M.~DeVos}, \hyperlink{Tomas Kaiser}{T.~Kaiser}, \hyperlink{Edita Macajova}{E.~M\'a\v{c}ajov\'a}, and \hyperlink{Reza Naserasr}{R.~Naserasr}.
\authrefspace

\author{Edita Rollov\'a, Michael Schubert, \& Eckhard Steffen}
\authorx{Edita Rollova, Michael Schubert, \& Eckhard Steffen}
\authorspace
\bsgitem{2014a}{}  Signed graphs with two negative edges.  In:  \emph{Bordeaux Graph Workshop 2014}, pp.\ 41--42.  LaBRI, Bordeaux, 2014.  
\\ URL \url{http://bgw.labri.fr/2014/bgw2014-booklet.pdf}
\annot{} Extended abstract of \xrefd{2018a}{Edita Rollova, Michael Schubert, \& Eckhard Steffen}.
\dated{19 Mar 2017} 
\codes{\CSG:~\CFlows}%
\itemspace

\bsgitem{2018a}{}  Flows in signed graphs with two negative edges.  \emph{Electronic J.\ Combin.}\ 25 (2018), no.\ 2, art.\ \#P2.40, 18 pp.
\MR{3814274}.  \Zbl{1388.05079}.  \arXiv{1604.08053}.				
\codes{\CSG:~\CFlows}%
\itemspace

\author{F.\ Rom\'a}							%ROMA
\authorx{F.\ Roma}
\authorspace
\authref  See also \hyperlink{A.J.\ Ramirez-Pastor}{A.J.~Ram{\'\i}rez-Pastor}.
\authrefspace

\author{F.\ Rom\'a, F.\ Nieto, A.J.\ Ramirez-Pastor, \& E.E.\ Vogel}
\authorx{F.\ Roma, F.\ Nieto, A.J.\ Ramirez-Pastor, \& E.E.\ Vogel}
\authorspace
\bsgitem{2005a}{}  Thermodynamic integration method applied to $\pm J$ Ising lattices.  \emph{Physica A} 348 (2005), 216--222.
\annot{} A signed toroidal square lattice with random signs.  Analytical and numerical methods to study functions of $x := \#E^+/\#E$ such as proportion of frustrated (negative) plaquettes (face boundaries).  (\Cf\ other papers of the authors.)
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CState}%
\itemspace

\bsgitem{2006a}{}  Novel order parameter to describe the critical behavior of Ising spin glass models.  \emph{Physica A} 363 (2006), 327--333.
\Zbl{1073.82565}.  \arXiv{cond-mat/0506767}.  
\annot{} Assuming (with little loss of generality) equally many edges of each sign, randomly distributed, in a square lattice.	
\dated{3 Jan 2015} 
\par
\codes{\CPhys:~\CSG:~\CRand:~\CState,~\CSw}%
\itemspace

\author{F.\ Rom\'a, F.\ Nieto, E.\ Vogel, \& A.J.\ Ramirez-Pastor}
\authorx{F.\ Roma, F.\ Nieto, E.\ Vogel, \& A.J.\ Ramirez-Pastor}
\authorspace
\bsgitem{2004a}{}  Ground-state entropy of $\pm J$ Ising lattices by Monte Carlo simulations.  \emph{J.\ Stat.\ Phys.}\ 114 (2004), no.\- 5-6, 1325-1341.
\Zbl{1073.82565}.  
\par
\codes{\CPhys:~\CSG:~\CRand:~\CState}%
\itemspace

\author{F.\ Rom\'a, S.\ Risau-Gusman, A.J.\ Ramirez-Pastor, F.\ Nieto, \& E.E.\ Vogel}
\authorx{F.\ Roma, S.\ Risau-Gusman, A.J.\ Ramirez-Pastor, F.\ Nieto, \& E.E.\ Vogel}
\authorspace
\bsgitem{2009a}{}  The ground state energy of the Edwards--Anderson spin glass model with a parallel tempering Monte Carlo algorithm.  \emph{Physica A} 88 (2009), 2821--2838.
\arXiv{0806.1054}.  
\annot{} The efficiency of a Monte Carlo method for finding ground state energies \etal, with either random pure signs (i.e., $\pm J$ with $J=1$, half the edges having each sign) or Gaussian randomly weighted random signs.
	
\dated{3 Jan 2015} 
\codes{\CSG:~\CFr,~\CSw,~\CPhys:~\CAlgor}%
\itemspace

\author{S.V.\ Roopa}							%ROOPA
\authorspace
\authref  See \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\authorx{P.\ Lawrence Rozario Raj}					%ROZARIO RAJ
\author{P.\ Lawrence Rozario Raj \& R.\ Lawrence Joseph Manoharan}
\authorspace
\bsgitem{2016a}{}  Face and total face signed product cordial labeling of planar graphs.  \emph{Asia Pacific J.\ Res.}\ 1 (2016), no.\ 41, 16--23.
\annot{} See \xref{Baskar Babujee and Shobana}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.  For a face of plane $\Gamma$, $\zeta^{**}(F) :=$ product of boundary vertex signs.  $\zeta$ is a ``face signed product cordial labeling'' if signed product cordial and $\#{\zeta^{**}}\inv(+) \approx \#{\zeta^{**}}\inv(-)$.  $\zeta$ is a ``total face signed product cordial labeling'' if total \# positive vertices, edges, and faces $\approx$ \# negative ones.
$K_{1,1,n-2}$, $n\geq5$, and another plane family are both.  
\dated{26 Dec 2020} 
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\author{Frances Rosamond}						%ROSAMOND
\authorspace
\authref  See \hyperlink{Hans L.\ Bodlaender}{H.L.~Bodlaender}.
\itemspace

\author{Milton J.\ Rosenberg} 						%ROSENBERG MJ
\authorspace
\authref   See also \hyperlink{Robert P.\ Abelson}{R.P.~Abelson}.
\authrefspace

\author{Milton J.\ Rosenberg \& Robert P.\ Abelson}
\authorspace
\bsgitem{1960a}{}  An analyisis of cognitive balancing.  In:  Milton J.\ Rosenberg \etal, eds., \emph{Attitude Organization and Change: An Analysis of Consistency Among Attitude Components}, Ch.\ 4, pp.\ 112--163.  Yale Univ.\ Press, New Haven, 1960.
\annot{} An attempt to test structural balance theory experimentally.  The test involves, in effect, a signed $K_4$ [an unusually large graph for such an experiment].  Conclusion: there is a tendency to balance but it competes with other forces.
\codes{\CPsS:~\CSG:~\Ckg}%
\itemspace

\author{Seymour Rosenberg} 						%ROSENBERG S
\authorspace
\bsgitem{1968a}{} Mathematical models of social behavior.  
In:  Gardner Lindzey and Elliot Aronson, eds., \emph{The Handbook of Social Psychology}, second ed., Vol.\ 1, Ch.\ 3, pp.\ 179--244.  Addison-Wesley, Reading, Mass., 1968.
\annot{} ``Balance model,'' pp.\ 196--199.  ``Congruity model,'' pp.\ 199--203.
\par
\codes{\CPsS,~\CSG:~\CBal:~\CExp,~\CRef}%
\itemspace

\author{M.\ Rosenfeld}						%ROSENFELD
\authorspace
\authref   See \hyperlink{William McCuaig}{W.D.~McCuaig}.
\itemspace

\author{Elissa Ross}							%ROSS
\authorspace
\bsgitem{2011a}{}  \emph{Geometric and Combinatorial Rigidity of Periodic Frameworks as Graphs on the Torus}.  Doctoral dissertation, York University, 2011.
\MR{2941979} (no rev).  
\annot{} Extending \xref{Whiteley}{1988a}{Walter Whiteley}.  \Cf\ \xrefd{2014a}{Elissa Ross}, \xrefd{2015a}{Elissa Ross}.
\par
\codes{\CGG:~\Csw,~\CCov,~\CTop,~\CGeom}
\itemspace

\bsgitem{2014a}{}  The rigidity of periodic frameworks as graphs on a fixed torus.  \emph{Contrib.\ Discrete Math.}\ 9 (2014), no.\ 1, 11--45.
\MR{3265750}.  \Zbl{1317.52028}.  \arXiv{1202.6652}.  
\annot{} $\bbZ^d$-gain graphs.  
Dictionary:  ``net gain'' = gain of closed walk, ``$T$-gain procedure'' = switching so $\phi^\zeta|_T \equiv 0$, ``local gain group'' = $\langle \Im \phi^\zeta \rangle$, ``derived graph'' = gain covering graph.
\codes{\CGG:~\CGeom,~\Csw,~\CCov}%
\itemspace

\bsgitem{2015a}{}  Inductive constructions for frameworks on a two-dimensional fixed torus.  \emph{Discrete Comput.\ Geom.}\ 54 (2015), 78--109.
\MR{3351759}.  \Zbl{1327.52038}.  \arXiv{1203.6561}.  
\annot{} $\bbZ^2$-gain graphs.  
Dictionary:  \xrefd{2014a}{Elissa Ross}.
\codes{\CGG:~\CGeom,~\Csw,~\CCov}%
\itemspace

\author{Philippe A.\ Rossignol}						%ROSSIGNOL
\authorspace
\authref  See \hyperlink{Jeffrey M.\ Dambacher}{J.M.~Dambacher}.
\itemspace

\author{Gian-Carlo Rota} 						%ROTA*
\authorspace
\authref   See \hyperlink{Peter Doubilet}{P.~Doubilet}.
\itemspace

\author{G\"unter Rote}						%ROTE
\authorspace
\authref  See \hyperlink{Herbert Edelsbrunner}{H.~Edelsbrunner}.
\itemspace

\authorx{Ron M.\ Roth}						%ROTH RM
\author{Ron M.\ Roth \& Krishnamurthy Viswanathan}
\authorspace
\bsgitem{2007a}{}  On the hardness of decoding the Gale--Berlekamp code.  
In:  \emph{Information Theory, 2007} (ISIT 2007, IEEE Int.\ Symp., Nice, 2007), pp.\ 1356--1360.  IEEE Press, 2007.
\annot{} See \xrefd{2008a}{Ron M.\ Roth \& Krishnamurthy Viswanathan}.
\codes{\Csg:~\Cfr,~\CAlgor}%
\itemspace

\bsgitem{2008a}{}  On the hardness of decoding the Gale--Berlekamp code.
\emph{IEEE Trans.\ Inform.\ Theory} 54 (2008), no.\ 3, 1050--1060. 
\MR{2445050} (2010h:94267).  
\annot{} Section III, ``Relaxed problem'':  The frustration index of a bipartite signed graph is NP-complete.  Thm.\ 4.1:  The frustration index of a signed $K_{n,n}$ (where $n$ is a variable) is NP-complete.  The proofs use the bipartite adjacency matrix of the signed graph.  The latter problem is polynomially reduced to the former by a construction using Kronecker product and a Hadamard matrix.  The problems are interpreted as nearest-neighbor decoding of the Gale--Berlekamp code of order $n$.
\annotctd{} Section V, ``Decoding algorithm over the BSC'':  A polynomial-time approximate decoding algorithm that is asymptotically reliable.
\dated{2 Sept 2009} 
\codes{\Csg:~\Cfr,~\CAlgor}%
\itemspace

\authorx{Uriel G.\ Rothblum}			 			%ROTHBLUM
\author{Uriel G.\ Rothblum \& Hans Schneider}
\authorspace
\bsgitem{1980a}{}  Characterizations of optimal scalings of matrices.  \emph{Math.\ Programming} 19 (1980), 121--136.    
\MR{0583274} (81j:65064).  \Zbl{437.65038}.
\codes{\Cgg:~\Cmatrd,~\CSw}%
\itemspace

\bsgitem{1982a}{}  Characterizations of extreme normalized circulations satisfying linear constraints.  \emph{Linear Algebra Appl.}\ 46 (1982), 61--72.
\MR{0664696} (84d:90047).  \Zbl{503.05032}.
\codes{\Cgg:~\Cmatrd}%
\itemspace

\author{Jianling Rou}							%ROU JL
\authorspace
\authref  See \hyperlink{Fenglei Tian}{F.L.~Tian}.
\itemspace

\author{Peter Rowlinson}						%ROWLINSON
\authorspace
\authref  See also \hyperlink{Domingos M.\ Cardoso}{D.M.~Cardoso}, \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}, and \hyperlink{Farzaneh Ramezani}{F.~Ramezani}.
\authrefspace

\author{Peter Rowlinson \& Zoran Stani\'c}
\authorx{Peter Rowlinson \& Zoran Stanic}
\authorspace
\bsgitem{2021a}{}  Signed graphs with three eigenvalues: Biregularity and beyond.  \emph{Linear Algebra Appl.}\ 621 (2021), 272--295.
\MR{4234244}.  \Zbl{1462.05240}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Signed graphs whose spectrum is bounded by $-2$.  \emph{Appl.\ Math.\ Computation} 423 (2022), art.\ 126991, 8 pp.
\MR{4381063}.  
\annot{} 
Builds on \xref{Yuan, Mao, and Liu}{2021a}{Xiying Yuan, Yanqi Mao, \& Lele Liu}.
\codes{\CSG,~\CLG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{20xxa}{}  Trees as foundations for signed graphs.  Submitted.  
\codes{\CSG:~\CKG:~\CLG:~\CAdj:~\CEig}%
\itemspace


\author{Bernard Roy} 						%ROY B
\authorspace
\bsgitem{1959a}{}  Contribution de la th\'eorie des graphes \`a l'\'etude de certains probl\`emes lin\'eaires.  
\emph{C.R.\ Acad.\ Sci.\ Paris} 248 (1959), 2437--2439.
\MR{0111631} (22 \#2493).  
\annot{} Given real arc weights $a$ on a digraph, there exists $t: V \to \bbR$ such that $t(w) - t(v) \geq a(e)$ for every arc $e{:}(v,w)$ iff every cycle has nonpositive weight sum.  [\Cf\ \xref{Afriat}{1963a}{S.N.\ Afriat}.]
\codes{\CWD:~\COG}%
\itemspace

\bsgitem{1970a}{}  \emph{Alg\`ebre moderne et th\'eorie des graphes, orient\'ees vers les sciences \'economiques et sociales.  Tome II:  Applications et probl\`emes sp\'ecifiques}.  Dunod, Paris, 1970.  
\MR{0260413} (41 \#5039).  \Zbl{238.90073}.
\annot{} \S~IX.B.3.b:  ``Flots multiplicatifs et non conditionnels, ou $k$-flots.''  \S~IX.E.1.b:  ``Extension du probl\`eme central aux $k$-flots.''  \S~IX.E.2.c:  ``Quelques utilisations concretes des $k$-flots.''
\codes{\CGN:~\Cmatrd({\bf circuit}):~\CExp}%
\itemspace

\author{Mithun Roy}							%ROY M
\authorspace
\authref  See \hyperlink{Julien Bensmail}{J.~Bensmail}.
\itemspace

\author{Roshni T Roy}						%ROY RT
\authorspace
\authref  See also \hyperlink{Albin Mathew}{A.~Mathew} and \hyperlink{Shahul Hameed K}{S.~Hameed}.
\authrefspace

\bsgitem{2023a}{}  \emph{A Study on Matrices and Spectra of Signed Graphs and Skew Gain Graphs}.  Doctoral dissertation, Central University of Kerala, 2023. 
\annot{} Ch.\ 1:  Basics of signed, gain, and skew gain graphs.  
Ch.\ 2, ``Signed distance Laplacian matrices for signed graphs'':  \xref{Roy, Germina, Hameed, and Zaslavsky}{20xxa}{Roshni T Roy, K A Germina, K Shahul Hameed, \& Thomas Zaslavsky}.  
Ch.\ 3, ``Normalized distance Laplacian matrices for signed graphs'':  \xref{Roy, Germina, and Hameed}{20xxa}{Roshni T.\ Roy, K.A.\ Germina, \& Shahul Hameed K}.
Ch.\ 4, ``On the characteristic polynomial of skew gain graphs'':  \xref{Hameed, Roy, Soorya, and Germina}{20xxa}{Shahul Hameed K, Roshni T Roy, Soorya P, \& Germina K A}.
Ch.\ 5, ``On two Laplacian matrices for skew gain graphs'':  \xref{Roy, Hameed, and Germina}{2021a}{Roshni T Roy, Shahul Hameed K., \& Germina K.A.}.
Ch.\ 6, ``On product of skew gain graphs'':  \xref{Roy, Germina, and Hameed}{20xxb}{Roshni T Roy, Germina K A, \& Shahul Hameed K}.
\S7.2:  Open problems.
\dated{16 Nov 2022} 
\codes{\CSG,~\CGG(\CGen):~\CAdj:~\CEig}%
\itemspace

\author{Roshni T.\ Roy, K.A.\ Germina, \& Shahul Hameed K}
\authorspace
\bsgitem{20xxa}{}  Normalized distance Laplacian matrices for signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ (to appear).
\codes{\CSG:~\CLap(\CGen)}%
\itemspace

\author{Roshni T Roy, Germina K A, \& Shahul Hameed K}
\authorspace
\bsgitem{20xxb}{}  On product of skew gain graphs.  Submitted.  
\codes{\CGG(\CGen):~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Roshni T Roy, K A Germina, K Shahul Hameed, \& Thomas Zaslavsky}
\authorspace
\bsgitem{20xxa}{}  Signed distance Laplacian matrices for signed graphs.  \emph{Linear Multilinear Algebra} (to appear).
\arXiv{2010.04204}.  
\annot{} \Cf\ \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}.  Laplacian $\Lap(\Sigma,w)$ and $\det \Lap(\Sigma,w)$ of positively weighted signed graph.  Balance iff $\nul \Lap(\Sigma,w)>0$.  Applied to $w :=$ min, max signed distance.  Balance iff $\Lap(\Sigma,w)$ cospectral with $\Lap(|\Sigma|,d)$, $d=$ distance in $|\Sigma|$.  
\dated{11 Oct 2020} 
\par
\codes{\CSG:~\CWG,~\CInc,~\CLap,~\CBal,~\CEig,~\CSw}%
\itemspace

\author{Roshni T Roy, Shahul Hameed K., \& Germina K.A.}
\authorspace
\bsgitem{2021a}{}  On two Laplacian matrices for skew gain graphs.  \emph{Electronic J.\ Graph Theory Appl.}\ 9 (2021), no\ 1, 125--135.
\MR{4253601}.  \Zbl{1468.05164}.  \arXiv{2009.10487}.  
\annot{} Skew gains (\cf\ \xref{Hage}{1999a}{Jurriaan Hage} {\etal}) in $F^\times$ for field $F$.  Sachs formula for characteristic polynomial of $\Lap(\Phi)$.  Modified ``$g$-Laplacian'' $\Lap_g(\Phi)$ if $F$ is ordered; incidence matrix, matrix-tree theorem.
\dated{14 Jun 2022} 
\codes{\CGG(\CGen):~\CLap,~\CIncid}%
\itemspace

\author{Gordon F.\ Royle}						%ROYLE
\authorspace
\authref   See \hyperlink{M.N.\ Ellingham}{M.N.~Ellingham} and \hyperlink{C.D.\ Godsil}{C.D.~Godsil}.
\itemspace

\author{G.\ Rozenberg}						%ROZENBERG
\authorspace
\authref  See \hyperlink{A.H.\ Deutz}{A.H.~Deutz}, \hyperlink{Andrzej Ehrenfeucht}{A.~Ehrenfeucht}, and \hyperlink{Tero Harju}{T.~Harju}.  
\itemspace

\author{Arthur L.\ Rubin}						%RUBIN
\authorspace
\authref   See \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\itemspace

\author{Jason D.\ Rudd}						%RUDD
\authorspace
\authref   See \hyperlink{Peter J.\ Cameron}{P.J.~Cameron}.
\itemspace

\author{Paul Ruet}							%RUET
\authorspace
\authref  See also \hyperlink{Anne Crumiere}{A.~Crumi\`ere} and \hyperlink{Elisabeth Remy}{\'E.~Remy}, and \hyperlink{Adrien Richard \& Paul Ruet}{A.~Richard}.
\authrefspace

\bsgitem{2017a}{}  Negative local feedbacks in Boolean networks.  \emph{Discrete Appl.\ Math.}\ 221 (2017), 1--17.
\MR{3612582}.  \Zbl{1357.05141}.  \arXiv{1512.01573}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Sarah Crown Rundell}						%RUNDELL
\authorspace
\authref  See \hyperlink{Benjamin Braun}{B.~Braun}.
\itemspace

\authorx{Philip J.\ Runkel}						%RUNKEL
\author{Philip J.\ Runkel \& David B.\ Peizer}
\authorspace
\bsgitem{1968a}{}  The two-valued orientation of current equilibrium theory.  \emph{Behavioral Sci.}\ 13 (1968), no.\ 1, 56--65.
\codes{\CPsS:~\Csg:~\CBal}%
\itemspace

\author{Richard Ruppert} 						%RUPPERT
\authorspace
\authref  See \hyperlink{James P.\ Quirk}{J.~Quirk}.
\itemspace

\author{Lucas J.\ Rusnak}						%RUSNAK
\authorx{Lucas Rusnak}
\authorspace
\authref  See also \hyperlink{Ghadeer Alabandi}{G.~Alabandi}, \hyperlink{Gina Chen}{G.~Chen}, \hyperlink{Vinciane Chen}{V.~Chen}, \hyperlink{Will Grilliette}{W.~Grilliette}, and \hyperlink{Nathan Reff}{N.~Reff}.
\authrefspace

\bsgitem{2010a}{\dag\dag}  \emph{Oriented Hypergraphs}.  Doctoral dissertation, Binghamton University, 2010.
\MR{2941411} (no rev).  
\annot{} Oriented hypergraphs generalize bidirected graphs:  Each incidence gets a direction, or sign.  Main interest:  The linear dependencies of columns of a $(0,\pm1)$-matrix, treated as the incidence matrix of an oriented hypergraph.  Techniques are generalizations of those of signed graphic matroids (\xref{Zaslavsky}{1982a}{Thomas Zaslavsky}) but more complicated.  The methods are most applicable to the matrices known as ``balanceable''.
\par
\codes{\CSH:~\CInc,~\CStr,~\CSG,~\COri}%
\itemspace

\bsgitem{2013a}{}  Oriented hypergraphs: Introduction and balance.  \emph{Electronic J.\ Combin.}\ 20 (2013), no.\ 3, art.\ P48, 29 pp.
\MR{3118956}.  \Zbl{1295.05169}.  \arXiv{1210.0943}.				
\codes{\CSH:~\CInc,~\CStr,~\CSG,~\COri}%
\itemspace

\author{Lucas J.\ Rusnak, Selena Li, Brian Xu, Eric Yan, \& Shirley Zhu}
\authorspace
\bsgitem{2022a}{}  Oriented hypergraphs: Balanceability.  \emph{Discrete Math.}\ 345 (2022), no.\ 6, art.\ 112832, 14 pp.
\MR{4385165}.  \Zbl{1486.05218}.  \arXiv{2005.07722}.  
\par
\codes{\CSH:~\COri,~\CSG,~\CInc}%
\itemspace

\author{Lucas J.\ Rusnak, Josephine Reynes, Skyler J.\ Johnson, \& Peter Ye}
\authorspace
\bsgitem{2021a}{}  Generalizing Kirchhoff laws for signed graphs.  \emph{Australasian J.\ Combin.}\ 81 (2021), no.\ 3, 388--411.
\MR{4333849}.  \Zbl{1483.05073}.  \arXiv{2009.12680}.				
\codes{\CSG,~\CSH,~\COri:~\CLap,~\CFlows(\CGen)}%
\itemspace

\author{Lucas J.\ Rusnak, Ellen Robinson, Martin Schmidt, \& Piyush Shroff}
\authorspace
\bsgitem{2019a}{}  Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals.  \emph{J.\ Algebraic Combin.}\ 49 (2019), no.\ 4, 461--473.
\MR{3954431}.  \Zbl{1416.05179}.  \arXiv{1709.04011}.				
\codes{\CSH:~\COri,~\CSG:~\CAdj,~\CLap,~\CSG}%
\itemspace

\author{Carrie Rutherford}
\authorspace
\authref  See \hyperlink{Murad Banaji}{M.~Banaji}.
\itemspace

\author{Joe Ryan}							%RYAN J
\authorspace
\authref  See \hyperlink{C.\ Dalf'o}{C.~Dalf'o}.
\itemspace

\author{K.\ Rybnikov [K.A.\ Rybnikov, Jr.; Konstantin Rybnikov]}		%RYBNIKOV
\authorx{Konstantin Rybnikov}
\authorspace
\authref  See also \hyperlink{S.S.\ Ryshkov}{S.S.~Ryshkov}.
\authrefspace

\bsgitem{1999a}{}  Stresses and liftings of cell-complexes.  \emph{ Discrete Comput.\ Geom.}\ 21 (1999), 481--517.
\MR{1681885} (2001a:52016).  \Zbl{941.52008}.					
\annot{} \S4, ``Quality transfer'', concerns the existence of a satisfied state (called ``quality translation'' in \xref{Ryshkov and Rybnikov}{1997a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}) in a permutation gain graph $\Phi$, where $\fG$ acts on a set $Q$.  P.\ 487, top:  A satisfied state exists iff $\Phi$ is balanced [\xref{Ryshkov and Rybnikov}{1997a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}; but necessity is incorrect].  Lemma 4.1 appears to mean that a satisfied state exists iff it exists on each member of an arbitrary basis of the binary cycle space.  [Not true, but interesting.  The following special case, also invalid in general, was the author's intention (as I was told, Oct.\ 2000):  $\Phi$ is balanced iff every member of a circle basis is balanced.  The special case Lemma 4.2 is correct, because it is essentially homotopic.]  [See \xref{Rybnikov and Zaslavsky}{2005a}{Konstantin Rybnikov \& Thomas Zaslavsky}, \xrefd{2006a}{Konstantin Rybnikov \& Thomas Zaslavsky}.]
\codes{\Cgg:~\Cbal,~\CCov}%
\itemspace

\author{Konstantin Rybnikov \& Thomas Zaslavsky}
\authorspace
\bsgitem{2005a}{}  Criteria for balance in abelian gain graphs, with an application to piecewise-linear geometry.  \emph{Discrete Comput.\ Geom.}\ 34 (2005), no.\ 2, 251--268.
\MR{2155721} (2006f:05086).  \Zbl{1074.05047}.					
\annot{}\S\S1--4:  A condition on binary cycles that implies balance but does not depend on having a fundmental system of circles; it requires an abelian gain group. 
\S5:  Satisfied states and balance of a permutation gain graph.
\par
\codes{\CGG:~\CBal}%
\annotctd{}\S6: The criterion is applied to calculate the dimension of the space of liftings of a piecewise-linear immersion of a $d$-cell complex in Euclidean $d$-space. 
\codes{\CGG:~\CBal,~\CGeom}%
\itemspace

\bsgitem{2006a}{}   Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups.  
\emph{J.\ Graph Theory} 51 (2006), no.\ 1, 1--21.
\MR{2184346} (2006i:05078).  \Zbl{1085.05033}.  					
\annot{} The class of $\Gamma$ such that the criterion of \xrefd{2005a}{Konstantin Rybnikov \& Thomas Zaslavsky} works for any gains on $\Gamma$ is minor-closed.  Some forbidden minors are given.  Which ones they are depends on the class of permitted gain groups in a way that is not understood. 
\codes{\CGG:~\CBal:~\CStr}%
\itemspace

\authorx{S.S.\ Ryshkov}						%RYSHKOV
\author{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}
\authorspace
\bsgitem{1996a}{}  Generatrissa.  The Maxwell and Vorono{\u\i} problems.  (In Russian.)  \emph{Dokl.\ Akad.\ Nauk} 349 (1996), no.\ 6, 743--746. 
\MR{1441201} (98b:52027).  \Zbl{906.51007}.
\annot{} Announcement of results in \xrefd{1997a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}.
[``Generatrissa'' corresponds to English ``generatrix''.]
\codes{\Cgg:~\CGeom}%
\itemspace

\bsgitem{1996b}{}  Generatrissa: The problems due to Maxwell and Voronoi.  \emph{Dokl.\ Math.}\ 54 (1996), no.\ 1, 614--617.
\Zbl{906.51007}.  
\annot{} Translation of \xrefd{1996a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}.  
\codes{\Cgg:~\CGeom}%
\itemspace

\bsgitem{1997a}{}  The theory of quality translations with applications to tilings.  \emph{European J.\ Combin.}\ 18 (1997), 431--444.
\MR{1444253} (98d:52031).  \Zbl{881.52015}.						
\annot{} Let $\Phi$ be a permutation gain graph, with gain group $\fG$ acting on a set $\fQ$, and with underlying graph the $d$-cell adjacency graph of a kind of simply connected polyhedral $d$-cell complex.  A ``quality translation'' is a satisfied state: a mapping $s : V \to \fQ$ such that $s(w) = s(v)\phi(e;v,w)$ for every edge.  A  ``circuit'' is a closed walk that is not trivially reducible.  Call a ``$d-2$-circle'' any circle contained in the star of a $d-2$-cell.  Assume $\fG$ and $\fQ$ fixed.  Thms.\ 1--2 can be stated:  In the free group on the edge set, the $d-2$-circles generate all circuits.  Also, $\Phi$ is balanced iff all $d-2$-circles have identity gain.  Thm.\ 3:  Identity gain of all $d-2$-circles is necessary and sufficient for the existence of a satisfied state.  [Necessity is incorrect because the action may have nontrivial kernel.]  The idea of quality transfer goes back to Vorono{\u\i} in 1908.  
[See \xref{Rybnikov}{1999a}{Konstantin Rybnikov} and \xref{Rybnikov and Zaslavsky}{2005a}{Konstantin Rybnikov \& Thomas Zaslavsky} for more.]
\annotctd{} Sufficiency in Thm.\ 3 is applied to lifting of tilings of Euclidean and spherical space.  Thm.\ 4 (\xrefd{1996a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}, Thm.\ 9):  Balance (``canonical definition'') of $\Phi$ is sufficient for lifting a tiling of $\bbR^d$.  Here the qualities are affine functions.  Thms.\ 5--6 (\xrefd{1996a}{S.S.\ Ryshkov \& K.A.\ Rybnikov, Jr.}, Thm.\ 10):  Balance within each $d-3$-cell star implies lifting.  
[See \xref{Rybnikov}{1999a}{Konstantin Rybnikov} and \xref{Rybnikov and Zaslavsky}{2005a}{Konstantin Rybnikov \& Thomas Zaslavsky} for more.]
\annotctd{} \S8: ``Applications to the colouring of tilings''.
\codes{\Cgg:~\Cbal,~\CGeom}%
\itemspace

\author{Herbert J.\ Ryser} 						%RYSER*
\authorspace
\authref  See \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}.
\itemspace

\author{Shinsei Ryu}							%RYU S
\authorspace
\authref  See \hyperlink{AtMa P.O.\ Chan}{A.P.O.~Chan}.
\itemspace

\author{Henry S.\ Rzepa}						%RZEPA
\authorspace
\bsgitem{2005a}{}  M\"obius aromaticity and delocalization.  \emph{Chem.\ Rev.}\ 105 (2005), no.\ 10, 3697--3715.
\codes{\CChem:~\Csg:~\Cbal:~\CExp,~\CRef}%
\itemspace


\label{LETTER S} \hypertarget{LETTER S}{}	% LETTER  S


\author{Rachid Saad}						%SAAD
\authorspace
\bsgitem{1996a}{}  Finding a longest alternating cycle in a $2$-edge-coloured complete graph is in RP.  \emph{Combin.\ Probab.\ Computing} 5 (1996), 297--306.
\MR{1411089} (97g:05156).  \Zbl{865.05054}.
\annot{} Thm.:  In a bidirected all-negative complete graph with a suitable extra hypothesis, the maximum length of a coherent circle equals the maximum order of a coherent degree-$2$ subgraph.  More or less generalizes part of \xref{B\'ankfalvi and B\'ankfalvi}{1968a}{M.\ Bankfalvi \& Zs.\ Bankfalvi} (\qv).  [Generalized in \xref{Bang-Jensen and Gutin}{1998a}{Jorgen Bang-Jensen \& Gregory Gutin}.]   
[\Problem.  Generalize to signed complete graphs or further.]%
\codes{\Cpar:~\Cori:~\CPaths,~\CAlgor}%
\itemspace

\authorx{Assieh Saadatpour}						%SAADATPOUR
\author{Assieh Saadatpour, Istv\'an Albert, \& R\'eka Albert}
\authorx{Assieh Saadatpour, Istvan Albert, \& Reka Albert}
\authorspace
\bsgitem{2010a}{}  Attractor analysis of asynchronous Boolean models of signal transduction networks.  \emph{J.\ Theor.\ Biol.}\ 266 (2010), 641--656.
\MR{2981575} (no rev).  \Zbl{1407.92058}.  
\par
\codes{\CSD,~\CBiol:~\CDyn}%
\itemspace

\author{Assieh Saadatpour, R\'eka Albert, \& Timothy C.\ Reluga}
\authorx{Assieh Saadatpour, Reka Albert, \& Timothy C.\ Reluga}
\authorspace
\bsgitem{2013a}{}  A reduction method for Boolean network models proven to conserve attractors.  \emph{SIAM J.\ Appl.\ Dyn.\ Syst.}\ 12 (2013), no.\ 4, 1997--2011.
\MR{3131470}.  \Zbl{1308.92040}.  
\codes{\CSD,~\CBiol:~\CDyn}%
\itemspace

\author{Eminjan Sabir}						%SABIR
\authorspace
\authref  See \hyperlink{Lili Yuan}{L.-L.~Yuan}.
\itemspace

\author{Mathieu Sablik}						%SABLIK
\authorspace
\authref  See \hyperlink{Anne Crumiere}{A.~Crumi\`ere}.
\itemspace

\author{Horst Sachs} 						%SACHS*
\authorspace
\authref  See \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}.
\itemspace

\author{Faiza Saeed}						%SAEED
\authorspace
\authref  See \hyperlink{Uzma Ahmad}{U.~Ahmad}.
\itemspace

\author{Julio Saez-Rodriguez}						%SAEZ-RODRIGUEZ
\authorspace
\authref  See \hyperlink{Steffen Klamt}{S.~Klamt}.
\itemspace

\author{Bruce Sagan} 						%SAGAN
\authorspace
\authref  See also \hyperlink{Curtis Bennett}{C.~Bennett}, \hyperlink{Anders Bjorner}{A.~Bj\"orner}, \hyperlink{Andreas Blass}{A.~Blass},  \hyperlink{Joshua Hallam}{J.~Hallam}, \hyperlink{Frank Harary \& Bruce Sagan}{F.~Harary}, and \hyperlink{Tadeusz Jozefiak}{T.~J\'ozefiak}.
\authrefspace

\bsgitem{1995a}{}  Why the characteristic polynomial factors.  \emph{S\'em.\ Lotharingien Combin.}\ 35 (1995) [1998], art.\ B35a, iii + 20 pp.
\MR{1399505} (98a:06006).  \Zbl{855.05012}.  \arXiv{math/9812136}.				
\annot{}  A shorter predecessor of \xrefd{1999a}{Bruce Sagan}.
\codes{\CSG,~\CGen:~N:~Col,~G:~\CExp}%
\itemspace

\bsgitem{1999a}{}  Why the characteristic polynomial factors.  \emph{Bull.\ Amer.\ Math.\ Soc.\ (N.S.)} 36 (1999), 113--133.
\MR{1659875} (2000a:06021).  \Zbl{921.06001}.						
\annot{} In \S4, coloring of a signed graph $\Sigma$, especially of $\pm K_n\full$ and $\pm K_n$, is used to calculate and factor the characteristic polynomial of $\bfF(\Sigma)$.  Presents the geometrical reinterpretation and generalization by \xref{Blass and Sagan}{1998a}{Andreas Blass \& Bruce Sagan}.  In \S\S5 and 6, other methods of calculation and factorization are applied to some signed graphs (in their geometrical representation).
\par
\codes{\CSG,~\CGen:~N:~Col,~G:~\CExp}%
\itemspace

\author{Barna Saha}							%SAHA B
\authorspace
\authref  See \hyperlink{Saba Ahmadi}{S.~Ahmadi}.
\itemspace

\author{Sahariya}							%SAHARIYA
\authorspace
\authref  See \hyperlink{K.A.\ Germina}{K.A.~Germina}.
\itemspace

\author{[Amine El Sahili]}						%SAHILI
\authorspace
\authref  See \hyperlink{Amine El Sahili}{A.~El~Sahili} (under `E').
\itemspace

\author{G.\ Sahoo}							%SAHOO
\authorspace
\authref  See \hyperlink{S.\ Barik}{S.~Barik}.
\itemspace

\authorx{Prabhat K.\ Sahu}						%SAHU
\author{Prabhat K.\ Sahu \& Shyi-Long Lee}
\authorspace
\bsgitem{2008a}{}  Net-sign identity information index: A novel approach towards numerical characterization of chemical signed graph theory.  \emph{Chem.\ Phys.\ Lett.}\ 454 (2008), 133--138.
\annot{} The ``net-sign identity information index'' $I_s$ is expressed [obscurely] in terms of $\#E^+$ and $\#E^-$ in a molecular structure graph.  The purpose is to correlate with chemical phenomena.  $I_s$ and $\sqrt I_s$ are compared with other indices.
\dated{6 Feb 2011} 
\codes{\CSG:~\CChem}%
\itemspace

\authorx{Shaik Sajana}						%SAJANA
\author{Shaik Sajana, D.\ Bharathi, \& K.K.\ Srimitra}
\authorspace
\bsgitem{2017a}{}  Signed intersection graph of ideals of a ring.  \emph{Int.\ J.\ Pure Appl.\ Math.}\ 113 (2017), no.\ 10, 175--183.
\annot{} Balance and homogeneity for ring $\bbZ_n$.
\dated{25 Apr 2019} 
\par
\codes{\CSG:~\CAlgeb:~\CBal}%
\itemspace

\author{Sina Sajjadi}							%SAJJADI
\authorspace
\authref  See \hyperlink{R.\ Masoumi}{R.~Masoumi}.
\itemspace

\author{Mateja \v{S}ajna}						%SAJNA
\authorspace
\authref  See \hyperlink{Primoz Potocnik}{P.~Poto\v{c}nik}.
\itemspace

\author{Michael Saks} 						%SAKS
\authorspace
\authref  See \hyperlink{Paul H.\ Edelman}{P.H.~Edelman}.
\itemspace

\author{Tadashi Sakuma}						%SAKUMA
\authorspace
\authref  See \hyperlink{Shuya Chiba}{Chiba}.
\itemspace

\authorx{M.C.\ Salas-Sol{\'\i}s}						%SALAS-SOLIS
\authorx{M.C.\ Salas-Solis}
\author{M.C.\ Salas-Sol{\'\i}s, F.\ Aguilera-Granja, E.E.\ Vogel, \& S.\ Contreras}
\authorx{M.C.\ Salas-Solis, F.\ Aguilera-Granja, E.E.\ Vogel, \& S.\ Contreras}
\authorspace
\bsgitem{2003a}{}  Order parameters in anisotropic two-dimensional $\pm J$ Ising lattices.  \emph{Physica A} 327 (2003), 477--490.
\Zbl{1031.82510}.  
\annot{} Toroidal square lattice with fixed weights $f_xJ$ horizontally and $f_yJ$ vertically ($f_x,f_y,J>0$), randomly signed with $\#E^+ = \frac12 \#E$ or with variable $x := \#E^+/\#E$.  Studies dependence of ``order parameters'' on $f := f_x/f_y$ and $x$.
\dated{3 Jan 2015} 
\codes{\CSG,~\CWG,~\CPhys:~\CFr}%
\itemspace

\author{Nicolau C.\ Saldanha}						%SALDANHA
\authorspace
\bsgitem{2002a}{}  Singular polynomials of generalized Kasteleyn matrices.  \emph{J.\ Algebraic Combin.}\ 16 (2002), no.\ 2, 195--207.
\MR{1943588} (2004c:05051).  \Zbl{1017.05077}.
\annot{} A generalized Kasteleyn matrix is the left-right adjacency matrix $B$ of a bipartite gain graph with the complex units as gain group.  (A Kasteleyn matrix has for gain group the sign group.)  The object is to interpret combinatorially the coefficients of the characteristic polynomial, or the eigenvalues, of $BB^*$.  The approach is cohomological (\cf\ \xref{Cameron}{1977b}{Peter J.\ Cameron}.
\dated{rev.\ 10 Jun 2017}
\codes{\CGG,~\CSG:~\CAdj,~\CEig,~\CSw}%
\itemspace

\author{Anwar Saleh}						%SALEH
\authorspace
\authref  See \hyperlink{Sophia Shalini G.B.}{S.~Shalini}.
\itemspace

\author{M.\ Ruby Salestina}						%SALESTINA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Lillian Salinas}						%SALINAS
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{Mahmoud Salmasizadeh}					%SALMASIZADEH
\authorspace
\authref  See \hyperlink{Siamak Fayyaz Shahandashti}{S.~Fayyaz Shahandashti}.
\itemspace

\author{Regina Samaga}						%SAMAGA
\authorspace
\authref  See \hyperlink{Ioannis N.\ Melas}{I.N.~Melas}.
\itemspace

\author{Robert \v{S}\'amal}						%SAMAL
\authorx{Robert Samal}
\authorspace
\authref  See \hyperlink{Matt DeVos}{M.~DeVos}.
\itemspace

\author{Aniruddha Samanta}						%SAMANTA
\authorspace
\authref  See also \hyperlink{Ranjit Mehatari}{R.~Mehatari}.
\authrefspace

\author{Aniruddha Samanta \& M.\ Rajesh Kannan}
\authorspace
\bsgitem{2021a}{}  Bounds for the energy of a complex unit gain graph.  \emph{Linear Algebra Appl.}\ 612 (2021), 1--29.
\MR{4188336}.  \Zbl{1459.05185}.  \arXiv{2005.08634}.  
\par
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Gain distance matrices for complex unit gain graphs.  \emph{Discrete Math.}\ 345 (2022) [2021], no.\ 1, art.\ 112634, 12 pp.
\MR{4318814}.  \Zbl{1476.05127}.  \arXiv{2101.11558}.  
\annot{} Generalizes \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky} to gain group of complex units.
\dated{20 Oct 2021} 
\codes{\CGG:~\CStr,~\CAdj(\CGen):~\CEig}%
\itemspace

\bsgitem{20xxa}{}  On the spectrum of complex unit gain graphs.  Submitted.  
\arXiv{1908.10668}.  
\par
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{20xxd}{}  On the multiplicity of $A_\alpha$-eigenvalues and the rank of complex unit gain graphs.  Submitted.  \emph{}\ 
\arXiv{2101.03752}.  
\codes{\CGG:~\CAdj(\CGen):~\CEig}%
\itemspace

\author{P.N.\ Samanta}						%SAMANTA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{U.\ Samee}							%SAMEE
\authorspace
\authref  See \hyperlink{Mushtaq A.\ Bhat}{M.A.~Bhat}.
\itemspace

\author{E.\ Sampathkumar} 						%SAMPATHKUMAR
\authorspace
\authref  See also \hyperlink{Chandrashekar Adiga}{C.~Adiga} and \hyperlink{S.\ Pirzada}{S.~Pirzada}.
\authrefspace

\bsgitem{1972a}{}  Point-signed and line-signed graphs.  Karnatak University Graph Theory Res.\ Rep.\ 1, 1972.
\annot{} See \emph{Graph Theory Newsletter} 2 (Nov., 1972), no.\ 2, Abstract No.\ 7.
	
\par
\codes{\CSG,~\CVS:~\CBal}%
\itemspace

\bsgitem{1984a}{}  Point signed and line signed graphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 7 (1984), no.\ 3, 91--93.
\Zbl{552.05051}.  
\annot{$\mu_\sigma$, $\partial\mu$} Consider a simple graph, an edge signature $\sigma$, and a vertex signature $\mu$.  
Define $\mu_\sigma(v) := \prod\{\sigma(e) : e \text{ incident with } v\}$ [later dubbed ``canonical marking''] and, for each component $X$, $\partial\mu(X) := \prod_{v \in X} \mu(v)$.  $\mu$ is ``p-balanced'' if $\partial\mu \equiv +$.   
Thm.\ 1:  $\partial\mu \equiv +$ iff $\mu = \mu_\sigma$ for some $\sigma$.  [An early homology theorem.]  
Thm.\ 2:  If $\sigma$ is balanced and $\partial\mu_\sigma \equiv +$, then there exist all-negative, pairwise edge-disjoint paths connecting the $\mu_\sigma$-negative vertices in pairs.  [Quick proof: $\partial\mu \equiv +$ iff $\mu$ has evenly many negative vertices in each component.  Negative vertices of $\mu_\sigma$ are odd-degree vertices of $\Sigma^-$.  Apply Listing's Theorem (independently discovered in stronger form by Sampathkumar) to $\Sigma^-$.]  
[It is interesting to base homology 0-chains like $\partial\mu$ on the components.]
\dated{rev.\ 27 Dec 2010}
\codes{\CSG,~\CVS:~\CBal}%
\itemspace

\bsgitem{2006a}{}  Generalized graph structures.  \emph{Bull.\ Kerala Math.\ Assoc.}\ 3, no.\ 2 (Dec., 2006), 67--123. 
\MR{2290946} (no rev).  
\annot{} Within the class of simple graphs, what is a complement of a signed graph?  An approach is to partition the edges of $K_n$ into 3 classes: $E^+$, $E^-$, and $E^c$ (the set of non-edges), and apply a specific permutation of these sets.  Each permutation of order 2 implies a kind of complementation.   Examination of self-complementarity.  Generalizations of balance.  Generalized to a graph $\Gamma$ with $k$ edge classes $R_i$ [i.e., $k$-edge-colored graphs].
\annotctd{} \S10, ``Balanced graph structures'':  ``$R_i$-balance'':  $(\exists X)\ R_i = E[X,X^c]$ (the cut between $X$ and $X^c$).  ``$R_1 \cdots R_r$-balance'': Similar for $R_1 \cup \cdots \cup R_r$.  ``Complete balance'':  $R_i$ balance for all $i$.  
``Arbitrary balance'':  $R_{i_1}\cdots R_{i_r}$-balance for every $i_1,\ldots,i_r \in [k]$.  \Problem\ 11:  Characterize this property.  
``$r$-relation balance'':  The same for fixed $r$.  \Problem\ 12:  Characterize this property.
Other, similar concepts based on partitioning $V$.
\dated{4 Sept 2010}
\codes{\CSG,~\CSGc:~\CGen:~\CBal}%
\itemspace

	
\bsgitem{2006b}{}  $4$-Sigraphs.  In: \emph{International Conference on Discrete Mathematics, ICDM 2006} (Lect.\ Notes, Bangalore, 2006), p.\ 288.
\codes{\CSG(\CGen):~\CGG}
\itemspace

\bsgitem{2011a}{}  Two new characterizations of consistent marked graph.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 21 (2011), no.\ 4, 437--439.
\MR{2885007} (2012j:05192).  \Zbl{1250.05056}.					
\codes{\CVS:~\CBal,~\CSG}%
\itemspace

\author{E.\ Sampathkumar \& V.N.\ Bhave} 
\authorspace
\bsgitem{1973a}{} Group valued graphs.  \emph{J.\ Karnatak Univ.\ Sci.}\ 18 (1973), 325--328.
\MR{0347675} (50 \#177).  \Zbl{284.05113}.					
\annot{} Group-weighted graphs, both in general and where the group has exponent $2$ (so all $x^{-1} = x$).  Analogs of elementary theorems of Harary and Flament.  Here balance of a circle means that the weight product around the circle, taking for each edge either $w(e)$ or $w(e)^{-1}$ arbitrarily, equals $1$ for some choice of where to invert.  [Hence, the graphs are not gain graphs.]
\codes{\CWG,~\CGG:~\CBal}%
\itemspace

\author{E.\ Sampathkumar \& L.\ Nanjundaswamy} 
\authorspace
\bsgitem{1973a}{}  Complete signed graphs and a measure of rank correlation.  \emph{J.\ Karnatak Univ.\ Sci.}\ 18 (1973), 308--311.
\MR{0423674} (54 \#11649) (\qv).  \Zbl{291.62066}.
\annot{} Given a permutation of $\{1, 2, \ldots, n\}$, sign $K_n$ so edge $ij$ is negative if the permutation reverses the order of $i$ and $j$ and is positive otherwise.  Kendall's measure $\tau$ of correlation of rankings (i.e., permutations) $A$ and $B$ equals $(\#E^+-\#E^-)/\#E$ in the signature due to $AB^{-1}$.
\codes{\CSG:~\CKG}%
\itemspace

\author{E.\ Sampathkumar, L.\ Pushpalatha, \& M.A.\ Sriraj}
\authorspace
\bsgitem{2016a}{}  Color matrices.  \emph{Indian J.\ Discrete Math.}\ 2 (2016), no.\ 2, 103--108.
\MR{4317596}.  
\annot{} Definitions at \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj} and \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.  The colors are unlabelled.  $G$ colored by $\pi(\Sigma^-)$ is ``color balanced'' if $\Sigma$ is balanced.  ``Color balanced path number'' [not clear].
\dated{27 Jul 2018, 4 Jan 2019} 
\codes{\CSG:~\Cclu,~\CAdj,~\CBal}%
\itemspace

\author{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}
\authorspace
\bsgitem{2008a}{}  Jump symmetric $n$-sigraph.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 11 (2008), no.\ 1, 89--95.
\MR{2429334} (2009j:05107).  \Zbl{1172.05028}.					
\annot{} In the $n$-fold sign group $\{+,-\}^n$ an element is ``symmetric'' if it is its own reverse.  A (symmetric) $n$-signed graph is a gain graph $\Phi = (\Gamma, \phi)$ which has (symmetric) gains $\phi(e) \in \{+,-\}^n$.  [Equivalent to having arbitrary gains in $\{+,-\}^{\lceil n/2 \rceil}$.]  Only symmetric $n$-signed graphs are treated.  
\annotctd{$\Sigma_\Phi$} [The mapping $\min: \{+,-\}^n \to \{+,-\}$ by $\min(a_1,\ldots,a_n) = +$ if all $a_i=+$ and $= -$ otherwise gives a signed graph $\Sigma_\Phi$ with signs $\sigma_\Phi(e) := \min(\phi(e))$.]
\annotctd{} Def.: $\Phi_1 \simeq \Phi_2$ (``cycle isomorphism'') if there is an isomorphism $\|\Phi_1\| \cong \|\Phi_2\|$ that preserves circle gains.  Prop.\ 3:  Symmetric $n$-signed graphs are cycle isomorphic iff they are switching isomorphic---generalizing $n=1$ due to \xref{Soza\'nski}{1980a}{Tadeusz Sozanski}, \xref{Zaslavsky}{1981b}{Thomas Zaslavsky}.  [The proof (omitted) applies iff the gain group has exponent 2.]  
\annotctd{$\phi_S$} Let $\phi_S(ef) := \phi(e)\phi(f)$ for $e,f \in E$.  [Generalizing $\sigma_{\times}$ of \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]  
\annotctd{$J_S$} The jump graph is $J_S(\Phi) := (\Lambda(\Gamma)^c, \phi_S)$.  
Solutions of $\Phi \simeq J_S(\Phi)$, $\Phi^t \simeq J_S(\Phi)$, $J_S(\Phi^t) \simeq J_S(\Phi)$, where $a^t := at$ for $a,t \in \{+,-\}^n$ and $t$ is one of three special $n$-signs.  [The last solution extends to arbitrary $t \in \{+,-\}^n$.]
\annotctd{} Dictionary:  ``identity balance'', ``$i$-balance'' = balance in $\Phi$; ``balance'' = balance in $\Sigma_\Phi$; $P(\vec C) := \phi(C)$ in the indicated direction.  
\annotctd{} [The results remain true without assuming symmetry.]  [Continued in \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010d}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xref{Sampathkumar, Subramanya, and Reddy}{2011a}{E.\ Sampathkumar, M.S.\ Subramanya, \& P.\ Siva Kota Reddy}, and papers of \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.]
\dated{2 Aug 2009, 20 Dec 2010}
\par
\codes{\CSG(\CGen),~\Cgg:~\CLG,~\CSw,~\CBal}%
\itemspace

\bsgitem{2008b}{}  $(3, d)$-Sigraph and its applications.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 17 (2008), no.\ 1, 57--67.
\MR{2428537} (2009g:05073).  
\annot{} The $n=3$ case of \xrefd{2010b}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
\dated{10 Apr 2009}
\par
\codes{\CSG(\CGen),~\Cgg:~\CBal,~\CSw}%
\itemspace

\bsgitem{2009a}{}  Directionally $n$-signed graphs.  II.  \emph{Int.\ J.\ Math.\ Combin.}\ 2009 (2009), vol.\ 4, 89--98 (2010).
\MR{2598676} (no rev).  \Zbl{1238.05125}.					
\annot{} \Cf\ \xrefd{2010b}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.
\codes{\CGG(\CGen):~\CBal}%
\itemspace

\bsgitem{2010a}{}  $(4, d)$-Sigraph and its applications.  \emph{Adv.\ Stud.\ Contemp.\ Math.\ (Kyungshang)} 20 (2010), no.\ 1, 115--124.
\MR{2597997} (2011i:05089).  \Zbl{1192.05067}.					
\annot{} The $n=4$ case of \xrefd{2010b}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
\dated{9 Sept 2010}
\par
\codes{\CSG(\CGen),~\Cgg:~\CBal,~\CSw}%
\itemspace

\bsgitem{2010b}{}  Directionally $n$-signed graphs.  In:  B.D.\ Acharya, G.O.H.\ Katona, and J.\ Nesetril, eds., \emph{Advances in Discrete Mathematics and Applications: Mysore, 2008} (Proc.\ Int.\ Conf.\ Discrete Math., ICDM-2008, Mysore, India, 2008), pp.\ 153--160.  Ramanujan Math.\ Soc.\ Lect.\ Notes Ser., No.\ 13.  Ramanujan Mathematical Soc., Mysore, India, 2010.
\MR{2766915} (2012g:05097).  \Zbl{1231.05119}.					
\annot{} The gain group is the $n$-fold sign group $\{+,-\}^n$, with reversing automorphism $(a_1,\ldots,a_n)^r := (a_n,\ldots,a_1)$.  The gains satisfy $\phi(e\inv) = \phi(e)^r$.  
For $t \in \{+,-\}^n$, the $t$-complement of $\Phi$ is $\|\Phi\|$ with gains $\phi^t(e) := t\phi(e)$.  
Elementary results on balance, $t$-complementation, switching, and isomorphism. 
Dictionary:  ``identity balance'' = ``$i$-balance'' = balance in $\Phi$; ``balance'' = balance in $\Sigma_\Phi$ defined at \xrefd{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}; $P(\vec C) := \phi(C)$ in the indicated direction.
[An interesting form of skew gain graph.  The ideas should be pursued in directions suggested by \xref{Hage and Harju}{2000a}{Jurriaan Hage \& Tero Harju} and \xref{Hage}{1999a}{Jurriaan Hage}.]
\dated{10 Apr 2009}
\codes{\CSG(\CGen),~\Cgg:~\CBal,~\CSw}%
\itemspace

\bsgitem{2010c}{}  The line $n$-sigraph of a symmetric $n$-sigraph.  \emph{Southeast Asian Bull.\ Math.}\ 34 (2010), no.\ 5, 953--958.
\MR{2746762} (2012a:05142).  \Zbl{1240.05142}.					
\annot{$\Lambda_S$} The line graph is $\Lambda_S(\Phi) := (\Lambda(\Gamma), \phi_S)$ [generalizing $\Lambda_{\times}$ of \xref{M.\ Acharya}{2009a}{Mukti Acharya}].  For other definitions and notation see \xrefd{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
\annotctd{} Line graphs and jump graphs in the sense of \xrefd{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya} are characterized, respectively, as balanced symmetric $n$-signings of (unsigned) line graphs and their complements.  [The characterizations remain true for unsymmetric $n$-signatures.]  
There are remarks about the $t$-complement $t\phi$ \xrefd{2010b}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya} for three $t \in  \{+,-\}^n$.
\annotctd{$\mu_\phi,\ \Phi^c$} The ``complement'' $\Phi^c$ is $(\Gamma^c, \phi^c)$ defined by $\mu_\phi(v) := \prod_{uv \in E} \phi(uv)$ (``canonical marking'') (\cf\ \xref{Sampathkumar}{1984a}{E.\ Sampathkumar}) and $\phi^c(uv) := \mu_\phi(u)\cdot\mu_\phi(v)$ [= product of gains of all edges incident in $\Phi$ to $u$ or $v$ but not both].  [Gains $\phi^c$ are clearly balanced.]  
Prop.\ 7:  A symmetric $n$-signed graph is a line graph iff it is a balanced, symmetric $n$-signature of an unsigned line graph.  [Because $\phi_S$ is arbitrary balanced gains.]  
Prop.\ 9:  $\Lambda_S(\Phi)^c \sim J_S(\Phi)$.  [Because both are balanced and the underlying graphs are the same.]  
Prop.\ 10 solves $\Lambda_S(\Phi) \simeq J_S(\Phi)$, generalizing \xref{M.~Acharya and Sinha}{2003a}{Mukti Acharya \& Deepa Sinha}.  
[The solutions to such graph equations, here and in related papers of \hyperlink{R.\ Rangarajan}{Rangarajan}, Sampathkumar, \hyperlink{P.\ Siva Kota Reddy}{Siva Kota Reddy}, \etal, are easy corollaries of the similar results for unsigned graphs.]
[All results remain true without assuming symmetry.]
\dated{10 Apr, 1 Aug 2009, 20 Dec 2010}
\codes{\CSG(\CGen),~\Cgg:~\CLG,~\CSw,~\CBal}%
\itemspace

\bsgitem{2010d}{}  Common-edge signed graph of a signed graph.  \emph{J.\ Indones.\ Math.\ Soc.}\ 16 (2010), no.\ 2, 105--112.
\MR{2752773} (no rev).  \Zbl{1236.05098}.						
\annot{$C_E$} See \xrefd{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xref{Sampathkumar, Subramanya, and Reddy}{2011a}{E.\ Sampathkumar, M.S.\ Subramanya, \& P.\ Siva Kota Reddy} for definitions.  The common-edge signed graph $C_E(\Sigma)$ is $\Lambda_{\times}^2(\Sigma)$.  
Prop.\ 4: $\Sigma_0$ is a common-edge signed graph iff it is balanced and $|\Sigma_0|$ is a common-edge graph.  [Incorrect.  $\Lambda_{\times}^2(\Sigma)$ does not have arbitrary balanced signs.  E.g., $|\Sigma| = C_4$.]
Equations solved [possibly incorrectly]:  $\Sigma \simeq C_E^k(\Sigma)$ and $\Sigma \simeq \Lambda_{\times}^k(\Sigma)$ [this includes the preceding].  $\Lambda_{\times}^k(\Sigma) \simeq C_E^r(\Sigma)$.  The jump graph \xrefd{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya} $J_S(\Sigma) \simeq C_E(\Sigma)$.
[$\Lambda_{\times}$ as in \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]
\dated{12 Apr 2009}
\annotctd{} ``Smarandanchely $k$-signed/marked graphs'' are defined as $k$-signed/\-marked graphs [and not used].  Signed/marked graphs are the case $k=2$ [correctly: $k=1$].  [Smarandanche has nothing to do with this.]
\dated{7 Jan 2011}
\codes{\CSG:~\CBal,~\CSw,~\CLG}%
\itemspace

\author{E.\ Sampathkumar, S.V.\ Roopa, K.A.\ Vidya, \& M.A.\ Sriraj}
\authorspace
\bsgitem{2015a}{}  Partition energy of a graph.  \emph{Proc.\ Jangjeon Math.\ Soc.}\ 18 (2015), no.\ 4, 473--493.
\MR{3444734}.  \Zbl{1332.05115}.  
\annot{} Definitions at \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj}.  Eigenvalues and energy of $A$.  Examples.
\dated{4 Jan 2019} 
\codes{\Csgw(\CGen):~\CAdj:~\CEig}%
\itemspace

\bsgitem{2018a}{}  Partition energy of complete product of circulant graphs and some new class of graphs. \emph{Adv.\ Stud.\ Contemp.\ Math., Kyungshang}\ 28 (2018), no.\ 2, 269--283.
\Zbl{1400.05203}.  
\annot{} Definitions at \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj}.  Equienergetic examples.  Circulant-type examples.
\dated{4 Jan 2019} 
\par
\codes{\Csgw(\CGen):~\CAdj:~\CEig}%
\itemspace

\bsgitem{2019a}{}  Partition Laplacian energy of a graph.   \emph{Palestine J.\ Math.}\ 8 (2019), no.\ 1, 272--284.
\MR{3884324}.  \Zbl{1406.05066}.  
\annot{} Definitions at \xref{Sampathkumar and Sriraj}{2013b}{E.\ Sampathkumar \& M.A.\ Sriraj}.  Eigenvalues and energy of Laplacian modification of $A$.  Examples.
\dated{4 Jan 2019} 
\codes{\Csgw(\CGen):~\CAdj:~\CEig}%
\itemspace

\author{E.\ Sampathkumar \& M.A.\ Sriraj}
\authorspace

\bsgitem{2013a}{}  $(2, d)$-Sigraphs.  \emph{Notes Number Theory Discrete Math.}\ 19 (2013), no.\ 4, 16--27.
\Zbl{1314.05091}.  
\annot{} $(2,d)$-sigraph = directionally 2-signed graph = bidirected graph $\Beta$.  A notable new observation is Prop.\ 13, refined in \xref{Sampathkumar, Sriraj, and Zaslavsky}{2012a}{E.\ Sampathkumar, M.A.\ Sriraj, \& Thomas Zaslavsky}.  \S3, ``Directional adjacency matrix'': = $A$ of the naturally corresponding signed digraph.  \S7, ``Clusterable $(2,d)$-sigraphs''.  \S9, ``Induced $(2,d)$-sigraph of an $(n,d)$-sigraph $G$'':  In a directionally $n$-signed graph, multiply the first $\lceil n/2 \rceil$ directional signs to get a single directional sign.
Dictionary:  Signed graph $is(\Beta) := -\Sigma(\Beta)$.
\dated{31 Jan, 29 Sept 2012} 
\codes{\CSG(\CGen):~\Cori,~\CSD:~\CGen:~\CBal,~\CAdj}%
\itemspace

\bsgitem{2013b}{}  Vertex labeled/colored graphs, matrices and signed graphs.  
\emph{J.\ Combin.\ Inform.\ System Sci.}\ 38 (2013), no.\ 1-2, 113--120.  
\Zbl{302.05162}.  
\annot{} Decompose $K_n$ into graphs $\Gamma_i$, $i=1,2,3,4$.  Let $w(e)=\alpha_i\in\bbZ$ for $e\in E_i$ and $A := \sum_i \alpha_i A(\Gamma_i)$.  
In this paper, $\Gamma$ is given, has labels $l:V\to S$ and partition $\pi$ of $V$ into color classes, or any partition $\pi$; $\alpha_i = 1,-1,2,0$ for $i=1,2,3,4$; $\Gamma_1=\Gamma(\pi)$, $\Gamma_2=\Gamma^c(\pi)$, $\Gamma_3=\Gamma{:}\pi$, and $\Gamma_4=\Gamma^c{:}\pi$.  The ``$L$-matrix'' of $(\Gamma,l)$ is $A$.  Thm.\ 1 characterizes $L$-matrices.
\S3:  If $l$ properly colors $\Gamma$, then $A = A(\Sigma)$ where $\Sigma := +\Gamma_1 \cup -\Gamma_2$.  Prop.\ 5 restated:  $\Sigma^-$ is a disjoint union of cliques.  Alternatively,  $-\Sigma$ is clusterable so the clusters are all-positive cliques.  
\Cf\ \xref{Adiga, Sampathkumar \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}, \xref{Sampathkumar, S.V.\ Roopa \etal\!}{2015a}{E.\ Sampathkumar, S.V.\ Roopa, K.A.\ Vidya, \& M.A.\ Sriraj}, \xrefd{2019a}{E.\ Sampathkumar, S.V.\ Roopa, K.A.\ Vidya, \& M.A.\ Sriraj}, \xref{Prakasha, Reddy, and Cangul}{2017a}{K.N.\ Prakasha, P.\ Siva Kota Reddy, \& Ismail Naci Cangul}.
[The purpose of choosing $w_3=2$ is not clear.]
\dated{14 Oct 2014, 4 Jan 2019} 
\par
\codes{\Csgw,~\CSG:~\CCol,~\CClu,~\CAdj}%
\itemspace

\author{E.\ Sampathkumar, M.A.\ Sriraj, \& L.\ Pushpalatha}
\authorspace
\bsgitem{2017a}{}  Strong signed graph structures labeled graph structures and vertex labeled graphs.  \emph{Indian J.\ Discrete Math.}\ 3 (2017), no.\ 3, 15--24.
\MR{4317598}.  
\par
\codes{\CSG:~\CAdj,~\Cclu}%
\itemspace

\bsgitem{2017b}{}  Notions of balance in signed and marked graphs.  \emph{Indian J.\ Discrete Math.}\ 3 (2017), no.\ 3, 25--32.
\MR{4317599}.  
\annot{} Given $\Sigma$ and $\mu: V \to \{pm\}$, let $\bar\sigma(uv) := \\sigma^\mu(uv)$.  Defs.:  ``$e$-balance'':  $\sigma^\mu \equiv +$.  ``Weak $e$-balance'':  $\sigma$ is balanced.  ``Total balance'':  $\sigma^\mu \equiv +$ and $\mu(x) = \sigma(ux)\sigma(vx)$ $\forall x\in V$ and distinct $u,v\in N(x)$; characterized in Prop.\ 5.7.
\dated{11 Jul 2022} 
\par
\codes{\CSG,~\CVS:~\CBal(\CGen),~\CClu}%
\itemspace

\author{E.\ Sampathkumar, M.A.\ Sriraj, \& Thomas Zaslavsky}
\authorspace

\bsgitem{2012a}{}  Directionally 2-signed graphs and bidirected graphs.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 373--377.
\Zbl{1301.05161}.  \arXiv{1303.3084}.  
\annot{} A formal treatment of \xref{Sampathkumar and Sriraj}{2013a}{E.\ Sampathkumar \& M.A.\ Sriraj}, Prop.\ 13, and the connection with sources, sinks, and switching in bidirected graphs.  Given a bidirected graph $\Beta$, its corresponding signed graph $\Sigma(\Beta)$ (\cf\ \xref{Zaslavsky}{1991b}{Thomas Zaslavsky}) is antibalanced iff every vertex in $\Beta$ is a source or sink. 
\dated{29 Sept 2012} 
\codes{\CSG,~\CGen:~\COri:~\CBal,~\CSw}%
\itemspace

\author{E.\ Sampathkumar, M.S.\ Subramanya, \& P.\ Siva Kota Reddy}
\authorspace
\bsgitem{2011a}{}  Characterization of line sidigraphs.  \emph{Southeast Asian Bull.\ Math.}\ 35 (2011), no.\ 2, 297--304.
\MR{2866547} (2012j:05193).  \Zbl{1240.05143}.					
\annot{} The line signed graph is $\Lambda_{\times}(\Sigma)$ [see \xref{M.\ Acharya}{2009a}{Mukti Acharya}].  
Prop.\ 3:  A signed graph is a line signed graph of this kind iff it is a line graph with balanced signs.
\annotctd{} The line signed digraph is $\Lambda_S(\vec\Gamma,\sigma) :=$ the Harary--Norman line digraph of $\vec\Gamma$, signed by $\sigma^c$ defined as $\phi^c$ in \xref{Sampathkumar, Reddy, and Subramanya}{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  
Prop.\ 11:  A signed digraph is a line signed digraph of this kind iff it is a Harary--Norman line digraph with (undirectedly) balanced signs.  $(\vec\Gamma,\sigma) $ is switching isomorphic to $\Lambda_S(\vec\Gamma,\sigma)$ iff each component is a balanced directed cycle.  
\dated{4 Sept 2010} 
\par
\codes{\CSG,~\CSD:~\CLG}%
\itemspace

\author{Rub\'en J.\ S\'anchez-Garc{\'\i}a}					%SANCHEZ-GARCIA
\authorspace
\authref  See \hyperlink{Raffaella Mulas}{R.~Mulas}.
\itemspace

\author{V.\ Sangeetha}						%SANGEETHA
\authorspace
\authref  See \hyperlink{Anisha Jean Mathias}{A.J.~Mathias}.
\itemspace

\author{Bernardo San Mart{\'\i}n}					%SAN MARTIN
\authorx{Bernardo San Martin}
\authorspace
\authref  See \hyperlink{Nair Maria Maia de Abreu}{N.M.M.~Abreu} and \hyperlink{Maria Robbiano}{M.~Robbiano}.
\itemspace

\author{Yoshio Sano}						%SANO
\authorspace
\authref  See \hyperlink{Taeyoung Chung}{T.Y~Chung}, \hyperlink{Alexander L.\ Gavrilyuk}{A.L.~Gavrilyuk}, \hyperlink{Gary Greaves}{G.~Greaves}, and \hyperlink{Akihiro Munemasa, Yoshio Sano, \& Tetsuji Taniguchi}{A.~Munemasa}.
\itemspace

\authorx{Santhi M}							%SANTHI
\author{Santhi.\ M \& J.\ James Albert}
\authorspace
\bsgitem{2015a}{}  Signed product cordial in cycle related graphs.  \emph{Int.\ J.\ Math.\ Comp.\ Appl.\ Res.}\ 5 (2015), no.\ 1, 29--36.
\annot{} More, as in \xref{Baskar Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\dated{11 Mar 2017} 
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\author{Santhosh G O}						%SANTHOSH
\authorspace
\authref  See \hyperlink{Shahul Hameed K}{S.~Hameed}.
\itemspace

\author{[Emilio De Santis]}						%SANTIS
\authorspace
\authref  See \hyperlink{Emilio De Santis}{E.~De~Santis} (under `D').
\itemspace

\author{Raman Sanyal}						%SANYAL
\authorspace
\authref  See \hyperlink{Laura Gellert}{L.~Gellert}.
\itemspace

\author{Mark Sapir}							%SAPIR
\authorspace
\authref  See \hyperlink{Victor Guba}{V.~Guba}.
\itemspace

\author{S.V.\ Sapunov}						%SAPUNOV
\authorspace
\bsgitem{2002a}{}  Equivalence of marked graphs.  [\emph{Or}:  Equivalence of labeled graphs.]  (In Russian.)  \emph{Proceedings of the Institute of Applied Mathematics and Mechanics} [\emph{Tr.\ Inst.\ Prikl.\ Mat.\ Mekh.}], Vol.\ 7, pp.\ 162--167.  Nats.\ Akad.\ Nauk Ukrainy Inst.\ Prikl.\ Mat.\ Mekh., Donetsk, 2002.
\MR{2141811} (2006c:05070).  \Zbl{1081.68074}.  
\annot{} Equivalence of signed graphs (?) that model languages.	
\dated{28 Dec 2011} 
\codes{\CSG?}%
\itemspace

\author{P.B.\ Sarasija}						%SARASIJA
\authorspace
\authref  See \hyperlink{P.\ Nageswari}{P.~Nageswari}.
\itemspace

\author{Kuldeep Sarma}						%SARMA
\authorspace
\authref  See \hyperlink{Debajit Kalita}{D.~Kalita}.
\itemspace

\author{Irasema Sarmiento}						%SARMIENTO
\authorspace
\authref  See also \hyperlink{Joanna A.\ Ellis-Monaghan}{J.A.~Ellis-Monaghan}.
\authrefspace

\bsgitem{1999a}{}  A characterisation of jointless Dowling geometries.  16th British Combinatorial Conf.\ (London, 1997).  \emph{Discrete Math.}\ 197/198 (1999), 713--731.
\MR{1674899} (99m:51020).  \Zbl{929.05016}.					
\annot{} They are 4-closed (determined by their flats of rank 4).  They are characterized, among all matroids, by the statistics of flats of rank $\leq 7$ and therefore by their Tutte polynomials.  There are exceptions in rank 3.
\codes{\CGG:~\CMatrd:~\CInvar}%
\itemspace

\author{Iwao Sato}							%SATO I
\authorspace
\authref  See also \hyperlink{Yusuke Higuchi}{Y.~Higuchi} and \hyperlink{Hirobumi Mizuno}{H.~Mizuno}.
\authrefspace

\bsgitem{2006a}{}  Weighted zeta functions of graph coverings.  \emph{Electronic J.\ Combin.}\ 13 (2006), art.\ R91, 12 pp.
\MR{2274306} (2007k:05138).  \Zbl{1114.05084}.  
\codes{\CGG,~\CSG:~\CCov}%
\itemspace

\bsgitem{2006b}{}  Edge zeta functions of graph coverings.  \emph{Ars Combin.}\ 81 (2006), 225--233.
\MR{2267814} (2007g:05082).  \Zbl{1174.05304}.  
\codes{\Cgg:~\CCov}%
\itemspace

\bsgitem{2008a}{}  The stochastic weighted complexity of a group covering of a digraph.  \emph{Linear Algebra Appl.}\ 429 (2008), 1905--1914. 
\MR{2446628} (2009h:05137).  \Zbl{1144.05322}.					
\annot{} \S3, ``Weighted zeta functions of group covering of digraphs'':  The covering graphs (``derived graphs'') of gain graphs (``voltage graphs'').
\par
\codes{\CGG:~\CCov}%
\itemspace

\author{Shun Sato}							%SATO S
\authorspace
\authref  See \hyperlink{Tatsuya Matsuoka}{T.~Matsuoka}.
\itemspace

\author{Roman V.\ Satyukov}						%SATYUKOV
\authorspace
\authref  See \hyperlink{Irina E.\ Bocharova}{I.E.~Bocharova}.
\itemspace

\authorx{Lawrence Saul}						%SAUL
\author{Lawrence Saul \& Mehran Kardar}
\authorspace
\bsgitem{1993a}{}  Exact integer algorithm for the two-dimensional $\pm J$ Ising spin glass.  \emph{Phys.\ Rev.\ E} 48 (1993), no.\ 5, R3221--R3224.
\annot{} Announcement of \xrefd{1994a}{Lawrence Saul \& Mehran Kardar} with some details, observations, and conclusions.
\dated{18 Aug 2012} 
\codes{\CSG:~\CPhys,~\CFr,~\Cstate:~\CAlgor}%
\itemspace

\bsgitem{1994a}{}  The 2D $\pm J$ Ising spin glass: exact partition functions in polynomial time.  \emph{Nuclear Phys.\ B} 432 [FS] (1994), 641--667.
\annot{} Algorithm for the energy distributions (the partition function) of the states of a randomly signed square, toroidal lattice graph.  Applied to find statistical properties of such a signed graph.
\dated{17 Aug 2012} 
\par
\codes{\CSG:~\CPhys,~\CFr,~\Cstate:~\CAlgor}%
\itemspace

\author{B.\ David Saunders}						%SAUNDERS
\authorspace
\authref  See also \hyperlink{Abraham Berman}{A.~Berman}.
\authrefspace

\author{B.\ David Saunders \& Hans Schneider} 
\authorspace
\bsgitem{1978a}{}  Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices.  \emph{Discrete Math.}\ 24 (1978), 205--220.  
\MR{0522929} (80e:15008).  \Zbl{393.94046}.						
\codes{\Cgg:~\CSw}%
\itemspace

\bsgitem{1979a}{}  Cones, graphs and optimal scalings of matrices.  \emph{Linear Multilinear Algebra} 8 (1979), 121--135.  
\MR{0552356} (80k:15036).  \Zbl{433.15005}.						
\codes{\Cgg:~\CSw}\codesctd{\CRef}%
\itemspace

\author{James Saunderson}						%SAUNDERSON
\authorspace
\authref  See \hyperlink{Tom Coleman}{T.~Coleman}.
\itemspace

\author{Saket Saurabh}						%SAURABH
\authorspace
\authref  See \hyperlink{Avinandan Das}{A.~Das} and \hyperlink{Geevarghese Philip}{G.~Philip}.
\itemspace

\author{D.\ Savithri}							%SAVITHRI D
\authorspace
\authref  See \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\itemspace

\author{H.C.\ Savithri}						%SAVITHRI HC
\authorspace
\authref  See \hyperlink{H.A.\ Malathi}{H.A.~Malathi} and \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Alex Schaefer}						%SCHAEFER
\authorspace
\bsgitem{2017a}{}  \emph{Permutable Matchings and Negative Cycle Vectors.}  Doctoral dissertation, Binghamton University, 2017.
\MR{3781741} (no rev).  
\annot{} Ch.\ 1, ``Signed graphs: background and miscellaneous results''.
\par
\codes{\CSG,~\CSw:~\CExp}
\annotctd{} Ch.\ 2, ``The dimension of the negative cycle vectors of a signed graph'':  Same as \xref{Schaefer and Zaslavsky}{2019a}{Alex Schaefer \& Thomas Zaslavsky}.
\annotctd{} Ch.\ 3, ``Graphs that contain multiply transitive matchings'':  Same as \xref{Schaefer and Swartz}{20xxa}{Alex Schaefer \& Eric Swartz}.
\dated{15 Nov 2017} 
\par
\codes{\CSG:~\CInvar,~\CSw,~\CGeom}%
\itemspace

\author{Alex Schaefer \& Eric Swartz}
\authorspace
\bsgitem{2021a}{}  Graphs that contain multiply transitive matchings.  \emph{European J.\ Combin.}\ 92 (2021), art.\ 103236, 22 pp.
\MR{4149160}.  \Zbl{1458.05219}.  \arXiv{1706.08964}.				
\codes{\CSG:~\CAut}%
\itemspace

\newcommand\NCV{\operatorname{NCV}}

\author{Alex Schaefer \& Thomas Zaslavsky}
\authorspace
\bsgitem{2019a}{}  The dimension of the negative cycle vectors of a signed graph.  \emph{Ars Math.\ Contemp.}\ 16 (2019), no.\ 2, 625--639.
\MR{3963226}.  \Zbl{1416.05133}.  \arXiv{1706.09041}.				
\annot{} Negative cycle vector $c^-(\Sigma) := (c_3^-(\Sigma),\ldots,c_n^-(\Sigma))$, $c_l^-(\Sigma) :=$ \# negative $l$-circles in $\Sigma$.  $\NCV(\Gamma) := \{ c^-(\Gamma,\sigma): \sigma$ signs $\Gamma\}$.  Trivially, $\dim\NCV \leq$ \# circle lengths in $\Gamma$.  Theorem:  Equality for graphs with a strongly permutable matching, including $K_n$, $K_{r,s}$.
\dated{30 Jul 2020} 
\codes{\CSG:~\CInvar,~\CSw,~\CGeom}%
\itemspace

\author{R.H.\ Schelp} 						%SCHELP
\authorspace
\authref   See \hyperlink{Paul Erdos}{P.~Erd\H{o}s} and \hyperlink{Ralph J.\ Faudree}{R.J.~Faudree}.
\itemspace

\author{Baruch Schieber}						%SCHIEBER
\authorspace
\authref   See \hyperlink{Leishen Cai}{L.~Cai}.
\itemspace

\author{Frank Schmidt}						%SCHMIDT F
\authorspace
\bsgitem{2004a}{}  Problems related to type-$A$ and type-$B$ matrices of chromatic joins.  \emph{Adv.\ Appl.\ Math.}\ 32 (2004), no.\ 1-2, 380--390.
\MR{2037637} (2004m:06007).  \Zbl{1050.06003}.  
\codes{\Csg:~\CMatrd}%
\itemspace

\author{Martin Schmidt}						%SCHMIDT M
\authorspace
\authref  See \hyperlink{Lucas Rusnak}{L.~Rusnak}.
\itemspace

\author{R\"udiger Schmidt}						%SCHMIDT R
\authorx{Rudiger Schmidt}
\authorspace
\bsgitem{1979a}{}  On the existence of uncountably many matroidal families.  \emph{Discrete Math.}\ 27 (1979), 93--97.  
\MR{0534956} (80i:05029).  \Zbl{427.05024}.
\annot{} The ``count'' matroids of graphs (see \xref{Whiteley}{1996a}{Walter Whiteley}) and an extensive further generalization of bicircular matroids that includes frame matroids of biased graphs.  His ``partly closed set'' is a linear class of circuits in an arbitrary ``count'' matroid.
\codes{\CGG:~\CMtrdF,~\CBic,~\CECyc:~\CGen}%
\itemspace

\author{Stephan Schmidt}						%SCHMIDT S
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Hans Schneider} 						%SCHNEIDER
\authorspace
\authref  See \hyperlink{Gernot M.\ Engel}{G.M.~Engel}, \hyperlink{Daniel Hershkowitz}{D.~Hershkowitz}, \hyperlink{Uriel G.\ Rothblum}{U.G.~Rothblum}, and \hyperlink{B.\ David Saunders}{B.D.~Saunders}.
\itemspace

\author{Irwin E.\ Schochetman}						%SCHOCHETMAN
\authorspace
\authref  See \hyperlink{Jerrold W.\ Grossman}{J.W.~Grossman}.
\itemspace

\author{Rainer Schrader}						%SCHRADER
\authorspace
\authref  See \hyperlink{Ulrich Faigle}{U.~Faigle}.
\itemspace

\author{Alexander Schrijver} 						%SCHRIJVER
\authorspace
\authref   See also \hyperlink{A.M.H.\ Gerards \& A.\ Schrijver}{A.M.H.~Gerards}.
\authrefspace

\bsgitem{1986a}{}  \emph{Theory of Linear and Integer Programming}.  Wiley, Chichester, 1986.
\MR{0874114} (88m:90090).  \Zbl{665.90063}.
\annot{} Remark 21.2 (p.\ 308) cites \xref{Truemper's}{1982a}{Klaus Truemper} definition of balance of a $0,\pm1$-matrix.
\codes{\Csg:~\Cpar:~\CInc:~\CExp}%
\itemspace

\bsgitem{1989a}{}  The Klein bottle and multicommodity flows.  \emph{Combinatorica} 9 (1989), 375--384.  
\MR{1054013} (92b:90083).  \Zbl{708.05019}.
\annot{} Assume $\Sigma$ embedded in the Klein bottle.  If $\Sigma$ is bipartite, negative girth = max.\ number of disjoint balancing edge sets.  If $\Sigma$ is Eulerian, frustration index = max.\ number of edge-disjoint negative circles.  Proved via polyhedral combinatorics. 
\codes{\CSG:~\CTop,~\CGeom,~\CFr}%
\itemspace

\bsgitem{1990a}{}  Applications of polyhedral combinatorics to multicommodity flows and compact surfaces.  
In:  William Cook and P.D.~Seymour, eds., \emph{Polyhedral Combinatorics} (Morristown, NJ., 1989), pp.\ 119--137.  DIMACS Ser.\ Discrete Math.\ Theor.\ Comp.\ Sci., Vol.\ 1. 
Amer.\ Math.\ Soc.\ and Soc.\ Indust.\ Appl.\ Math., Providence, R.I., 1990. 
\MR{1105122} (92d:05057).  \Zbl{727.90025}.
\annot{} \S2:  ``The Klein bottle,'' surveys \xrefd{1989a}{Alexander Schrijver}.	
\codes{\CSG:~\CTop,~\CGeom,~\CFr:~\CExp}%
\itemspace

\bsgitem{1990b}{}  Homotopic routing methods.  
In:  B.~Korte, L.~Lov\'asz, H.J.~Pr\"omel, and A.~Schrijver, eds., \emph{Paths, Flows, and VLSI-Layout}, pp.\ 329--371.  Algorithms and Combinatorics, Vol.\ 9.  Springer-Verlag, Berlin, 1990. 
\MR{1083385} (92f:\-68139).  \Zbl{732.90087}.
\annot{} \S4:  ``Edge-disjoint paths in planar graphs,'' pp.\ 342--345, ``The projective plane and the Klein bottle,'' surveys \xrefd{1989a}{Alexander Schrijver}.
\par
\codes{\CSG:~\CTop,~\CGeom,~\CFr:~\CExp}%
\annotctd{} \S3:  ``Edge-disjoint paths and multicommodity flows,'' pp.\ 334 ff.  [This work suggests there may be a signed-graph generalization with the theorems discussed corresponding to all-negative signatures.]   
\par
\codes{\Cpar:~\CPaths:~\CExp}%
\itemspace

\bsgitem{1991a}{}  Disjoint circuits of prescribed homotopies in a graph on a compact surface.  \emph{J.\ Combin.\ Theory Ser.\ B} 51 (1991), 127--159.  
\MR{1088630} (92a:05048).  \Zbl{723.05050}.
\annot{} \S2:  ``An auxiliary theorem on linear inequalities,'' concerns feasibility of inequalities with coefficient matrix containing incidence matrix of $-\Gamma$.  [See \xref{Hurkens}{1989a}{C.A.J.\ Hurkens}.]
\codes{\Cecyc:~\CInc}%
\itemspace

\bsgitem{1991b}{}  (As ``A.\ Skhre\u{\i}ver'')  \emph{Teoriya line{\u\i}nogo i tselochislennogo programmirovaniya}, Vols.\ 1 and 2.  Mir, Moscow, 1991.
\MR{1224001} (94c:90003), \MR{1240318} (94g:90005).  
\annot{} Russian translation of \xrefd{1986a}{Alexander Schrijver}.
\codes{\Csg:~\Cpar:~\CInc:~\CExp}%
\itemspace

\bsgitem{2002a}{}  A short proof of Guenin's characterization of weakly bipartite graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 85 (2002), 255--260.
\MR{1912966} (2003e:05119).  \Zbl{1024.05079}.					
\annot{} A streamlined proof of the theorem of \xref{Guenin}{2001a}{Bertrand Guenin}.
\par
\codes{\CSG:~\CGeom,~\CStr}%
\itemspace

\bsgitem{2003a}{}  \emph{Combinatorial Optimization: Polyhedra and Efficiency}.  Vol.\ A, \emph{Paths, Flows, Matchings}.  Vol.\ B, \emph{Matroids, Trees, Stable Sets}.  Vol.\ C, \emph{Disjoint Paths, Hypergraphs}.
Algor.\ Combin., Vol.\ 24 A, B, C.  Springer, Berlin, 2003.
\MR{1956924} (2004b:90004a), \MR{1956925} (2004b:90004b), \MR{1956926} (2004b:90004c).  \Zbl{1041.90001}, \Zbl{1072.90030}.								
\annot{} Vol.\ A, Ch.\ 36, ``Bidirected graphs''.  
\annotctd{} Vol.\ B, \S68.6b, ``Bidirected graphs'', includes incidence matrix and signed graph.  
\S68.6c, ``Characterizing odd-$K_4$-free graphs by mixing stable sets and vertex covers'':  ``Odd-$K_4$'' = $-K_4$ minor of $-\Gamma$.
\annotctd{} Vol.\ C, Ch.\ 75, ``Cuts, odd circuits, and multiflows''.  Signed graphs, weakly and strongly balanced (``bipartite'') signed graphs.  \S75.2, ``Signed graphs''.	
Ch.\ 78, ``Ideal hypergraphs''.  
\S80.4, ``On characterizing binary ideal hypergraphs''.  
\annotctd{} Dictionary:  ``positive, negative'' edge = extraverted or introverted negative edge, ``directed'' edge = positive edge, ``even, odd'' = positive, negative (edge or circle), ``bipartite'' = balanced.  
\dated{9 Jun 2011, 31 Dec 2020} 
\codes{\Csg:~\COri:~\CInc,~\CGeom}%
\annotctd{} Vol.\ C, Ch.\ 76, ``Homotopy and graphs on surfaces''.  
\dated{9 Jun 2011} 
\codes{\Cgg:~\CTop}%
\itemspace

\author{Konrad Schr{\o}der}						%SCHRODER
\authorx{Konrad Schroder}
\authorspace
\bsgitem{1995a}{}  Mixed-sign conductor networks.  REU paper, University of Washington, 1995.  URL \url{http://www.math.washington.edu/~reu/papers/1993/schroder/sign.pdf}
\annot{} Partial treatment of the problem in \xref{W.~Johnson}{2012a}{Will Johnson}.
\dated{26 Dec 2012} 
\codes{\Csg:~\CWG:~\CAdj}%
\itemspace

\author{Michael W.\ Schroeder}						%SCHROEDER
\authorspace
\authref  See \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}.
\itemspace

\author{Michael Schubert}						%SCHUBERT
\authorspace
\authref  See also \hyperlink{You Lu}{Y.~Lu}, \hyperlink{Edita Rollova}{E.~Rollov\'a}, and \hyperlink{Eckhard Steffen}{E.~Steffen}.
\authrefspace

\author{Michael Schubert \& Eckhard Steffen}
\authorspace
\bsgitem{2015a}{}  Nowhere-zero flows on signed regular graphs.  \emph{European J.\ Combin.}\ 48 (2015), 34--47.
\MR{3339010}.  \Zbl{1315.05070}.  \arXiv{1307.1562}.				
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\author{Michelle Schultz} 						%SCHULTZ
\authorspace
\authref   See \hyperlink{Gary Chartrand}{G.~Chartrand}.
\itemspace

\author{Felix Schwagereit}						%SCHWAGEREIT
\authorspace
\authref  See \hyperlink{Jerome Kunegis}{J.~Kunegis}.
\itemspace

\author{Gary K.\ Schwartz}						%SCHWARTZ GK
\authorspace
\bsgitem{2002a}{}  On the automorphism groups of Dowling geometries.  \emph{Combin.\ Probab.\ Comput.}\ 11 (2002), no.\ 3, 311--321.
\MR{1909505} (2004c:20005).  \Zbl{1008.06007}.
\annot{} $\Aut Q_n(\fG)$ factors in a certain natural way if, but also only if, $\fG$ factors.  [Succeeds \xref{Bonin}{1995a}{Joseph E.\ Bonin}.  \Cf\ \xref{Sikiri\'c, Felikson, and Tumarkin}{2011a}{Mathieu Dutour Sikiric, Anna Felikson, \& Pavel Tumarkin} for (mostly) more restricted related results.]
\dated{rev.\ 9 Apr 2016}
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\author{Roy Schwartz}						%SCHWARTZ R
\authorspace
\authref  See \hyperlink{Moses Charikar}{M.~Charikar}.
\itemspace

\authorx{W.\ Schw\"arzler}						%SCHWARZLER
\authorx{W.\ Schwarzler}
\author{W.\ Schw\"arzler \& D.J.A.\ Welsh}
\authorx{W.\ Schwarzler \& D.J.A.\ Welsh}
\authorspace
\bsgitem{1993a}{} Knots, matroids and the Ising model.  \emph{Math.\ Proc.\ Cambridge Philos.\ Soc.}\ 13 (1993), 107--139.
\MR{1188822} (94c:57019).  \Zbl{797.57002}.
\annot{} Tutte and dichromatic polynomials of signed matroids, generalized from \xref{Kauffman}{1989a}{Louis H.\ Kauffman}; this is the $2$-colored case of \xref{Zaslavsky's}{1992b}{Thomas Zaslavsky} strong Tutte functions of colored matroids.  [For terminology see \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}.]  Applications to knot theory.
\annotctd{} \S2, ``A matroid polynomial'', is foundational.  
Prop.\ 2.1 characterizes strong Tutte functions of signed matroids by two equations connecting their parameters and their values on signed coloops and loops.  [If the function is $0$ on positive coloops, the proof is incomplete and the functions = $0$ except on $M = \eset$ are missed.]  
Prop.\ 2.2:  The Tutte (basis-expansion) polynomial of a function $W$ of signed matroids is well defined iff $W$ is a strong Tutte function.  Eq.\ (2.8) says $W =$ the rank generating polynomial $Q_\Sigma$ (here also called $W$) if certain variables are nonzero; (2.9) shows there are only $3$ essential variables since, generically, only the ratio of parameters is essential [an observation that applies to general strong Tutte functions].
Prop.\ 2.5 computes $Q_\Sigma$ of a $2$-sum.
\annotctd{} \S3 adapts $Q_\Sigma$ to \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} and \xref{Murasugi's}{1989a}{Kunio Murasugi} signed-graph polynomials and simplifies some of the latter's results (esp.\ his chromatic degree).  \S4, ``The anisotropic Ising model'', concerns the Hamiltonian of a state of a signed graph.  The partition function is essentially an evaluation of $Q_\Sigma$.  \S5, ``The bracket polynomial'', and \S6, ``The span of the bracket polynomial'': Certain substitutions reduce $Q_\Sigma$ to $1$ variable; its properties are examined, esp.\ in light of knot-theoretic questions.  Thm.\ 6.4 characterizes signed matroids with ``full span'' (a degree property).  \S7, ``Adequate and semi-adequate link diagrams'', generalizes those notions to signed matroids.  \S8, ``Zero span matroids'': when does span(bracket) = $0$?  Yes if $M = M(\Sigma)$ where $\Sigma$ reduces by Reidemeister moves to $K_1$, but the converse is open (and significant if true).
\par
\codes{\CSc(\CMatrd),~\CSGc:~\CInvar,~\CKnot,~\CPhys}%
\itemspace

\author{Allen J.\ Schwenk} 						%SCHWENK
\authorspace
\authref  See \xref{Harary, Palmer, Robinson, and Schwenk}{1977a}{Frank Harary, Edgar M.\ Palmer, Robert W.\ Robinson, \& Allen J.\ Schwenk}.
\itemspace

\authorx{Thomas Schweser}						%SCHWESER
\author{Thomas Schweser \& Michael Stiebitz}
\authorspace
\bsgitem{2017a}{}  Degree choosable signed graphs.  \emph{Discrete Math.}\ 340 (2017), no.\ 5, 882--891.
\MR{3612419}.  \Zbl{1357.05055}.  \arXiv{1507.04569}.				
\codes{\CSG:~\CCol}%
\itemspace

\author{Irene Sciriha}						%SCIRIHA
\authorspace
\authref  See also \hyperlink{Nino Basic}{N.~Basic} and \hyperlink{Francesco Belardo}{F.~Belardo}.
\authrefspace

\author{Irene Sciriha \& Luke Collins}
\authorspace
\bsgitem{2019a}{}  Two-graphs and NSSDs: An algebraic approach.  \emph{Discrete Appl.\ Math.}\ 266 (2019), 92--102.
\MR{3991601}.  \Zbl{1464.05250}.  
\annot{} Signed $K_n$ with two eigenvalues, especially negatives of each other, leading to properties of conference graphs.  ``NSSD'' = non-singular graph with a singular deck.
\dated{4 Jun 2018, 11 Jul 2022} 
\par
\codes{\Csg:~\Ckg,~\CAdj:~\CTG,~\CSw,~\CEig}%
\itemspace

\author{Matt Scobee}						%SCOBEE
\authorspace
\authref  See \hyperlink{Jon Lee \& Matt Scobee}{J.~Lee}.
\itemspace

\authorx{Alexander D.\ Scott}						%SCOTT
\author{Alexander D.\ Scott \& Alan D.\ Sokal}
\authorspace
\bsgitem{2009a}{}  Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model).  \emph{S\'em.\ Lotharingien Combin.}\ 61A (2009), art.\ B61Ae, 33 pp.
\MR{2529396} (2010i:05167).  \Zbl{1283.05138}.  
\annot{} \Cf\ \xref{Sokal}{2005a}{Alan D.\ Sokal}.
\codes{\CSGw:~\CGen:~\CInvar}%
\itemspace

\author{Andr\'as Seb\"o}						%SEBO
\authorx{Andras Sebo}
\authorspace
\authref  See also \hyperlink{Frederic Meunier}{F.~Meunier} and \hyperlink{Beth Novick}{B.~Novick}. 
\authrefspace

\bsgitem{1990a}{}  Undirected distances and the postman-structure of graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 49 (1990), 10--39.
\MR{1056818} (91h:05049).  \Zbl{638.05032}.					
\annot{} See \xref{A.\ Frank}{1996a}{Andras Frank}.
\codes{\CSGw:~\CStr}%
\itemspace

\authorx{Etsuo Segawa}						%SEGAWA
\author{Etsuo Segawa \& Yusuke Yoshie}
\authorspace
\bsgitem{2021a}{}  Quantum search of matching on signed graphs.  \emph{Quantum Inform.\ Process.}\ 20 (2021), no.\ 5, art.\ 182, 23 pp.
\MR{4259185}.  
\arXiv{2007.07223}.  
\annot{} \Cf\ \xref{Brown, Godsil, Mallory, Raz, Tamon}{2013a}{John Brown, Chris Godsil, Devlin Mallory, Abigail Raz, \& Christino Tamon}.
\codes{\CSG:~\CKG}%
\itemspace

\authorx{Deepak Sehrawat}						%SEHRAWAT	%DEEPAK
\author{Deepak Sehrawat \& Bikash Bhattacharjya}
\authorspace
\bsgitem{2019a}{}  Some bounds on the double domination of signed generalized Petersen graphs and signed I-graphs.  \emph{Indian J.\ Discrete Math.}\ 5 (2019), no.\ 2, 63--75.
\MR{4317623}.  \arXiv{1907.11099}.  
\annot{} \Cf\ \xref{Ashraf and Germina}{2016a}{P.K.\ Ashraf \& K.A.\ Germina}.  Cubic:  $\gamma_{\times2}(\Sigma) \geq n/2$.  $k$-step generalized Petersen graph $P_{m,k}$: upper bounds for $k=1$, $k>1$ and $\gcd(m,k)=1$ or $>1$, are $\approx m+2,\ \frac32 m,\ \frac43m$.  Similar bounds for I-graphs.  
``I-graph'': bipartite, union of 3 particular matchings.
\dated{26 Dec 2020} 
\codes{\CSG:~\CSw,~\CDom}%
\itemspace

\bsgitem{2019b}{}  Maximum frustration in signed generalized Petersen graphs.  \emph{Indian J.\ Discrete Math.}\ 5 (2019), no.\ 2, 77--93.
\MR{4317624}.  \arXiv{1905.05548}.  
\annot{} $D(P_{m,k}) \leq 1+\lfloor m/2 \rfloor$ if $d:=\gcd(m,k) = 1$, $\leq 1+ d(1+\lfloor m/2d \rfloor)$ if $d>1$, thus improving \xref{Sivaraman}{2012a}{Vaidy Sivaraman} for these graphs.
\dated{26 Dec 2020} 
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{2020a}{}  Signed complete graphs on six vertices and their frustration indices.  \emph{Adv.\ Appl.\ Discrete Math.}\ 24 (2020), no.\ 2, 129--142.
\arXiv{1812.08383}.								
\annot{} Thm.\ 4.2:  There are 16 switching isomorphism types.  [Previously found as ``two-graphs'', \cf\ \xref{Bussemaker, Mathon, and Seidel}{1979a}{F.C.\ Bussemaker, R.A.\ Mathon, \& J.J.\ Seidel}, \xrefd{1981a}{F.C.\ Bussemaker, R.A.\ Mathon, \& J.J.\ Seidel}.]  New:  Proof, unique minimal representatives, and (Thms.\ 5.1, 5.2) frustration indices $l$ and frustration numbers $l_0$.  E.g., $\max l = 6$ and $\max l_0 = 4$, uniquely for $E^- = C_3 \cupdot\ C_3$.  [This is useful information.]
\dated{16 Apr 2019, 28 Dec 2020} 
\codes{\CSG:~\Ctg:~\CSw,~\CFr}%
\itemspace

\bsgitem{2021a}{}  Non-isomorphic signatures on some generalised Petersen graph.  \emph{Electronic J.\ Graph Theory Appl.}\ 9 (2021), no.\ 2, 235--255.
\MR{4331324}.  \Zbl{1482.05141}.  \arXiv{1812.11360}.				
\annot{} Number of switching isomorphism types of $(P_{2n+1,1},\sigma)$.
\dated{16 Apr 2019} 
\codes{\CSG:~\CSw:~\CEnum}%
\itemspace

\bsgitem{2022a}{}  Chromatic polynomials of signed book graphs.  \emph{Theory Appl.\ Graphs} 9 (2022), no.\ 1, art.\ 4, 11 pp.
\MR{4398619}.  \Zbl{1486.05103}.  \arXiv{2206.08580} (\arXiv{1812.08382}).		
\annot{} Book graph $B(m,n) :=$ union of $n$ circles $C_m$ at a single common edge.  \# switching isomorphism types $=n+1$ (depending on number of negative $C_m$'s).  Explicit formulas for chromatic and zero-free chromatic polynomials.
\dated{16 Apr 2019, rev 24 Mar 2022} 
\codes{\CSG:~\CSw,~\CInvar}%
\itemspace

\bsgitem{20xxa}{}  Some bounds on the double domination of signed generalized petersen graphs and signed I-graphs.  \emph{}\ 
\arXiv{1907.11099}.							
\codes{\CSG:~\CDom}%
\itemspace

\bsgitem{20xxb}{}  RNA number of some parity signed generalized Petersen graphs.  Submitted.  \emph{}\ 
\arXiv{2110.03264}.  
\annot{} See \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara} for \emph{rna} number.
\codes{\CLab:~\CSG}%
\itemspace

\bsgitem{20xxc}{}  On the \emph{rna} number of generalized Petersen graphs.  \emph{Commun.\ Combin.\ Optim.}\ (to appear)
\annot{} See \xref{Acharya and Kureethara}{2021a}{Mukti Acharya \& Joseph Varghese Kureethara} for \emph{rna} number.
\codes{\CLab:~\CSG}%
\itemspace

\bsgitem{20xxd}{}  Enumeration of switching non-isomorphic signed wheels.  \emph{}\ 
\arXiv{2106.01265}.							
\codes{\CSG:~\CSw:~\CEnum}%
\itemspace

\author{J.J.\ Seidel} 							%SEIDEL*
\authorspace
\authref  See also \hyperlink{F.C.\ Bussemaker}{F.C.~Bussemaker}, \hyperlink{Peter J.\ Cameron}{P.J.~Cameron}, \hyperlink{P.W.H.\ Lemmens}{P.W.H.~Lemmens}, and \hyperlink{J.H.\ van Lint}{J.H.~van Lint}.
\authrefspace

\bsgitem{1968a}{} Strongly regular graphs with $(-1,1,0)$ adjacency matrix having eigenvalue 3.  \emph{Linear Algebra Appl.}\ 1 (1968), 281--298.  
\MR{0234861} (38 \#3175).  \Zbl{159.25403} (159, p.\ 254c).  
Reprinted in \xrefd{1991a}{J.J.\ Seidel}, pp.\ 26--43.					
\codes{\Ctg}
\itemspace

\bsgitem{1969a}{}  Strongly regular graphs.  
In:  W.T.~Tutte, ed., \emph{Recent Progress in Combinatorics} (Proc.\ Third Waterloo Conf.\ on Combinatorics, 1968), pp.\ 185--198.  Academic Press, New York, 1969.  
\MR{0253935} (54 \#10047).  \Zbl{191.55202} (191, p.\ 552b).				
\codes{\CTG}%
\itemspace

\bsgitem{1974a}{} Graphs and two-graphs.  
In:  F.~Hoffman \etal, eds., \emph{Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory, and Computing} (Boca Raton, 1974), pp.\ 125--143.
Congressus Numerantium X.  Utilitas Math.\ Publ.\ Inc., Winnipeg, Man., 1974.  
\MR{0364028} (51 \#283).  \Zbl{308.05120}.					
\annot{} \Cf\ \xref{Mallows and Sloane}{1975a}{C.L.\ Mallows \& N.J.A.\ Sloane}.
	
\codes{\CTG}%
\itemspace

\bsgitem{1976a}{\dag}  A survey of two-graphs.  
In: \emph{Colloquio Internazionale sulle Teorie Combinatorie} (Roma, 1973), Tomo I, pp.\ 481--511.  Atti dei Convegni Lincei, No.\ 17.  Accad.\ Naz.\ Lincei, Rome, 1976.  
\MR{0550136} (58 \#27659).  \Zbl{352.05016}.					
Reprinted in \xrefd{1991a}{J.J.\ Seidel}, pp.\ 146--176.
\codes{\CTG:~\CAdj,~\CEig,~\CCov,~\CAut}%
\itemspace

\bsgitem{1978a}{}  Eutactic stars.  
In:  A.~Hajnal and Vera T.\ S\'os, eds., \emph{Combinatorics} (Proc.\ Fifth Hungar.\  Colloq., Keszthely, 1976), Vol.\ 2, pp.\ 983--999.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 18.  North-Holland, Amsterdam, 1978.  
\MR{0519322} (80d:05016).  \Zbl{391.05050}.						
\itemspace

\bsgitem{1979a}{}  The pentagon.  In:  Allan Gewirtz and Louis V.\ Quintas, eds., Second Int.\ Conf.\ Combin.\ Math.\ (New York, 1978).  \emph{Ann.\ New York Acad.\ Sci.}\ 319 (1979), 497--507.  
\MR{0556060} (81e:05004).  \Zbl{417.51005}.   
\codes{\CTG:~\CAdj,~\CEig}%
\itemspace

\bsgitem{1979b}{}  The pentagon.  
In:  P.C.\ Baayen \etal, eds., \emph{Proceedings, Bicentennial Congress, Wiskundig Genootschaap} (Amsterdam, 1978), Part I, pp.\ 80--96.  
Mathematical Center Tracts, 100.  Mathematisch Centrum, Amsterdam, 1979.  
\MR{0541389} (80f:51008).  \Zbl{417.51005}.
\annot{} Same as \xrefd{1979a}{J.J.\ Seidel}, with photograph. 
\codes{\CTG:~\CAdj,~\CEig}%
\itemspace

\bsgitem{1991a}{}  \emph{Geometry and Combinatorics:  Selected Works of J.J.~Seidel}.
D.G.~Corneil and R.~Mathon, eds.  Academic Press, Boston, 1991.
\MR{1116326} (92m:01098).  \Zbl{770.05001}.
\annot{} Reprints many articles on two-graphs and related systems.
\par
\codes{\CTG:~\CSw,~\CAdj,~\CEig,~\CGeom}%
\itemspace

\bsgitem{1992a}{}  More about two-graphs.  
In:  Jaroslav Ne\v{s}et\v{r}il and Miroslav Fiedler, eds., \emph{Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity} (Prachatice, 1990), pp.\ 297--308.  Ann.\ Discrete Math., Vol.\ 51.  North-Holland, Amsterdam, 1992.  
\MR{1206283} (94h:05040).  \Zbl{764.05036}.						
\codes{\CTG:~\CExp,~\CRef}%
\itemspace

\bsgitem{1995a}{}  Geometric representations of graphs.  \emph{Linear Multilinear Algebra} 39 (1995), 45--57.  
\MR{1374470} (97e:05149a).  \Zbl{832.05079}.						
Errata.  \emph{Ibid.}\ 39 (1995), 405.
\MR{1399446} (97e:05149b).  \Zbl{843.05078}.
\annot{} \S4, ``Signed graphs'': The ``intersection matrix'' $A+2I$ of a signed simple graph is the Gram matrix of a set of ``root vectors'' with respect to an ``inner product'' that may not be positive definite.  Explains origin of local switching (\cf\ \xref{Cameron, Seidel, and Tsaranov}{1994a}{P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov} and \xref{Bussemaker, Cameron, Seidel, and Tsaranov}{1991a}{F.C.\ Bussemaker, P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov}).  For a signed complete graph, $A+3I$ represents lines at angles $\cos\inv 1/3$; it is positive semidefinite only for few graphs, which are classified (implicit in \xref{Lemmens and Seidel}{1973a}{P.W.H.\ Lemmens \& J.J.\ Seidel}).
\codes{\CSG:~\CAdj,~\CEig,~\CGeom:~\CExp}%
\itemspace

\bsgitem{1995b}{}  Discrete non-Euclidean geometry.  
In:  F.~Buekenhout, ed., \emph{Handbook of Incidence Geometry: Buildings and Foundations}, Ch.\ 15, pp.\ 843--920.  
North-Holland (Elsevier), Amsterdam, 1995.  
\MR{1360730} (96m:52001).  \Zbl{826.51012}.
\annot{} \S3.2: ``Equidistant sets in elliptic $(d - 1)$-space.''  \S3.3: ``Regular two-graphs.''
\codes{\CTG:~\CAdj,~\CEig,~\CGeom:~\CExp}%
\itemspace

\author{J.J.\ Seidel \& D.E.\ Taylor} 
\authorspace
\bsgitem{1981a}{} Two-graphs, a second survey.  
In:  L.~Lov\'asz and Vera T.\ S\'os, eds., \emph{Algebraic Methods in Graph Theory} (Proc.\ Int.\ Colloq., Szeged, 1978), Vol.\ II, pp.\ 689--711.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 25.  J\'anos Bolyai Math.\ Soc., Budapest, and North-Holland, Amsterdam, 1981.  
\MR{0642068} (83f:05070).  \Zbl{475.05073}.					
Reprinted in \xrefd{1991a}{J.J.\ Seidel}, pp.\ 231--254.
\codes{\CTG}%
\itemspace

\author{J.J.\ Seidel \& S.V.\ Tsaranov} 
\authorspace
\bsgitem{1990a}{}  Two-graphs, related groups, and root systems.  Algebra, Groups and Geometry.  \emph{Bull.\ Soc.\ Math.\ Belg.\ Ser.\ A} 42 (1990), 695--711.  
\MR{1316218} (95m:20046).  \Zbl{736.05048}.
\annot{} A group $\Ts(\Sigma)$ is defined from a signed complete graph $\Sigma$: its generators are the vertices and its relations are $(uv^{-\sigma(uv)})^2 = 1$ for each edge $uv$.  It is invariant under switching, hence determined by the two-graph of $\Sigma$.  A certain subgraph of a Coxeter group of a tree $T$ is isomorphic to $\Ts(\Sigma)$ for suitable $\Sigma_T$ constructed from $T$.  [Generalized in \xref{Cameron, Seidel, and Tsaranov}{1994a}{P.J.\ Cameron, J.J.\ Seidel, \& S.V.\ Tsaranov}.  More on $\Sigma_T$ under \xref{Tsaranov}{1992a}{S.V.\ Tsaranov}.  The construction of $\Sigma_T$ is simplified in \xref{Cameron}{1994a}{Peter J.\ Cameron}.]
\codes{\CTG:~\CAdj,~\CGeom}%
\itemspace

\author{Chelliah Selvaraj}						%SELVARAJ
\authorspace
\authref  See also \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\authrefspace

\bsgitem{2007a}{}  Factor algebras of signed Brauer's algebras.  \emph{Kyungpook Math.\ J.}\ 47 (2007), no.\ 4, 549--568.
\MR{2397479} (2009b:16076).  \Zbl{1187.16013}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\authorx{Charles Semple}						%SEMPLE
\authorspace
\authref  See also \hyperlink{Tara Fife}{T.~Fife}.
\authrefspace

\author{Charles Semple \& Geoff Whittle}
\authorspace
\bsgitem{1996a}{}  Partial fields and matroid representation.  \emph{Adv.\ Appl.\ Math.}\ 17 (1996), 184--208.  
\MR{1390574} (97g:05046).  \Zbl{859.05035}.						
\annot{} \S7: ``Dowling group geometries''.  A Dowling geometry of a group $\fG$ has a partial-field representation iff $G$ is abelian and has at most one involution.  [The condition is necessary but insufficient; see \xref{Vertigan}{2015a}{Dirk Vertigan}, 
or \xref{Pendavingh and van Zwam}{2013a}{R.A.\ Pendavingh \& S.H.M.\ van Zwam}, p.\ 225.]
\codes{\Cgg:~\CMatrd:~\CInc}%
\itemspace

\author{Parongama Sen}						%SEN P
\authorspace
\authref  See \hyperlink{Bikas K.\ Chakrabarti}{B.K.~Chakrabarti}.
\itemspace

\author{Sagnik Sen}							%SEN S
\authorspace
\authref  See also \hyperlink{Julien Bensmail}{J.~Bensmail}, \hyperlink{Sandip Das}{S.~Das}, \hyperlink{Pascal Ochem}{P.~Ochem} and \hyperlink{Reza Naserasr}{R.~Naserasr}.
\authrefspace

\bsgitem{2014a}{}  \emph{A Contribution to the Theory of Graph Homomorphisms and Colorings}.  Doctoral dissertation, Univ.\ de Bordeaux, 2014. 
\HAL{tel-00960893}.  
\annot{} Ch.\ 5, ``Signified graphs''.  \S5.3, ``Signified coloring'' [coloring is strange; unrelated to \xref{Zaslavsky}{1982b}{Thomas Zaslavsky}].  Ch.\ 6, ``Signed graphs''.  \S6.3, ``Signed coloring'' (mostly homomorphisms).
Dictionary:  Very strange terminology due in part to avoiding use of signs.  ``Signified'' = signed, ``signed'' = switching class, ``unbalanced path'' = negative path, ``odd'' = antibalanced, ``even'' = bipartite, etc.
\dated{13 Oct 2021} 
\par
\codes{\CSG:~\CCol,~\CHom,~\CSw}%
\itemspace

\author{Sylvain Sen\'e}						%SENE
\authorx{Sylvain Sene}
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot} and \hyperlink{Mathilde Noual}{M.~Noual}.
\itemspace

\author{Masakazu Sengoku}						%SENGOKU
\authorspace
\bsgitem{1974a}{}  On hybrid tree graphs. 
\emph{Electronic Commun.\ Japan} 57 (1974), no.\ 5, 18--23. 
\MR{0456991} (56 \#15210).  
\annot{} A signed graph derived from trees and cotrees is balanced.
	
\dated{24 July 2010} 
\codes{\CSG:~\CBal}%
\itemspace

\author{Seunghyun Seo}						%SEO
\authorspace
\bsgitem{2012a}{}  Shi threshold arrangement.  \emph{Electronic J.\ Combin.}\ 19 (2012), no.\ 3, art.\ P39, 9 pp.
\MR{2988861}.  \Zbl{1257.52009}.  
\annot{} The characteristic polynomial of the Shi threshold arrangement $\{x_i+x_j=0,1: i<j\}$, computed modulo a large prime (the ``finite field method'', \xref{Athanasiadis}{1996a}{Christos A.\ Athanasiadis}).
	
\dated{14 Mar 2013, corr 22 Jan 2020} 
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2017a}{}  The Catalan threshold arrangement.  \emph{J.\ Integer Seq.}\ 20 (2017), art.\ 17.1.1, 12 pp.
\MR{3606971}.  \Zbl{1354.52026}.  
\annot{} The characteristic polynomial of the Catalan threshold arrangement $\{x_i+x_j = 0, \pm 1: i \neq j\}$, computed modulo a large prime.
\dated{20 Feb 2020} 
\codes{\Cgg:~\CGeom,~\CInvar}%
\itemspace

\author{B.\ Seoane}							%SEOANE
\authorspace
\authref  See \hyperlink{L.A.\ Fernandez}{L.A.~Fern\'andez}.
\itemspace

\author{Han-Guk Seol}						%SEOL HG
\authorspace
\authref  See \hyperlink{Sang-Gu Lee}{S.-G.~Lee}.
\itemspace

\author{Mark R.\ Sepanski}						%SEPANSKI
\authorspace
\authref  See \hyperlink{Isaac B.\ Michael}{I.B.~Michael}.
\itemspace

\author{Jean-Se\'bastien Sereni}					%SERENI
\authorx{Jean-Sebastien Sereni}
\authorspace
\authref  See \hyperlink{Daniel Kral'}{D.~Kr\'al'}.
\itemspace

\author{\'Akos Seress}						%SERESS*
\authorx{Akos Seress}
\authorspace
\authref  See \hyperlink{Peter Brooksbank}{P.~Brooksbank}.
\itemspace

\author{Ritesh Seth}							%SETH R
\authorspace
\authref  See \hyperlink{Sandip Das}{S.~Das}.
\itemspace

\author{Anshu Sethi}							%SETHI
\authorspace
\authref  See \hyperlink{Deepa Sinha \& Anshu Sethi}{D.~Sinha}.
\itemspace

\author{James P.\ Sethna}						%SETHNA
\authorspace
\bsgitem{2006a}{}  \emph{Statistical Mechanics:  Entropy, Order Parameters, and Complexity}.  Oxford Master Ser.\ in Physics, Vol.\ 14.  Oxford Univ.\ Press, Oxford, 2006.
\Zbl{1140.82004}.  
\annot{} Textbook.  
P.\ 12, fn.\ 16:  Frustration index (``spin-glass ground states'') is polynomially equivalent to graph coloring.  
\S12.3.4, ``Glassy systems: random but frozen'', mentions frustration due to negative circles (``a loop with an odd number of antiferromagnetic couplings'').  It is not yet known how many equilibrium [ground?] states exist.  Fig.\ 12.17, ``Frustration'':  An all-negative triangle with Ising spins ($\pm1$).  
\dated{28 Aug 2012} 
\codes{\CPhys:~\CSG:~\CFr,~\CState(\Cfr):~\CExp}%
\itemspace

\author{E.C.\ Sewell}							%SEWELL
\authorspace
\bsgitem{1996a}{}  Binary integer programs with two variables per inequality.  \emph{Math.\ Programming} 75 (1996), Ser.\ A, 467--476.
\MR{1422181} (97m:90059).  \Zbl{874.90138}.						
\annot{} See \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg} for definitions.
\S2, ``Equivalence to stable set problem'':  Optimization on the bidirected stable set polytope is reduced to optimization on a stable set polytope with no more variables.  Results of \xref{Bourjolly}{1988a}{Jean-Marie Bourjolly} and \xref{Hochbaum, Megiddo, Naor, and Tamir}{1993a}{Dorit S.\ Hochbaum, Nimrod Megiddo, Joseph (Seffi) Naor, \& Arie Tamir} can thereby be explained.
\S3, ``Perfect bigraphs'', proves the conjectures of \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg}: a transitively closed bidirection of a simple graph is perfect iff its underlying graph is perfect.  
[Also proved by \xref{Ikebe and Tamura}{20xxa}{Yoshiko T.\ Ikebe \& Akihisa Tamura}.]
Dictionary: ``Bigraph'' = bidirected graph $B$.  ``Stable'' set in $B$ = vertex set inducing no introverted edge.
\codes{\CSG:~\COri:~\CInc,~\CGeom,~\Csw}%
\itemspace

\author{E.C.\ Sewell \& L.E.\ Trotter, Jr.}
\authorspace
\bsgitem{1993a}{}  Stability critical graphs and even subdivisions of $K_4$.  \emph{J.\ Combin.\ Theory Ser.\ B} 59 (1993), 74--84.
\MR{1234384} (94f:05122).  \Zbl{793.05133}.  
\annot{} ``Even subdivision of $K_4$'' = $|\Sigma|$ where $\Sigma$ is an all-negative subdivision of $-K_4$.
	
\codes{\Csg:~\Cpar:~\CStr}%
\itemspace

\bsgitem{1995a}{}  Stability critical graphs and ranks facets of the stable set polytope.  \emph{Discrete Math.}\ 147 (1995), 247--255.
\MR{1364517} (96g:05077).  \Zbl{838.05068}.  
\codes{\Csg:~\Cpar:~\CStr}%
\itemspace

\author{P.D.\ Seymour} 						%SEYMOUR
\authorspace
\authref   See also \hyperlink{Maria Chudnovsky}{M.~Chudnovsky}, \hyperlink{Jim Geelen}{J.~Geelen}, \xref{Gerards, Lov\'asz, \etal{}}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}, \hyperlink{William McCuaig}{W.~McCuaig}, \hyperlink{Bojan Mohar}{B.~Mohar}, and \hyperlink{Neil Robertson}{N.~Robertson}.
\authrefspace

\bsgitem{1974a}{}  On the two-colouring of hypergraphs.  \emph{Quart.\ J.\ Math.\ Oxford} (2) 25 (1974), 303--312.
\MR{0371710} (51 \#7927).  \Zbl{299.05122}.					
\codes{\Csd:~\CPar:~\Cbal}%
\itemspace

\bsgitem{1977a}{}  The matroids with the max-flow min-cut property.  \emph{J.\ Combin.\ Theory Ser.\ B} 23 (1977), 189--222.
\MR{0462996} (57 \#2960).  \Zbl{375.05022}.					
\annot{} The central example is $Q_6 = \cC^-(-K_4)$, the clutter of (edge sets of) negative circles in $-K_4$.  P.\ 199: the extended lift matroid $\bfL_0(-K_4) = F_7^*$, the dual Fano matroid.  Result (3.4) readily generalizes (by the negative-subdivision trick) to: every $\cC^-(\Sigma)$ is a binary clutter, that is, a port of a binary matroid.  [This is also immediate from the construction of $\bfL_0(\Sigma)$.]
\annotctd{} P.\ 200, (i)--(iii): Amongst minor-minimal binary clutters without the ``weak MFMC property'' are the circuit clutter of $F_7^*$ and $\cC^-(-K_5)$ and its blocker.
\annotctd{} Main Thm.\ (\S5):  A binary clutter is ``Mengerian'' (I omit the definition) iff it does not have $\cC^-(-K_4)$ as a minor.  (See p.\ 200 for the antecedent theorem of Gallai.)
\annotctd{} [See \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}, \xref{Guenin}{2001a}{Bertrand Guenin} for more.]
\par
\codes{\Csg,~\CPar:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{1980a}{}  Decomposition of regular matroids.  \emph{J.\ Combin.\ Theory Ser.\ B} 28 (1980), 305--359.
\MR{0579077 (82j:05046)}.  \Zbl{443.05027}.  
\annot{} The crucial matroid $R_{10} = \bfF(-K_5)$.
\dated{3 Aug 2019} 
\par
\codes{\Csg:~\Cpar:~\CMatrd}%
\itemspace

\bsgitem{1981a}{}  Matroids and multicommodity flows.  \emph{European J.\ Combin.}\ 2 (1981), 257--290. 
\MR{0633121} (82m:05030).  \Zbl{479.05023}.					
\annot{} \Conjecture\ (based on \xrefd{1977a}{P.D.\ Seymour}).  A binary clutter has the weak MFMC property iff no minor is either the circuit clutter of $F_7$ or $\cC^-(-K_5)$ or its blocker.
	
\codes{\Csgnd(\CMatrd),~\Csg:~\CMatrd}%
\itemspace

\bsgitem{1995a}{\dag}  Matroid minors.  
In:  R.L.~Graham, M.~Gr\"otschel, and L.~Lov\'asz, eds., \emph{Handbook of Combinatorics}, Vol.\ I, Ch.\ 10, pp.\ 527--550.  
North-Holland (Elsevier), Amsterdam, and MIT Press, Cambridge, Mass., 1995.
\MR{1373666} (97a:05055).  \Zbl{960.24825}.
\annot{} In Thm.\ 6.6, p.\ 546, interpreting $G$ as a signed graph and an ``odd-$K_4$'' as a subdivision of $-K_4$ gives the signed graph generalization, due to \xref{Gerards and Schrijver}{1986a}{A.M.H.\ Gerards \& A.\ Schrijver} [also \xref{Gerards}{1990a}{A.M.H.\ Gerards}, Thm.\ 3.2.3].  Let $\Sigma$ be a signed simple, 3-connected graph in which no 3-separation has $> 4$ edges on both sides.  Then $\Sigma$ has no $-K_4$ minor iff either (i) deleting some vertex makes it balanced (the complete lift matroid of this type is graphic); or (ii) it is cylindrical: it can be drawn on a cylindrical surface that has a lengthwise red line so that an edge is negative iff it crosses the red line an odd number of times [Note: the extended lift matroid of this type is cographic, as observed by, I think, Gerards and Schrijver or by Lov\'asz].  [See \xref{Pagano}{1998a}{Steven R.\ Pagano} for another use of cylindrical signed graphs.]  [\Problem.  Find the forbidden topological subgraphs, link minors, and $Y\Delta$ graphs for cylindrical signed graphs.]  [\Question.  Embed a signed graph in the plane with $k$ distinguished faces so that a circle's sign is the parity of the number of distinguished faces it surrounds.  Cylindrical embedding is $k = 1$.  For each $k$, which signed graphs are so embeddable?]
\codes{\CSG:~\CStr,~\CTop}%
\annotctd{} Thm.\ 6.7, pp.\ 546--547, generalizes to signed graphs, interpreting $G$ as a signed graph and an ``odd cycle'' as a negative circle.  Take a signed simple, 3-connected, internally 4-connected graph.  It has no two vertex-disjoint negative circles iff it is one of four types: (i) deleting some vertex makes it balanced; (ii) deleting the edges of an unbalanced triangle makes it balanced; (iii) it has order $\leq 5$; (iv) it can be orientation-embedded in the projective plane.  This is due to Lov\'asz; see, if you can, \xref{Gerards, Lov\'asz, \etal{}}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}.  [A 2-connected $\Sigma$ has no vertex-disjoint negative circles iff $\bfF(\Sigma)$ is binary iff $\bfF(\Sigma)$ is regular iff the lift matroid $\bfL(\Sigma)$ is regular.  See \xref{Pagano}{1998a}{Steven R.\ Pagano} for classification of $\Sigma$ with vertex-disjoint negative circles according to representability of the frame matroid.] 
\par
\codes{\CSG:~\CStr,~\Cmatrd,~\CTop}%
\itemspace

\author{Paul Seymour \& Carsten Thomassen}
\authorspace
\bsgitem{1987a}{}  Characterization of even directed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 42 (1987), 36--45.
\MR{0872406} (88c:05089).  \Zbl{607.05037}.					
\annot{} ``Even'' means every signing contains a positive cycle.  A digraph is even iff it contains a subdigraph that is obtained from a symmetric odd-circle digraph by subdivision and a vertex-splitting operation.  [\Cf\ \xref{Thomassen}{1985a}{Carsten Thomassen}.]
\codes{\Csd:~\Cpar:~\CStr}%
\itemspace

\author{L.\ de S\`eze} 						%SEZE
\authorx{L.\ de Seze}
\authorspace
\authref  See \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\itemspace

\author{Bryan L.\ Shader} 						%SHADER
\authorspace
\authref  See \hyperlink{Richard A.\ Brualdi}{R.A.~Brualdi}, \hyperlink{Steve Butler}{S.~Butler}, and \hyperlink{David A.\ Gregory}{D.A.~Gregory}. 
\itemspace

\author{Nisarg Shah}							%SHAH
\authorspace
\authref  See \hyperlink{Manas Joglekar}{M.~Joglekar}.
\itemspace

\author{[Siamak Fayyaz Shahandashti]}					%SHAHANDASHTI
\authorspace
\authref  See \hyperlink{Siamak Fayyaz Shahandashti}{S.~Fayyaz Shahandashti} (under `F').
\itemspace

\author{Mohsen Shahriari}						%SHAHRIARI
\authorspace
\authref  See also \hyperlink{Saeed Reza Shahriary}{S.R.~Shahriary}.
\authrefspace

\author{Mohsen Shahriari \& Ralf Klamma}
\authorspace
\bsgitem{2015a}{}  Signed social networks: Link prediction and overlapping community detection.  In:  \emph{Proceedings of the 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining} (ASONAM '15, Paris, 2015), pp.\ 1608--1609.  ACM, New York, 2015.
\codes{\CSG:~\CClu(\CGen),~\CFr(\CGen),~\CPred:~\CAlgor}%
\itemspace

\author{Mohsen Shahriari, Omid Askari Sichani, Joobin Gharibshah, \& Mahdi Jalili}
\authorspace
\bsgitem{2016a}{}  Sign prediction in social networks based on users reputation and optimism.  \emph {Social Network Analysis Mining} 6 (2016), art.\ 91, 10 pp.
\annot{} Ranking vertices by ``reputation'', RR$(v_i)) := (d^+_\text{in}(v_i)-d^-_\text{in}(v_i))/(d^+_\text{in}(v_i) +d^-_\text{in}(v_i))$, and ``optimism'', OP$(v_i)) := (d^+_\text{out}(v_i)-d^-_\text{out}(v_i))/(d^+_\text{out}(v_i) +d^-_\text{out}(v_i))$, produces better predictions of arc signs.
\dated{22 Sept 2018} 
\par
\codes{\CSD:~\CPred:~\CAlgor}%
\itemspace

\authorx{Saeed Reza Shahriary}					%SHAHRIARY
\author{Saeed Reza Shahriary, Mohsen Shahriari, \& Rafidah MD Noor}
\authorspace
\bsgitem{2015a}{}  A community-based approach for link prediction in signed social networks.  \emph{Sci.\ Programming} 2015 (2015), art.\ 602690, 10 pp.
\codes{\CSG:~\CPred:~\CAlgor,~\CPsS}%
\itemspace

\author{Naomi Shaked-Monderer}					%SHAKED-MONDERER
\authorspace
\bsgitem{2016a}{}  SPN graphs: when copositive = SPN.  \emph{Linear Algebra Appl.}\  509 (2016), 82--113.
\MR{3546402}.  \Zbl{1345.05111}.  \arXiv{1604.02172}, \arXiv{1712.05115}.		
\codes{\CSG:~\CQM}%
\itemspace

\bsgitem{2018a}{}  Corrigendum to ``SPN graphs: when copositive = SPN''.  \emph{Linear Algebra Appl.}\ 541 (2018), 285--286.
\MR{3742623}.  \Zbl{1430.05127}.  \arXiv{1712.05115}.  
\par
\codes{\CSG:~\CQM}%
\itemspace

\author{Sophia Shalini G.B.}						%SHALINI
\author{Sophia Shalini G.B., Anwar Saleh, \& Dhananjayamurthy B.V}
\authorspace
\bsgitem{2019a}{}  On the Seidel energy of certain mesh derived networks.  \emph{Int.\ J.\ Engineering Appl.\ Tech.}\ 9 (2019), no.\ 2, 1905--1910.
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Tahir Shamsher}						%SHAMSHER
\authorspace
\authref  See also \hyperlink{S.\ Pirzada}{S.~Pirzada}.
\itemspace

\author{Tahir Shamsher, Mushtaq A.\ Bhat, \& S.\ Pirzada}
\authorspace
\bsgitem{20xxa}{}  Unicyclic signed graphs with first eleven minimal energies.  Submitted.  \emph{}\ 
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Tahir Shamsher, S.\ Pirzada, \& Mushtaq A.\ Bhat}
\authorspace
\bsgitem{20xxa}{}  On adjacency and Laplacian cospectral switching non-isomorphic signed graphs.  \emph{Ars Math.\ Contemp.}\ (to appear)
\arXiv{2205.08705}.  
\annot{} Cospectrality of $A(\Sigma_1)$ with $\Lap(\Sigma_2)$.	
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Hai-Ying Shan}						%SHAN HY
\authorspace
\authref  See also \hyperlink{Jia-Yu Shao}{J.-Y.~Shao} and \hyperlink{Lihua You}{L.~You}.
\authrefspace

\author{Hai-Ying Shan \& Jia-Yu Shao}
\authorspace
\bsgitem{2004a}{}  Matrices with totally signed powers.  \emph{Linear Algebra Appl.}\ 376 (2004), 215--224.
\MR{2015535} (2004i:15030).  \Zbl{1055.15036}.  
\codes{\CQM:~\CSD}%
\itemspace

\author{Ronghua Shang}						%SHANG RH
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.S.~Wu}.
\itemspace

\author{Jia-Yu Shao} 						%SHAO JY
\authorspace
\authref  See also \hyperlink{Yue Liu}{Y.~Liu}, \hyperlink{Rachel Manber}{R.~Manber}, \hyperlink{Hai-Ying Shan}{H.Y.~Shan}, and \hyperlink{Lihua You}{L.~You}.
\authrefspace

\bsgitem{1998a}{}  On digraphs and forbidden configurations of strong sign nonsingular matrices.  \emph{Linear Algebra Appl.}\ 282 (1998), 221--232.
\MR{1648336} (99h:05086).  \Zbl{940.05044}.					
\codes{\CSD:~\CSol}%
\itemspace

\bsgitem{2000a}{}  On the digraphs of sign solvable linear systems.  \emph{Linear Algebra Appl.}\ 313 (2000), 115--126.
\MR{1770361} (2001e:05083).  \Zbl{958.15003}.					
\annot{} Forbidden subgraphs are used to characterize the signed digraphs.	
\dated{6 Mar 2011} 
\codes{\CSD:~\CSol}%
\itemspace

\author{Jia-Yu Shao, Jin-Ling He, \& Hai-Ying Shan}
\authorspace
\bsgitem{2003a}{}  Number of nonzero entries of $S^2NS$ matrices and matrices with signed generalized inverses.  \emph{Linear Algebra Appl.}\ 373 (2003), 223--239.
\MR{1648336} (99h:05086).  \Zbl{1036.15005}.					
\codes{\CSG:~\CSol}%
\itemspace

\author{Yanling Shao}						%SHAO YL
\authorspace
\authref  See also \hyperlink{Yu-Bin Gao}{Y.-B.~Gao}.
\authrefspace

\author{Yanling Shao \& Yubin Gao}
\authorspace
\bsgitem{2009a}{}  The local bases of primitive non-powerful signed symmetric digraphs with loops.  \emph{Ars Combin.}\ 90 (2009), 357--369.
\MR{2489538} (2010c:05054).  \Zbl{1224.05223}.					
\par
\codes{\CSD,~\Csg:~\Cqm}%
\itemspace

\author{Yanling Shao, Jian Shen, \& Yubin Gao}
\authorspace
\bsgitem{2009a}{}  The $k$th upper bases of primitive non-powerful signed digraphs.  \emph{Discrete Math.}\ 309 (2009), no.\ 9, 2682--2686.
\MR{2523775} (2010h:05144).  \Zbl{1207.05073}.					
\par
\codes{\CSD:~\Cqm}%
\itemspace

\bsgitem{2012a}{}  The $k$th upper and lower bases of primitive nonpowerful minimally strong signed digraphs.  \emph{Linear Multilinear Algebra} 60 (2012), no.\ 9, 1093--1113.
\MR{2966153}.  \Zbl{1252.05077}.  
\codes{\CSD:~\Cqm}%
\itemspace

\authorx{Zhenan Shao}						%SHAO ZN
\author{Zhenan Shao \& Xiying Yuan}
\authorspace
\bsgitem{2022a}{}  Some signed graphs whose eigenvalues are main.  \emph{Appl.\ Math.\ Comput.}\ 423 (2022), art.\ 127014, 9 pp.
\MR{4385150}.  
\arXiv{2106.07878}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Deepakshi Sharma}						%SHARMA D 
\authorspace
\authref  See \hyperlink{Deepa Sinha \& Deepakshi Sharma}{D.~Sinha}.
\itemspace

\author{[Pranjali Sharma]}						%SHARMA P
\authorx{Pranjali Sharma [Pranjali]}
\authref  See \hyperlink{Pranjali}{Pranjali}.
\itemspace

\authorx{Ram Parkash Sharma}					%SHARMA RP
\author{Ram Parkash Sharma \& Vikram Singh Kapil}
\authorspace
\bsgitem{2011a}{}  Irreducible $\overarrow{S_n}$-modules and a cellular structure of the signed Brauer algebras.  \emph{Southeast Asian Bull.\ Math.}\ 35 (2011), no.\ 3, 497--522,
\MR{2856396} (2012h:16003).  \Zbl{1240.20010}.  
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\authorx{Tushar Sharma}						%SHARMA T
\author{Tushar Sharma, Ankit Charls, \& P.K.\ Singh}
\authorspace
\bsgitem{2011a}{}  Community mining in signed social networks --An automated approach.  In:  \emph{2009 International Conference on Computer Engineering and Applications} (ICCEA 2009, Manila), art.\ 28, pp.\ 152--157.  Int.\ Proc.\ Computer Sci.\ Inform.\ Tech.\ (IPCSIT), Vol.\ 2.  IACSIT Press, Singapore, 2011.  URL \url{http://www.ipcsit.com/vol2/28-A217.pdf}
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{John Shawe-Taylor}						%SHAWE-TAYLOR
\authorspace
\authref   See \hyperlink{Tomaz Pisanski}{T.~Pisanski}.
\itemspace

\author{Bo Shen}							%SHEN B
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai}.
\itemspace

\author{Jian Shen}							%SHEN JA
\authorspace
\authref  See \hyperlink{Yu-Bin Gao}{Y.-B.~Gao} and \hyperlink{Yanling Shao}{Y.L.~Shao}.
\itemspace

\authorx{Xiao Shen}							%SHEN X
\author{Xiao Shen \& Fu-Lai Chung}
\authorspace
\bsgitem{2020a}{}  Deep network embedding for graph representation learning in signed networks.  \emph{IEEE Trans.\ Cybernetics} 50 (2020), no.\ 4, 1556--1568.
\arXiv{1901.01718}.  
\annot{} Algorithm for reconstruction assuming few negative edges.
\dated{5 Feb 2022} 
\codes{\CSG:~\CAlgor}%
\itemspace

\authorx{Yi-Huang Shen}						%SHEN YH
\author{Yi-Huang Shen \& Guangjun Zhu}
\authorspace
\bsgitem{2023a}{}  Regularity of powers of (parity) binomial edge ideals.  \emph{J.\ Algebraic Combin.}\ 57 (2023), 75--100.
\arXiv{2111.14175}.  
\annot{} A hint of $\bfF(-\Gamma)$ in the parity binomial edge ideal.
\dated{5 Feb 2023} 
\par
\codes{\CAlgeb:~\Csg:~\Cpar}%
\itemspace

\authorx{Jia Sheng}							%SHENG JA
\author{Jia Sheng \& MiaoLin Ye}
\authorspace
\bsgitem{2010a}{}  The spectral radius of signless Laplacian of a connected graph with given independence number.  \emph{Math.\ Appl.\ (Wuhan)} 23 (2010), no.\ 4, 709--712.
\MR{2765865} (no rev).  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Stephen H.\ Shenker}						%SHENKER
\authorspace
\authref  See \hyperlink{Eduardo Fradkin}{E.~Fradkin}.
\itemspace

\author{F.B.\ Shepherd}						%SHEPHERD
\authorspace
\authref   See \hyperlink{A.M.H.\ Gerards \& F.B.\ Shepherd}{A.M.H.~Gerards} and \hyperlink{T.R.\ Jensen \& F.B.\ Shepherd}{T.R.~Jensen}.
\itemspace

\author{Laura Sheppardson}						%SHEPPARDSON
\authorspace
\authref  See \hyperlink{Torina Lewis}{T.~Lewis}.
\itemspace

\author{R.\ Sherkati}							%SHERKATI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Steven J.\ Sherman}						%SHERMAN
\authorspace
\authref  See \hyperlink{Robert B.\ Zajonc}{R.B.~Zajonc}.
\itemspace

\authorx{David Sherrington}						%SHERRINGTON
\author{David Sherrington \& Scott Kirkpatrick}
\authorspace
\authref  See also \hyperlink{Scott Kirkpatrick}{S.~Kirkpatrick}.
\authrefspace

\bsgitem{1975a}{}  Solvable model of a spin-glass.  \emph{Phys.\ Rev.\ Lett.}\ 35 (1975), no.\ 26, 1792--1796.
Repr.\ in M.~M\'ezard, G.~Parisi, and M.A.~Virasoro, \emph{Spin Glass Theory and Beyond}, pp.\ 104--108.  World Scientific Lect.\ Notes in Physics, Vol.\ 9.  World Scientific, Singapore, 1987.
\annot{} Introduces the Sherrington--Kirkpatrick spin-glass model, a randomly signed and (usually) weighted $K_n$.  Announcement of part of \xref{Kirkpatrick and Sherrington}{1978a}{Scott Kirkpatrick \& David Sherrington} (\qv). 
\dated{22 Aug 2012} 
\codes{\CPhys:~\Csg:~\CFr}%
\itemspace

\author{Ronald G.\ Sherwin}						%SHERWIN
\authorspace
\bsgitem{1975a}{}  Structural balance and the sociomatrix:  Finding triadic valence structures in signed adjacency matrices.  \emph{Human Relations} 28 (1975), 175--189.  
\annot{} A very simple [but not efficient] matrix algorithm for counting different types of circles in a signed (di)graph.  [``Valence'' means sign, unfortunately.]
\codes{\Csg,~\CSD:~\CBal,~\CCircles:~\CInvar:~\CAlgor}%
\itemspace

\author{Jeng-Horng Sheu}						%SHEU JH
\authorspace
\authref   See \hyperlink{Ivan Gutman}{I.~Gutman}.
\itemspace

\author{Chuan-Jin Shi}						%SHI CJ
\authorspace
\bsgitem{1992a}{}  A signed hypergraph model of constrained via minimization.  In: \emph{VLSI, 1992.  Proceedings of the Second Great Lakes Symposium on VLSI} (Kalamazoo, Mich., 1992), pp.\ 159--166.  IEEE, 1992.
\codes{\CSH:~\CAppl}%
\itemspace

\bsgitem{1992b}{}  A signed hypergraph model of constrained via minimization.  \emph{Microelectronics J.}\ 23 (1992), no.\ 7, 533--542.
\codes{\CSH:~\CAppl}%
\itemspace

\bsgitem{1993a}{}  Constrained via minimization and signed hypergraph partitioning.  In: D.T.\ Lee and M.\ Sarrafzadeh, eds., \emph{Algorithmic Aspects of VLSI Layouts}, pp.\ 337--356.  World Scientific, Singapore, 1993.
\codes{\CSH:~\CAppl:~\CExp}%
\itemspace

\bsgitem{1993b}{}  \emph{Optimum Logic Encoding and Layout Wiring for VLSI Design: A Graph-Theoretic Approach}.  Ph.D.\ thesis, University of Waterloo, 1993.
\par
\codes{\CSH:~\CInc,~\CBal,~\CAlgor,~\CSG,~\CAppl}%
\itemspace

\author{C.-J.\ Shi \& J.A.\ Brzozowski}
\authorspace
\bsgitem{1999a}{}  A characterization of signed hypergraphs and its applications to VLSI via minimization and logic synthesis.  \emph{Discrete Appl.\ Math.}\ 90 (1999), no.\ 1-3, 223--243.
\MR{1666019} (99m:68155).  \Zbl{913.68104}.					
\annot{} A signed hypergraph $H=(V,E,\psi)$ is a hypergraph $(V,E)$ with an incidence signature $\psi: V\times E \to \{-1,0,1\}$.  
``Underlying graph'' = bipartite incidence graph with edge signs $\psi$.  Sign of a path [or walk] = product of incidence signs.
Motivation: via minimization, i.e., minimize the number of connections between different planar layers of a two-layer circuit.
[See \xref{Rusnak}{2010a}{Lucas Rusnak} for a different development of the same definitions.  Path signs are different; the normal sign for signed graphs has an extra factor $-1$ for each edge.]
$e$ is ``balanced'' by a bipartition $V = V_1 \cupdot V_2$ when incidences of $e$ are in the same $V_i$ iff they have the same sign.  $H$ is ``balanced'' if some bipartition balances every edge.  Thm.\ 3.1:  $H$ is balanced iff every circle is positive.  [I.e., antibalance, since walk signs are different from the norm.]  Proof:  Constructive [similar to but less exact than algorithms for signed graphs as in \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}], yielding Cor.\ 3.1:  Testing balance takes linear time.  Thm.\ 3.2:  $H$ is balanced iff its incidence dual is balanced.  ``Maximum balance problem'':  Minimize the number of unbalanced edges.  Thm.\ 4.1:  This is NP-complete, even for cubic graphs.  [Known, as it contains the max-cut problem.]  Thm.\ 4.2:  NP-complete for planar signed hypergraphs with maximum degree $>3$.  (For max degree $\leq 3$, polynomial-time algorithms are given in \xref{Shi}{1993b}{Chuan-Jin Shi}.)
\Problem: Minimum Covering:  Find the minimum number of bipartitions of $V$ such that every edge is balanced by one of the bipartitions.  Equivalently, decompose $H$ into the smallest number of balanced subhypergraphs.  [See \xref{Zaslavsky}{1987b}{Thomas Zaslavsky} for signed graphs.]  Thm.\ 5.1:  NP-complete.  Proof:  Reduction to graph colorability via decomposability of a graph into bipartite subgraphs [special case of signed-graph decomposition as in \xref{Zaslavsky}{1987b}{Thomas Zaslavsky}].  
\annotctd{} \S6, ``Constrained via minimization'', summarizes connection with signed hypergraphs, based on \xref{Shi}{1992a}{Chuan-Jin Shi}, \xrefd{1992b}{Chuan-Jin Shi}.  
\S7, ``Constrained logic encoding''.  
\annotctd{} \S8, ``Related notions:  Signed graphs and $(0,\pm1)$-matrices''.  \S8.1, ``Hara\-ry's signed graphs'', compares their work with \xref{Harary}{1953a}{Frank Harary} [no mention of \xref{Harary and Kabell}{1980a}{Frank Harary \& Jerald A.\ Kabell}].  \S8.2, ``Restricted unimodularity and balanced $(0,\pm1)$ matrices'':  The incidence matrix of $\Eta(H)$ if $H$ is a graph [$\Eta(-H)$ in the normal definition] is totally unimodular iff $-H$ is balanced [essentially, \xref{Heller and Tompkins}{1956a}{I.\ Heller \& C.B.\ Tompkins}].
\annotctd{} [All problems and methods are equivalent to the similar problems for the signed graph derived by replacing each hyperedge by a balanced complete graph with Harary bipartition given by the sign bipartition of the hyperedge's incidences.]
\dated{4 Nov 2010}
\par
\codes{\CSH:~\CInc,~\CBal,~\CAlgor,~\CSG}%
\itemspace

\author{C.-J.\ Shi, A.\ Vannelli, \& J.\ Vlach}
\authorspace
\bsgitem{1990a}{}  A hypergraph partitioning approach to the via minimization problem.  In: \emph{Proceedings of the Canadian Conference on VLSI} (1990), pp.\ 2.7.1--2.7.8. 
\par
\codes{\CSH:~\Csg:~\CBal}%
\itemspace

\bsgitem{1997a}{}  Performance-driven layer assignment by integer linear programming and path-constrained hypergraph partitioning.  
\emph{J.\ Heuristics} 3 (1997), no.\ 3, 225--243.
\Zbl{1071.90584}.  
\codes{\CSH:~\Csg:~\CBal,~\CAlgor:~\CAppl}%
\itemspace

\authorx{Guodong Shi}						%SHI GD
\author{Guodong Shi, Claudio Altafini, \& John S.\ Baras}
\authorspace
\bsgitem{2017a}{}  Algebraic-graphical approach for signed dynamical networks.  In:  \emph{2017 IEEE 56th Annual Conference on Decision and Control (CDC)} (CDC2017, Melbourne, 2017), pp.\ 2009--2014.  IEEE, 2017.
\codes{\CSG:~\CDyn}%
\itemspace

\author{Guodong Shi, Alexandre Proutiere, Mikael Johansson, John S.\ Baras, \& Karl H.\ Johansson}
\authorspace
\bsgitem{2013a}{}  Emergent behaviors over signed random networks in dynamical environments.  Manuscript, 2013.
\arXiv{1309.5488}.  
\codes{\CSG:~\CRand}%
\itemspace

\bsgitem{2015a}{}  Emergent behaviors over signed random dynamical networks: State-flipping model.  \emph{IEEE Trans.\ Control Network Syst.}\ 2 (2015), no. 2, 142--153.
\MR{3361856}.  \Zbl{1370.90061}.  \arXiv{1411.0074}.				
\codes{\CSG:~\CRand}%
\itemspace

\bsgitem{2016a}{}  The evolution of beliefs over signed social networks.  \emph{Operations Res.}\ 64 (2016), no. 3, 585--604.
\MR{3515199}.  \Zbl{1348.91239}.  \arXiv{1307.0539}.				
\codes{\CSG:~\CRand}%
\itemspace

\bsgitem{2017a}{}  Emergent behaviors over signed random dynamical networks: Relative-state-flipping model.  \emph{IEEE Trans.\ Control Network Syst.}\ 4 (2017), no.\ 2, 369--379.
\MR{3667952}.  \Zbl{1370.90062}.  \arXiv{1412.1990}.				
\codes{\CSG:~\CRand}%
\itemspace

\author{Jinsong Shi}							%SHI JS 
\authorspace 
\authref  See \hyperlink{Ruilin Li}{R.L.~Li}.
\itemspace

\authorx{Lingsheng Shi}						%SHI LS
\author{Lingsheng Shi \& Zhang Zhang}
\authorspace
\bsgitem{2018a}{}  Signed cycle double covers.  \emph{Electronic J.\ Combin.}\ 25 (2018), no.\ 4, art.\ P4.63, 12 pp.
\MR{3907794}.  \Zbl{1409.05097}.  
\codes{\CSG:~\CCircles}%
\itemspace

\author{Yongtang Shi}						%SHI YT
\authorspace
\authref  See \hyperlink{Bofeng Huo}{B.F.~Huo} and \hyperlink{Jiaao Li}{J.-A.~Li}.
\itemspace

\author{Kazuki Shibata}						%SHIBATA
\authorspace
\authref  See \hyperlink{Hidefumi Ohsugi}{H.~Ohsugi}.
\itemspace

\authorx{V.S.\ Shigehalli}						%SHIGEHALLI
\author{V.S.\ Shigehalli \& Kenchappa S.\ Betageri}
\authorspace
\bsgitem{2015a}{}  Color Laplacian energy of graphs.  \emph{J.\ Computer Math.\ Sci.}\ 6 (2015), no.\ 9, 485--494.
\annot{} Laplacian energy of the signed graph of \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\bsgitem{2016a}{}  A note on color energy and color Laplacian energy of graphs.  \emph{Int.\ J.\ Math.\ Archive} 7 (2016), no.\ 10, 199--204.
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\bsgitem{2016b}{}  A note on color energy of graphs.  \emph{Bull.\ Math.\ Stat.\ Res.}\ 4 (2016), no.\ 4, 47--50.
\annot{} \Cf\ \xref{Adiga, Sampathkumar, \etal\!}{2013a}{Chandrashekar Adiga, E.\ Sampathkumar, M.A.\ Sriraj, \& Shrikanth A.S.}.
\codes{\Csg:~\CAdj:~\CEig}%
\itemspace

\author{Ching-Hsien Shih}						%SHIH CH
\authorspace
\bsgitem{1982a}{\dag}  \emph{On Graphic Subspaces of Graphic Spaces}.  Doctoral dissertation, Ohio State University, 1982.
\MR{2632480} (no rev).  
\annot{} \Cf\ \xref{Ferchiou and Guenin}{2020a}{Zouhaier Ferchiou \& Bertrand Guenin}.
\codes{\Csg:~\Cmatrd}%
\itemspace

\author{Wei-Kuan Shih}						%SHIH WK
\authorspace
\authref  See also \xref{Kuo--Chern--Shih}{1988a}{Y.S.\ Kuo, T.C.\ Chern, \& Wei-kuan Shih}.
\authrefspace

\author{Wei-Kuan Shih, Sun Wu, \& Y.S.\ Kuo}
\authorspace
\bsgitem{1990a}{}  Unifying maximum cut and minimum cut of a planar graph.  \emph{IEEE Trans.\ Computers} 39 (1990), no.\ 5, 694--697.
\MR{1059768} (91h:05117).  \Zbl{1395.05173}.  
\annot{} Real edge weights make max and min cut equivalent.  A faster algorithm than before.  [This problem includes frustration index $l(\Sigma)$.]  Dictionary:  ``positive'', ``negative'' cut means total weight.
\dated{19 Dec 2014} 
\par
\codes{\CWG,~\Csg:~\Cfr:~\CAlgor}%
\itemspace

\author{Shijin T V [T.V.\ Shijin]}						%SHIJIN
\authorx{Shijin T V}
\authorspace
\authref  See also \hyperlink{Albin Mathew}{A.~Mathew} and \hyperlink{Shahul Hameed K}{S.~Hameed}.
\authrefspace

\bsgitem{2023a}{}  \emph{The Notion of Signed Distance in Signed Graphs}.  Doctoral dissertation, Central University of Kerala, 2023. 
\annot{} \S1.2, ``Signed graphs''.  
Ch.\ 2, ``Signed distance and distance compatibility in signed graphs'': \xref{Hameed, Shijin, Soorya, Germina, and Zaslavsky}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}.  
Ch.\ 3, ``On the distance compatibility in product of signed graphs'': \xref{Shijin, Soorya, Hameed, and Germina}{20xxb}{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}.  
Ch.\ 4, ``On the distance spectra of product of signed graphs'': \xref{Shijin, Soorya, Hameed, and Germina}{20xxc}{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}.  
Ch.\ 5, ``On the powers of signed graphs'':  \xref{Shijin, Germina, and Hameed}{2022a}{Shijin T V, Germina K A, \& Shahul Hameed K}.  
Ch.\ 6, ``Signed distance in weighted signed graphs'':  \xref{Shijin and Germina}{20xxa}{Shijin T V \& Germina K A}.  
\S7.2, ``Scope for further research and some open problems'':  Problems about distance compatibility, signed distance matrices, signed-graph products and powers, and their connections.
\dated{16 Nov 2022} 
\codes{\CSG:~\CStr,~\CEig,~\CWG}%
\itemspace

\author{Shijin T V \& Germina K A}
\authorspace
\bsgitem{20xxa}{}  Signed distance in weighted signed graphs.  Submitted.  
\annot{} Weights treated as edge lengths.  Generalizes \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}.
\dated{2 Nov 2022} 
\par
\codes{\CSG:~\CWG,~\CAdj(\CGen),~\CBal,~\CEig,~\CSw}%
\itemspace

\author{Shijin T V, Germina K A, \& Shahul Hameed K}
\authorspace
\bsgitem{2022a}{}  On the powers of signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ 7 (2022), no.\ 1, 45--51.
\MR{4337200}.  
\arXiv{2009.10486}.  
\annot{} \Cf\ \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}.  $\Sigma^k_{\max}$ ($\Sigma^k_{\min}$) has edge $uv$ signed $\sigma_{\max\,(\min)}(u,v)$ when $d(u,v)\leq k$.  $\exists\ \Sigma^k$ if $\Sigma^k_{\max} = \Sigma^k_{\min}$.  Theorems on balance and distance compatibility.
\dated{27 Sept 2020} 
\par
\codes{\CSG:~\CBal,~\CAdj(\CGen)}%
\itemspace

\author{Shijin T V, Soorya P, Shahul Hameed K, \& Germina K A}
\authorspace
\bsgitem{2020a}{}  On signed distance in product of signed graphs.  Manuscript, 2020.
\arXiv{2009.08707}.  
\annot{} \Cf\ \xref{Hameed, Shijin, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}.  Thm.\ 2.2 characterizes distance-compatible signed graphs, somewhat more simply than the definition.  \S3 characterizes distance compatibility for Cartesian and lexicographical products and partially for tensor product.  \S4: Formulas for signed distance matrices of the former two.  \S5: Examples.
\dated{19 Mar 2021} 
\annotctd{} Superseded by \xrefd{2023a}{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}, \xrefd{2023b}{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}.
\codes{\CSG}\codesctd{\CSG:~\CAdj(\CGen),~\CEig}%
\itemspace

\author{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}
\bsgitem{2023a}{}  On the distance compatibility in product of signed graphs.  \emph{Asian-European J.\ Math.}\ 16 (2023), no.\ 2, art.\ 2350015, 10 pp.
\MR{4526099}.  
\codes{\CSG}%
\itemspace

\author{T.V.\ Shijin, P.\ Soorya, K.\ Shahul Hameed, \& K.A.\ Germina}
\authorspace
\bsgitem{2023b}{}  On the distance spectra of the product of signed graphs.  \emph{Commun.\ Combin.\ Optim.}\ 8 (2023), no.\ 1, 67--76.
\MR{450365}.  
\annot{} Only for distance-compatible signed graphs and products (\xref{Hameed, Shijin, Soorya, \etal\!}{2021a}{Shahul K.\ Hameed, T.V.\ Shijin, P.\ Soorya, K.A.\ Germina, \& Thomas Zaslavsky}).  Thms.\ 4, 5:  Distance matrices of Cartesian and lexicographic products.  \S3, ``Distance spectra of some compatible signed graphs and their products'':  Signed Petersen graphs; Cartesian products of $(K_n,\sigma_1)$ with $(K_n,\sigma_2)$ or with $-\Gamma$; $\Sigma[(K_2,\sigma)]$.
\dated{22 Apr 2023} 
\codes{\CSG:~\CAdj(\CGen),~\CEig}%
\itemspace

\author{M.\ Ashwin Shijo}							%SHIJO
\authorspace
\authref  See \hyperlink{C.\ Jayasekaran}{C.~Jayasekaran}.
\itemspace

\author{Akihiro Shikama}						%SHIKAMA
\authorspace
\authref  See \hyperlink{Atsushi Funato}{A.~Funato} and \hyperlink{Takayuki Hibi}{T.~Hibi}.
\itemspace

\author{Young-hee Shin}						%SHIN YH
\authorspace
\authref  See \hyperlink{Jin Ho Kwak}{J.H.~Kwak}.
\itemspace

\authorx{Guy Shinar}							%SHINAR
\author{Guy Shinar \& Martin Feinberg}
\authorspace
\bsgitem{2013a}{}  Concordant chemical reaction networks and the species-reaction graph.  \emph{Math.\ Biosci.}\ 241 (2013), 1--23.
\MR{3019690}.  \Zbl{1309.92094}.  \arXiv{1203.6560}.				
\annot{} The ``c-pair'' (``complex pair'') edges act like a negative edge.
\dated{21 Jan 2015} 
\codes{\Csd,~\Csg:~\CDyn,~\CChem}%
\itemspace

\author{Alana Shine}							%SHINE
\authorspace
\authref  See \hyperlink{Matthias Beck}{M.~Beck}.
\itemspace

\authorx{J.\ Shiozaki}						%SHIOZAKI
\author{J.\ Shiozaki, H.\ Matsuyama, E.\ O'Shima, \& M.\ Iri}
\authorspace
\bsgitem{1985a}{}  An improved algorithm for diagnosis of system failures in the chemical process.  \emph{Computers and Chem.\ Eng.}\ 9 (1985), 285--293.
\annot{} Continuation of \xref{Iri, Aoki, O'Shima, and Matsuyama}{1979a}{Masao Iri, Katsuaki Aoki, Eiji O'Shima, \& Hisayoshi Matsuyama}.
\par
\codes{\CSD,~\CVS:~\CAppl,~\CAlgor}%
\itemspace

\author{A.H.\ Shirazi}							%SHIRAZI
\authorspace
\authref  See \hyperlink{R.\ Masoumi}{R.~Masoumi}.
\itemspace

\author{H.\ Shirazi}							%SHIRAZI
\authorspace
\authref  See \hyperlink{G.\ Coutinho}{G.~Coutinho}.
\itemspace

\author{Shailaja S.\ Shirkol}						%SHIRKOL
\authorspace
\authref  See \hyperlink{Prabhakar Ramrao Hampiholi}{P.R.~Hampiholi}.
\itemspace

\author{Keisuke Shiromoto}						%SHIROMOTO
\authorspace
\authref  See \hyperlink{Yoshitaka Koga}{Y.~Koga}.
\itemspace

\author{Wai Chee Shiu}						%SHIU WC
\authorspace
\authref  See \hyperlink{Ji Ming Guo}{J.M.~Guo}.
\itemspace

\author{[Shivakumar Swamy C.S.]}					%SHIVAKUMAR SWAMY
\authorspace
\authref  See \hyperlink{Shivakumar Swamy C.S.}{S.~Swamy~C.S.}.
\itemspace

\author{K.\ Shivashankara}						%SHIVASHANKARA
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{S.B.\ Shlosman}						%SHLOSMAN
\authorspace
\authref  See \xref{Dobrushin and Shlosman}{1985a}{R.L.\ Dobrushin \& S.B.\ Shlosman}.
\itemspace

\author{L.\ Shobana [Shobana Loganathan]}				%SHOBANA
\authorx{L.\ Shobana}
\authorspace
\authref  See also \hyperlink{Jayapal Baskar Babujee}{J.~Baskar Babujee} and \hyperlink{B.\ Vasuki}{B.~Vasuki}.
\authrefspace

\author{L.\ Shobana \& B.\ Vasuki}
\authorspace
\bsgitem{2017a}{}  Signed product cordial labeling on some connected graphs.  \emph{Int.\ J.\ Pure Appl.\ Math.}\ 113 (2017), no.\ 13, 198--207.
\annot{} More, as in \xref{Baskar Babujee and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\par
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\authorx{Elizabeth G.\ Shrader}			 			%SHRADER E
\author{Elizabeth G.\ Shrader \& David W.\ Lewit}
\authorspace
\bsgitem{1962a}{}  Structural factors in cognitive balancing behavior.  
\emph{Human Relations} 15 (1962), 265--276.  
\annot{} For $\Gamma \subset K_n$ and signing $\sigma$ of $\Gamma$, ``plausibility'' = mean and ``differentiability'' = standard deviation of $f(K_n, \sigma')$ over all extensions of  $\sigma$  to  $K_n$, where $f$ is any function that measures degree of balance.  Proposed: tendency toward  balance is high when plausibility and differentiability are high.  A specific $f$, based on triangles and quite complicated, is studied for $n = 4$, with experiments.
\codes{\Csg,~\Cfr,~\CPsS}%
\itemspace

\authorx{S.R.\ Shreyas}						%SHREYAS
\author{S.R.\ Shreyas \& M.\ Joseph}
\authorspace
\bsgitem{2020a}{}  Characterization of signed paths and cycles admitting minus dominating function.  \emph{Commun.\ Combin.\ Optim.}\ 5 (2020), no.\ 1, 61--68.
\MR{4048279}.  \Zbl{1449.05129}.  
\annot{} \Cf\ \xref{B.D.\ Acharya}{2012a}{B.\ Devadas Acharya}.  Characterizations are largely in terms of the lengths of positive and negative sections.
\dated{25 Dec 2020} 
\par
\codes{\CSG:~\CLab,~\CDom}%
\itemspace

\author{A.S.\ Shrikanth}						%SHRIKANTH
\authorspace
\authref  See \hyperlink{Chandrashekar Adiga}{C.~Adiga}.
\itemspace

\author{Mohan S.\ Shrikhande}						%SHRIKHANDE
\authorspace
\authref  See \hyperlink{Yuri J.\ Ionin \& Mohan S.\ Shrikhande}{Y.J.~Ionin}.
\itemspace

\author{Shrikanth A.S.}						%SHRIKANTH
\authorspace
\authref  See \hyperlink{Chandrashekar Adiga}{C.~Adiga}.
\itemspace

\author{Piyush Shroff}						%SHROFF
\authorspace
\authref  See \hyperlink{Lucas J.\ Rusnak}{L.~Rusnak}.
\itemspace

\author{Jinlong Shu}							%SHU JL
\authorspace
\authref  See \hyperlink{Guanglong Yu}{G.L.~Yu} and \hyperlink{Mingqing Zhai}{M.Q.~Zhai}.
\itemspace

\author{Alan Shuchat} 						%SHUCHAT
\authorspace
\authref   See \hyperlink{Randy Shull}{R.~Shull}.
\itemspace

\authorx{Randy Shull}						%SHULL 
\author{Randy Shull, James B.\ Orlin, Alan Shuchat, \& Marianne L.\ Gardner}
\authorspace
\bsgitem{1989a}{}  The structure of bases in bicircular matroids.  \emph{Discrete Appl.\ Math.}\ 23 (1989), 267--283.  
\MR{0996138} (90h:05040).  \Zbl{698.05022}.
\annot{} [See \xref{Coullard, del Greco, and Wagner}{1991a}{Collette R.\ Coullard, John G.\ del Greco, \& Donald K.\ Wagner}.]  
\codes{\CBic:~\CBases}%
\itemspace

\author{Randy Shull, Alan Shuchat, James B.\ Orlin, \& Marianne Lepp} 
\authorspace
\bsgitem{1993a}{}  Recognizing hidden bicircular networks.  \emph{Discrete Appl.\ Math.}\ 41 (1993), 13--53.  
\MR{1197249} (94e:90122).  \Zbl{781.90089}.
\codes{\CGN:~\CBic:~\CInc,~\CAlgor}%
\itemspace

\bsgitem{1997a}{}  Arc weighting in hidden bicircular networks.  Proc.\ Twenty-eighth Southeastern Int.\ Conf.\ on Combinatorics, Graph Theory and Computing (Boca Raton, Fla., 1997).  \emph{Congressus Numer.}\ 125 (1997), 161--171.
\MR{1604964} (98m:05181).  \Zbl{902.90157}.
\codes{\CGN:~\CBic:~\CInc,~\CAlgor}%
\itemspace

\author{E.E.\ Shult} 							%SHULT
\authorspace
\authref  See \hyperlink{Peter J.\ Cameron}{P.J.~Cameron}.
\itemspace

\author{Robert Shwartz}						%SHWARTZ
\authorspace
\authref  See \hyperlink{Meirav Amram, Robert Shwartz, \& Mina Teicher}{M.~Amram}, \hyperlink{Vadim Bugaenko}{V.~Bugaenko}, and \hyperlink{Yonah Cherniavsky}{Y.~Cherniavsky}.
\itemspace

\author{Jana \v{S}iagiov\'a}						%SIAGIOVA
\authorx{Jana Siagiova}
\authorspace
\authref  See also \hyperlink{Jozef Siran}{J.~\v{S}ir\'a\v{n}}.
\authrefspace

\bsgitem{2001a}{}  A note on the McKay--Miller--\v{S}ir\'a\v{n} graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 81 (2001), 205--208.
\MR{1814904} (2001k:05110).  \Zbl{1024.05039}.  
\annot{} \Cf\ \xref{McKay, Miller, and \v{S}ir\'a\v{n}}{1998a}{Brendan D.\ McKay, Mirka Miller, \& Jozef Siran}.
\codes{\CGG:~\CCov}%
\itemspace

\author{M.\ Siami}							%SIAMI
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.

\author{Omid Askari Sichani}						%SICHANI
\authorspace
\authref  See \hyperlink{Mohsen Shahriari}{M.~Shahriari}.
\itemspace

\author{V.M.\ Siddalingaswamy}					%SIDDALINGASWAMY
\authorspace
\authref  See \xref{Rajendra, Reddy, and Siddalingaswamy}{2018a}{R.\ Rajendra, P.\ Siva Kota Reddy, \& V.M.\ Siddalingaswamy} and \xref{Reddy, Nagaraja, and Siddalingaswamy}{2015a}{P.\ Siva Kota Reddy, K.M.\ Nagaraja, \& V.M.\ Siddalingaswamy}.
\itemspace

\author{Heike Siebert}						%SIEBERT
\authorspace
\authref  See also \hyperlink{Alexander Bockmayr}{A.~Bockmayr}.
\authrefspace

\bsgitem{2008a}{}  Local structure and behavior of Boolean bioregulatory networks.  
In:  Katsuhisa Horimoto \etal, eds., \emph{Algebraic Biology} (Third Int.\ Conf., AB 2008, Castle of Hagenberg, Austria, 2008), pp.\ 185--199.  
\LNCS 5147.  Springer, Berlin, 2008.
\Zbl{1171.92303}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2009a}{}  Deriving behavior of Boolean bioregulatory networks from subnetwork dynamics.  \emph{Math.\ Computer Sci.}\ 2 (2009), no.\ 3, 421--442.
\MR{2507427} (2010g:92009).  \Zbl{1205.37097}.					
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2009b}{}  Dynamical and structural modularity of discrete regulatory networks.  In: Ralph-Johan Back, Ion Petre, and Erik de Vink, eds., \emph{Proceedings Second International Workshop on Computational Models for Cell Processes} (CompMod 2009, Eindhoven, Netherlands), pp.\ 109--124.  Electronic Proc.\ Theoretical Computer Sci., Vol.\ 6, 2009.
\arXiv{0910.1412}.  
\annot{} The local and global interaction graphs are signed digraphs (p.\ 111).  
\par
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2011a}{}  Analysis of discrete bioregulatory networks using symbolic steady states.  \emph{Bull.\ Math.\ Biol.}\ 73 (2011), no.\ 4, 873--898.
\\ \MR{2785148} (2012c:92006).  \Zbl{1214.92033}.					
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Heike Siebert \& Alexander Bockmayr}
\authorspace
\bsgitem{2006a}{}  Incorporating time delays into the logical analysis of gene regulatory networks.  
In:  Corrado Priami, ed.,\emph{Computational Methods in Systems Biology} (Proc.\ Int.\ Conf.\ CMSB 2006, Trento, Italy), pp.\ 169--183.  
\LNCS 4210.  Springer, Berlin, 2006.
\MR{2288350} (2007k:92067).  
\par
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2007a}{}  Context sensitivity in logical modeling with time delays.  
In:  Muffy Calder and Stephen Gilmore, eds., \emph{Computational Methods in Systems Biology} (Proc.\ Int.\ Conf.\ CMSB 2007, Edinburgh, 2007), pp.\ 64--79.  
\LNCS 4695.  Springer, Berlin, 2007.
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2008a}{}  Temporal constraints in the logical analysis of regulatory networks.  \emph{Theoretical Computer Sci.}\ 391 (2008), no.\ 3, 258--275.
\MR{2386791} (2008k:92035).  \Zbl{1133.68041}.					
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2008b}{}  Relating attractors and singular steady states in the logical analysis of bioregulatory networks.  
In:  Hirokazu Anai \etal, eds.,  \emph{Algebraic Biology} (Second Int.\ Conf., AB 2007, Castle of Hagenberg, Austria, 2007), pp.\ 36--50.  \LNCS 4545.  Springer, Berlin, 2008.
\Zbl{1127.92002}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{David Siegel}						%SIEGEL
\authorspace
\authref  See \hyperlink{H.\ Kunze}{H.~Kunze}.
\itemspace

\author{Mark Siggers}						%SIGGERS
\authorspace
\authref  See \hyperlink{Richard C.\ Brewster}{R.C.~Brewster}.
\itemspace

\authorx{Mathieu Dutour Sikiri\'c}					%SIKIRIC
\authorx{Mathieu Dutour Sikiric}
\author{Mathieu Dutour Sikiri\'c, Anna Felikson, \& Pavel Tumarkin}
\authorx{Mathieu Dutour Sikiric, Anna Felikson, \& Pavel Tumarkin}
\authorspace
\bsgitem{2011a}{}  Automorphism groups of root system matroids.  \emph{European J.\ Combin.}\ 32 (2011), 383--389.
\MR{2764801} (2012a:05063).  \Zbl{1229.05068}.  \arXiv{0711.4670}.			
\annot{} \xref{Schwartz}{2002a}{Gary K.\ Schwartz} is more general but excepts the exceptional root systems.	
\dated{9 Apr 2016} 
\codes{\Cgg:~\CMatrd:~\CAut}%
\itemspace

\author{Arlei Silva}							%SILVA A
\authorspace
\authref  See \hyperlink{Zexi Huang}{Z.-X.~Huang}.
\itemspace

\author{Ilda P.F.\ da Silva}						%SILVA I	%DA SILVA
\authorspace
\bsgitem{1997a}{}  Note on inseparability graphs of matroids having exactly one class of orientations.  \emph{Discrete Math.}\ 171 (1997), 77--87.
\MR{1454442} (98c:05044).  \Zbl{}874.05016.  
\codes{\CSG,~\CMatrd}%
\itemspace

\author{Pedro V.\ Silva}						%SILVA PV
\authorspace
\authref  See \hyperlink{Stuart Margolis}{S.~Margolis}.
\itemspace

\author{Bruno Simeone}						%SIMEONE
\authorspace
\authref  See also \hyperlink{C.\ Benzaken}{C.~Benzaken}, \hyperlink{Jean-Marie Bourjolly}{J.-M.~Bourjolly}, \hyperlink{Peter L.\ Hammer}{P.L.~Hammer}, and \hyperlink{Pierre Hansen}{P.~Hansen}.
\authrefspace

\bsgitem{2011a}{}  Quadratic functions.  In: Yves Crama and Peter L.\ Hammer, eds., \emph{Boolean Functions: Theory, Algorithms, and Applications}, Ch.\ 5.  Encyc.\ Math.\ Appl., Vol.\ 142.  Cambridge University Press, Cambridge, 2011.
\MR{2742439} (book).  \Zbl{1237.06001} (book).  
\codes{\CSG:~\CBal,~\CAlgor:~\CExp}%
\itemspace

\author{Luca Simeoni}						%SIMEONI
\authorspace
\authref  See \hyperlink{Steffen Klamt}{S.~Klamt}.
\itemspace

\author{Slobodan K.\ Simi\'c} 						%SIMIC*
\authorx{Slobodan K.\ Simic}
\authorspace
\authref  See also \hyperlink{Milica Andelic}{M.~An\'deli\'c}, \hyperlink{Francesco Belardo}{F.~Belardo}, \hyperlink{Domingos M.\ Cardoso}{D.M.~Cardoso}, \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}, and \hyperlink{Xianya Geng}{X.Y.~Geng}.
\authrefspace

\bsgitem{1980a}{}  Graphs which are switching equivalent to their complementary line graphs I.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 27(41) (1980), 229--235.  
\MR{0621954} (82m:05077).  \Zbl{531.05050}.					
\codes{\CTG:~\CLG}%
\itemspace

\bsgitem{1982a}{}  Graphs which are switching equivalent to their complementary line graphs II.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 31(45) (1982), 183--194.  
\MR{0710958} (85d:05207).  \Zbl{531.05051}.					
\codes{\CTG:~\CLG}%
\itemspace

\author{Slobodan K.\ Simi\'c, Milica Andeli\'c, Carlos M.\ da Fonseca, \& Dejan \v{Z}ivkovi\'c}
\authorx{Slobodan K.\ Simic, Milica Andelic, Carlos M.\ da Fonseca, \& Dejan Zivkovic}
\authorspace
\bsgitem{2015a}{}  Notes on the second largest eigenvalue of a graph.  \emph{Linear Algebra Appl.}\ 465 (2015), 262--274.
\MR{3274675}.  \Zbl{1302.05112}.  
\annot{} Let $H_i'$ denote the bridges of a cutpoint $u$ in $\Gamma$ with each edge subdivided once.  Order so $\epsilon_i := \lambda_1(H_i'\setm u)$ is decreasing.  Cor.\ 3.4 (restated):  $\epsilon_2^2 \leq \lambda_2(\Gamma) \leq \epsilon_1^2$, with $=$ iff $\epsilon_1 = \epsilon_2$.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Slobodan K.\ Simi\'c \& Zoran Stani\'c}
\authorx{Slobodan K.\ Sim'c \& Zoran Stanic}
\authorspace
\bsgitem{2008a}{}  $Q$-integral graphs with edge-degrees at most five.  \emph{Discrete Math.}\ 308 (2008), 4625--4634.
\MR{2438168} (2010g:05229).  \Zbl{1156.05037}.					
\annot{} Determines all $\Gamma$ with edge degree $\leq 4$ and integral $\Spec \Lap(-\Gamma)$, and partially for edge degree 5.  An important paper for integral spectra of $\Lap(-\Gamma)$.
\dated{29 Apr 2022} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2009a}{}  On some forests determined by their Laplacian or signless Laplacian spectrum.  \emph{Computers Math.\ Appl.}\ 58 (2009), no.\ 1, 171--178.
\MR{2535979} (2010j:05252).  \Zbl{1189.05106}.					
\annot{} \S4, ``Determination by the signless Laplacian spectrum''.  Thm.\ 4.1:  Among the forests whose trees are Smith trees (excluding a few), the three minimal graphs not determined amongst all graphs by $\Spec \Lap(-\Gamma)$.
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2016a}{}  Polynomial reconstruction of signed graphs.  \emph{Linear Algebra Appl.}\ 501 (2016), 390--408.
\MR{3485074}.  \Zbl{1334.05056}.  
\annot{} Reconstructing the characteristic polynomial (of $A(\Sigma)$) from vertex-deleted subgraphs: solved for cyclomatic number 0 [known: same as for graphs] and 1.  A trivial counterexample: positive and negative $C_n$.  \Question:  Is there a counterexample with nonisomorphic $|\Sigma_1|, |\Sigma_2|$?
\dated{18 Dec 2016} 
\codes{\CSG:~\CAdj}%
\itemspace

\bsgitem{2016b}{}  Polynomial reconstruction of signed graphs whose least eigenvalue is close to $-2$.  \emph{Electronic J.\ Linear Algebra} 31 (2016), Article 52, 740--753.
\MR{3603936}.  \Zbl{1352.05119}.  
\codes{\CSG:~\CAdj}%
\itemspace

\author{Rodica Simion}						%SIMION*
\authorspace
\bsgitem{1995a}{}  On $q$-analogues of partially ordered sets.  \emph{J.\ Combin.\ Theory Ser.\ A} 72 (1995), no.\ 1, 135--183.
\MR{1354971} (97h:06011).  \Zbl{834.06006}.  
\annot{} \S 6, ``Dowling lattices'':  They are an example, thus having an EL-labelling induced from $\Pi_n$.
\dated{9 Apr 2016} 
\codes{\Cgg:~\CMatrd,~\CInvar}%
\itemspace

\bsgitem{2000a}{}  Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations  \emph{Electronic J.\ Combin.}\ 7 (2000), Research Paper R9, 27 pp.
\MR{1741331} (2000k:05013).  \Zbl{938.05003}.						
\annot{} ``Type-B noncrossing partitions'' are certain signed partial partitions of the ground set; i.e., certain elements of the Dowling lattice of $\{\pm\}$.
\par
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{R.\ Simion \& D.-S.\ Cao}
\authorspace
\bsgitem{1989a}{}  Solution to a problem of C.\ D.\ Godsil regarding bipartite graphs with unique perfect matching.  \emph{Combinatorica} 9 (1989), 85--89.  
\MR{1010303} (90f:05113).  \Zbl{688.05056}.						
\annot{} Answering \xref{Godsil}{1985a}{C.D.\ Godsil}:  $|\Sigma| = \Gamma$ iff $\Gamma$ consists of a bipartite graph with a pendant edge attached to every vertex.  
[Surely there is a signed-graphic generalization of Godsil's and this theorem in which bipartiteness becomes balance or something like it.]
\codes{\Csg:~\CAdj,~\Cbal}%
\itemspace

\authorx{Aron Simis}							%SIMIS
\author{Aron Simis, Wolmer V.\ Vasconcelos, \& Rafael H.\ Villarreal}
\authorspace
\bsgitem{1999a}{}  The integral closure of subrings associated to graphs.  \emph{J.\ Algebra} 199 (1998), 281--289.
\MR{1489364} (99c:13004).  \Zbl{902.13004}.  
\par
\codes{\Csg:~\CPar:~\CInc~\CAlgeb,~\Cmatrd}%
\itemspace

\author{J.M.S.\ Sim\~oes-Pereira}					%SIMOES-PEREIRA
\authorx{Jos\'e Manuel dos Santos Sim\~oes-Pereira}
\authorx{J.M.S.\ Simoes-Pereira}
\authorspace
\bsgitem{1972a}{} On subgraphs as matroid cells.  \emph{Math.\ Z.}\ 127 (1972), 315--322.  
\MR{0317973} (47 \#6522).  \Zbl{226.05016}, (\Zbl{243.05022}).			
\annot{} ``Cell'' = circuit.  Along with \xref{Klee}{1971a}{Victor Klee}, invents the bicircular matroid (here, for finite graphs) (Thm.\ 1).  Suppose we have matroids on the edge sets of all [simple] graphs, such that the class of circuits is a [nonempty] union of homeomorphism classes of connected graphs.  Thm.\ 2:  The circle and bicircular matroids [and free matroids] are the only such matroids.
\codes{\CMtrdF,~\CBic}%
\itemspace

\bsgitem{1973a}{} On matroids on edge sets of graphs with connected subgraphs as circuits.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 38 (1973), 503--506.  
\MR{0314663} (47 \#3214).  \Zbl{241.05114}, \Zbl{264.05126}.				
\annot{} A family of (isomorphism types of) [simple] connected graphs is ``matroidal'' if for any $\Gamma$ the class of subgraphs of $\Gamma$ that are in the family constitute the circuits of a matroid on $E(\Gamma)$.  Bicircular and even-cycle matroids are the two nicest examples.  A referee contributes the even-cycle matroid [\cf\ \xref{Tutte}{1981a}{W.T.\ Tutte}, \xref{Doob}{1973a}{Michael Doob}].  Thm.: The family cannot be finite [unless it is void or consists of $K_2$].  [See \xref{Marcu}{1987a}{Danut Marcu} for a valuable new viewpoint.]
\codes{\CMtrdF,~\CBic,~\CECyc,~\CGen}%
\itemspace

\bsgitem{1975a}{} On matroids on edge sets of graphs with connected subgraphs as circuits II.  \emph{Discrete Math.}\ 12 (1975), 55--78.  
\MR{0419275} (54 \#7298).  \Zbl{307.05129}.					
\annot{} Partial results on describing matroidal families of simple, connected graphs.  Five basic types: free [omitted in the paper], cofree, circle, bicircular, and even-cycle.  If the family does not correspond to one of these, then every member has $\geq 3$ independent circles and minimum degree $\geq 3$.
\codes{\CMtrdF,~\CBic,~\CECyc:~\CGen}%
\itemspace

\bsgitem{1978a}{}  A comment on matroidal families.  
In: \emph{Probl\`emes Combinatoires et Th\'eorie des Graphes} (Colloq.\ Int., Orsay, 1976), pp.\ 385--387.  Colloques Int.\ du CNRS, 260.  Editions du C.N.R.S., Paris, 1978.  
\MR{0540020} (81b:05031).  \Zbl{412.05023}.
\annot{} Two small additions to \xrefd{1973a}{J.M.S.\ Simoes-Pereira}, \xrefd{1975a}{J.M.S.\ Simoes-Pereira}; one is that a matroidal family not one of the five basic types must contain $K_{p,q(p)}$ for each $m \geq 3$, with $q(p) \geq p$.
\codes{\CMtrdF,~\CBic,~\CECyc:~\CGen}%
\itemspace

\bsgitem{1992a}{}  Matroidal families of graphs.  
In:  Neil White, ed., \emph{Matroid Applications}, Ch.\ 4, pp.\ 91--105.  Encycl.\ Math.\ Appl., Vol.\ 40.  Cambridge Univ.\ Press, Cambridge, Eng., 1992.  
\MR{1165541} (93c:05036).  \Zbl{768.05024}.					
\annot{} ``Count'' matroids (see \xref{N.\ White}{1986a}{Neil L.\ White}) in \S4.3; \xref{Schmidt's}{1979a}{Rudiger Schmidt} remarkable generalization in \S4.4.
\par
\codes{\CGG:~\CMtrdF,~\CBic,~\CECyc:~\CGen:~\CExp,~\CExr,~\CRef}%
\itemspace

\bsgitem{2016a}{}  (as Jos\'e Manuel dos Santos Sim\~oes-Pereira)  An existence problem for matroidal families.  In:  Karim Adiprasito \etal, eds., \emph{Convexity and Discrete Geometry Including Graph Theory} (Mulhouse, 2014), pp.\ 261--262.  Springer Proc.\ Math.\ Stat., Vol.\ 148.  Springer, Cham, 2016.
\MR{3516717}.  \Zbl{1379.05023}.  
\annot{} \Cf\ \xref{Schmidt}{1979a}{Rudiger Schmidt}.  Are there unknown matroidal families?
\dated{6 Jul 2022} 
\codes{\CMtrdF}%
\itemspace

\author{Klaus Simon}						%SIMON
\authorspace
\authref   See \hyperlink{Thomas Raschle}{T.~Raschle}.
\itemspace

\author{[C.\ De Simone]}						%SIMONE
\authorspace
\authref  See \hyperlink{C.\ De Simone}{C.~De~Simone} (under `D').
\itemspace

\author{M.\ Simonovits} 						%SIMONOVITS
\authorspace
\authref   See \hyperlink{Bela Bollobas}{B.~Bollob\'as}, \hyperlink{J.A.\ Bondy}{J.A.~Bondy}, and \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\itemspace

\author{Daniel Simson}						%SIMSON
\authorspace
\authref  See also \hyperlink{Rafal Bocian}{R.~Bocian}, \hyperlink{Mariusz Felisiak}{M.~Felisiak}, \hyperlink{Marcin Gasiorek}{M.~G\c{a}siorek}, \hyperlink{Stanislaw Kasjan}{S.~Kasjan}, \hyperlink{Grzegorz Marczak}{G.~Marczak}, and \hyperlink{Agnieszka Polak}{A.~Polak}.
\authrefspace

\bsgitem{2013a}{}  A Coxeter-Gram classification of positive simply laced edge-bipartite graphs.  \emph{SIAM J.\ Discrete Math.}\ 27 (2013), no.\ 2, 827--854.
\MR{3048204}.  \Zbl{1272.05072}.  
\codes{\CSG}%
\itemspace

\bsgitem{2013b}{}  A framework for Coxeter spectral analysis of edge-bipartite graphs, their rational morsifications and mesh geometries of root orbits.  \emph{Fundamenta Inform.}\ 124 (2013), no.\ 3, 309--338.
\MR{3100347}.  \Zbl{1269.05073}.  
\codes{\CSG}%
\itemspace

\bsgitem{2013c}{}  Toroidal algorithms for mesh geometries of root orbits of the Dynkin diagram $\bbD_4$.  \emph{Fundamenta Inform.}\ 124 (2013), no.\ 3, 339--364.
\MR{3100348}.  \Zbl{1269.05105}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Daniel Simson \& Katarzyna Zaj\c{a}c}
\authorx{Daniel Simson \& Katarzyna Zajac}
\authorspace
\bsgitem{2013a}{}  An inflation algorithm and a toroidal mesh algorithm for edge-bipartite graphs.  Combinatorics 2012 (Perugia, 2012).  \emph{Electronic Notes in Discrete Math.}\ 40 (2013), 377--383.
\MR{3155275} (volume).  \Zbl{1292.05005} (volume).  
\par
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2017a}{}  Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two..  \emph{Linear Algebra Appl.}\ 524 (2017), 109--152.
\MR{3630181}.  \Zbl{1361.05061}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Alistair Sinclair}						%SINCLAIR
\authorspace
\authref  See \hyperlink{Mark Jerrum \& Alistair Sinclair}{M.~Jerrum}.
\itemspace

\author{Amit Singer}							%SINGER A
\authorspace
\authref  See \hyperlink{Afonso S.\ Bandeira}{A.S.~Bandeira}.
\itemspace

\author{Ambuj Singh}						%SINGH A
\authorspace
\authref  See \hyperlink{Zexi Huang}{Z.-X.~Huang}.
\itemspace

\author{Amrik Singh}							%SINGH A
\authorspace
\authref  See \hyperlink{Bibhas Adhikari}{B.~Adhikari}.
\itemspace

\author{Manjeet Singh}						%SINGH M
\authorspace
\authref  See \hyperlink{Seema Mehra}{S.~Mehra}.
\itemspace

\author{P.K.\ Singh}							%SINGH PK
\authorspace
\authref  See \hyperlink{Tushar Sharma}{T.~Sharma}.
\itemspace

\author{Rajiv R.P.\ Singh}						%SINGH RRP
\authorspace
\authref  See also \hyperlink{Michael E.\ Fisher}{M.E.~Fisher}.
\authrefspace

\author{Rajiv R.P.\ Singh and Sudip Chakravarty}
\authorspace
\bsgitem{1986a}{}  Critical behavior of an Ising spin-glass.  \emph{Phys.\ Rev.\ Lett.}\ 56 (1986), no.\ 2, 245--248.
\codes{\CPhys,~\CSG:~\CFr}%
\itemspace

\author{Ranveer Singh}						%SINGH R
\authorspace
\bsgitem{2019a}{}  On eigenvector structure of weakly balanced networks.  \emph{Physica A} 527 (2019), art.\ 121093, 10 pp.
\MR{3947751}.  
\annot{} ``Weakly balanced'' = clusterable.  Examples.
\dated{11 Jul 2022} 
\par
\codes{\CSG:~\CClu:~\CAdj:~\CEig}%
\itemspace

\author{Ranveer Singh \& Bibhas Adhikari}
\authorspace
\bsgitem{2017a}{}  Measuring the balance of signed networks and its application to sign prediction.  \emph{J.\ Stat.\ Mech.}\ 2017 (2017), no.\ 6, art.\ 063302, 16 pp.
\MR{3673443}.  \Zbl{1457.90046}.  
\codes{\CSG:~\CFr,~\CPred:~\CAdj:~\CEig}%
\itemspace

\author{Ranveer Singh \& Ravindra B.\ Bapat}
\authorspace
\bsgitem{2017a}{}  Eigenvalues of weakly balanced signed graphs and graphs with negative cliques.  Manuscript, 2017.  
\arXiv{1702.06322}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2018a}{}  $\cB$-partitions, determinant and permanent of graphs.  \emph{Trans.\ Combin.}\ 7 (2018), no.\ 3, 29--47.
\MR{3841400}.  \Zbl{1474.05325}.  \arXiv{1705.02517}.				
\codes{\CSG,~\CWG:~\CAdj}%
\itemspace

\author{Tarkeshwar Singh}						%SINGH T
\authorspace
\authref  See also \hyperlink{Mukti Acharya}{M.~Acharya} and \hyperlink{S.B.\ Rao}{S.B.~Rao}.
\authrefspace

\bsgitem{2003a}{}  \emph{Advances in the Theory of Signed Graphs}.  Doctoral dissertation, University of Delhi, India, 2003.
\annot{} Fairly complete accounts of \hyperlink{Mukti Acharya \& Tarkeshwar Singh}{M.\ Acharya \& Singh} (various) and \xref{Singh}{2008b}{Tarkeshwar Singh}, supplemented with background, appendix, etc.  
Ch.\ II, ``Graceful signed graphs'', is in \xref{M.\ Acharya and Singh}{2003a}{Mukti Acharya \& Tarkeshwar Singh}, \xrefd{2004a}{Mukti Acharya \& Tarkeshwar Singh}, \xrefd{2005a}{Mukti Acharya \& Tarkeshwar Singh}, \xrefd{2004b}{Mukti Acharya \& Tarkeshwar Singh}. 
Ch.\ III, ``Skolem graceful sigraphs'':  Announced in \xref{M.\ Acharya and Singh}{2003b}{Mukti Acharya \& Tarkeshwar Singh}.  Thm.\ 3.12:  See \xref{M.\ Acharya and Singh}{2010a}{Mukti Acharya \& Tarkeshwar Singh}.  Thm.\ 3.13:  A necessary condition for Skolem-gracefulness of signed multiple stars.  Thm.\ 3.14:  A sufficient condition for two signed stars.
Ch.\ IV, ``Negation-switching invariant sigraphs'':  See \xref{M.\ Acharya and Singh}{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  Also:  A binary encoding of signed circles.
App., ``A catalog of assorted labelled sigraphs''.
\dated{20 July 2009} 
\par
\codes{\CSGc:~\CLab}\codesctd{\CSG:~\CSw,~\CLG}%
\itemspace

\bsgitem{2008a}{}  Skolem and hooked Skolem graceful sigraphs.  
In:  B.D.\ Acharya, S.\ Arumugam, and Alexander Rosa, eds., \emph{Labelings of Discrete Structures and Applications} (Mananthavady, Kerala, 2006), pp.\ 155--164.  Narosa, New Delhi, 2008.
\MR{2391786} (2009e:05281) (book).  \Zbl{1161.05340}.
\annot{} [\Cf\ \xref{M.\ Acharya and Singh}{2004a}{Mukti Acharya \& Tarkeshwar Singh}, \xrefd{2003b}{Mukti Acharya \& Tarkeshwar Singh}.  Generalizing the definition:  Given: a graph with $r$-colored edges, $m_i$ of color $i$; a list $L$ of $n$ integers.  Required:  A bijection $\lambda: V \to  L$ such that, if $f(vw) := |\lambda(v)-\lambda(w)|$, then $f$ restricted to color class $i$ is a bijection to $[m_i]$.]  Signed graphs are the case $r=2$.  Skolem gracefulness is the case where $\lambda$ exists for $L = [n]$.  Hooked Skolem gracefulness is the case where $\lambda$ exists for $L = [n+1] \setminus \{n\}$.  Results from \xref{M.\ Acharya and Singh}{2010a}{Mukti Acharya \& Tarkeshwar Singh} and \xref{Singh}{2008b}{Tarkeshwar Singh}, examples, some proofs.
\codes{\CSGc:~\CLab:~\CExp}%
\itemspace

\bsgitem{2009a}{}  Graceful signed graphs on $C_3^k$.  Fifth Int.\ Workshop on Graph Labelings (IWOGL 2009) (Krishnankoil, 2009).  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 6 (2009), no.\ 1, 201--208.
\MR{2533200} (2010g:05330).  \Zbl{1210.05155}.					
\annot{} ``Graceful'' means $(1,1)$-graceful, $r=1$, as at \xref{M.\ Acharya and Singh}{2004a}{Mukti Acharya \& Tarkeshwar Singh}.  $C_3^k$ is the windmill with $k$ blades.  Let $\Sigma$ have $\nu$ negative rim edges, $1\leq\nu\leq k/2$, and no other negative edges.
Thm.\ 10:  $\Sigma$ is graceful if $k\equiv0 \mod 4$ and $\nu$ is even.
Thms.\ 11, 12:  $\Sigma$ is graceful if $k\equiv1,2 \mod 4$.
\dated{21 July 2010} 
\codes{\CSGc:~\CLab}%
\itemspace

\bsgitem{2008b}{}  A note on hooked Skolem graceful sigraphs and its application.  Manuscript, n.d.
\annot{} See \xrefd{2008a}{Tarkeshwar Singh}.  Thm.:  A signed $k$-edge matching is hooked Skolem graceful iff $k \equiv 0 \pmod 4$ and $\#E^-$ is odd, or $k \equiv 2 \pmod 4$ and $\#E^-$ is even, or $k \equiv 3 \pmod 4$.  Curiously complementary to the theorem of \xref{M.\ Acharya and Singh}{2010a}{Mukti Acharya \& Tarkeshwar Singh}.
\codes{\CSGc:~\CLab}%
\itemspace

\author{Tarkeshwar Singh \& Natasha D'Souza}
\authorspace
\bsgitem{2010a}{}  Some results in graceful signed tree.  [Abstract.]  In:  \emph{International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTBC-2010)} (Cochin, 2010) [Summaries], p.\ 169.  
Dept.\ of Math., Cochin Univ.\ of Science and Technology, 2010.
\annot{} Some graceful signed trees (see \xref{M.\ Acharya and Singh}{2004a}{Mukti Acharya \& Tarkeshwar Singh}).  Every signed tree is an induced subgraph of a graceful signed tree.
\dated{31 Aug 2010}
\codes{\CSGc:~\CLab}%
\itemspace

\author{N.M.\ Singhi} 						%SINGHI
\authorspace
\authref  See also \hyperlink{S.B.\ Rao}{S.B.~Rao}, \hyperlink{D.K.\ Ray-Chaudhuri, N.M.\ Singhi, \& G.R.\ Vijayakumar}{D.K.~Ray-Chaudhuri}, and \hyperlink{G.R.\ Vijayakumar}{G.R.~Vijayakumar}.
\authrefspace

\author{N.M.\ Singhi \& G.R.\ Vijayakumar} 
\authorspace
\bsgitem{1992a}{}  Signed graphs with least eigenvalue $< -2$.  \emph{European J.\ Combin.}\ 13 (1992), 219--220.
\MR{1164766} (93e:05069).  \Zbl{769.05065}.						
\annot{} A short proof that every such signed simple graph contains an induced subgraph with least eigenvalue $= -2$.  [Their $M := 2I + A(\Sigma)$ is the Laplacian matrix of $-\Sigma$.]
\codes{\CSG:~\Cadj}%
\itemspace

\author{Deepa Sinha}						%SINHA
\authorspace
\authref  See also \hyperlink{Mukti Acharya}{M.~Acharya}.
\authrefspace

\bsgitem{2005a}{}  \emph{New Frontiers in the Theory of Signed Graphs}.  Doctoral dissertation, University of Delhi, 2005.
\annot{} [Partial description]  
$\Sigma$ is ``sign compatible'' if $\exists\ X \subseteq V$ such that $E^- = E\ind X$.  It is ``canonically sign compatible'' if $X = \mu_\sigma\inv(-)$ (\cf\ \xref{Sampathkumar}{1984a}{E.\ Sampathkumar}).  
\dated{12 Oct 2010}
\codes{\CSG:~\CBal(\CGen)}	
\itemspace

\author{Deepa Sinha \& Mukti Acharya}
\authorspace
\bsgitem{2016a}{}  Characterization of signed graphs whose iterated signed line graphs are balanced or $S$-consistent.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\  39 (2016), no.\ 1, suppl., S297--S306.  
\MR{3509081}.  \Zbl{1339.05171}.  
\annot{} Extensions to $\Lambda_{BC}^k(\Sigma)$ of \xref{M.\ Acharya and Sinha's}{2002a}{Mukti Acharya \& Deepa Sinha} characterization of balance in the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph $\Lambda_{BC}(\Sigma)$ and to $\Lambda^k(\Sigma)$ of \xref{Acharya, Acharya, and Sinha's}{2009a}{Belmannu Devadas Acharya, Mukti Acharya, \& Deepa Sinha} criterion for consistency of $\Lambda(\Sigma)$.
\codes{\CSG,~\CVS:~\CLG(\CGen):~\CBal}%
\itemspace

\author{Deepa Sinha \& Ayushi Dhama}
\authorspace
\bsgitem{2012a}{}  Sign-compatibility of some derived signed graphs.  \emph{Mapana J.\ Sci.}\ 11 (2012), no.\ 4, 1--14.
\MR{3086508}.  
\annot{} \Cf\ \xref{Sinha}{2005a}{Deepa Sinha}.	
\codes{\CSG:~\CVS,~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2013a}{}  Sign-compatibility of some derived signed graphs.  \emph{Indian J.\ Math.}\ 55 (2013), no.\ 1, 23--40.
\MR{3086508}.  \Zbl{1274.05209}.  
\annot{} \Cf\ \xref{Sinha}{2005a}{Deepa Sinha}.	
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2013b}{}  Canonical sign-compatibility of some signed graphs.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 38 (2013), 129--138.
\Zbl{1316.05056}.  
\annot{} \Cf\ \xref{Sinha}{2005a}{Deepa Sinha}.	
\codes{\CSG:~\CBal(\CGen)}%
\itemspace

\bsgitem{2013c}{}  Sign-compatibility of common-edge sigraphs and 2-path sigraphs.  \emph{Graph Theory Notes N.Y.}\ 65 (2013), 55--61.
\MR{3204940}.  
\annot{} \Cf\ \xref{Sinha}{2005a}{Deepa Sinha}.	
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2013d}{}  On the unitary Cayley ring signed graphs $S_n^\oplus$.  \emph{J.\ Interconnection Networks} 14 (2013), no.\ 4, art.\ 1350020, 20 pp.
\annot{} Characterizes balance, clusterability, and for some $n$ also canonical consistency and sign compatibility.
\dated{11 Apr 2016} 
\par
\codes{\CAlgeb:~\CSG:~\CBal,~\CClu,~\CVS}%
\itemspace

\bsgitem{2014a}{}  Negation switching invariant signed graphs.  \emph{Electronic J.\ Graph Theory Appl.}\ 2 (2014), no.\ 1, 32--41.
\MR{3199369}.  \Zbl{1306.05088}.  
\annot{} Thm.\ 2.2:  Fairly elementary characterization of $\Sigma \simeq -\Sigma$.  Thm.\ 2.3:  Same for $\Sigma \cong -\Sigma$.  [\Problem.  Find complete structural characterizations.]
\dated{5 May 2014, 15 May 2018} 
\codes{\CSG:~\CSw,~\CAut(\CGen):~\CStr}%
\itemspace

\bsgitem{2014b}{}  Canonical sign compatibility of semi-total and total signed graphs.  \emph{Bull.\ Calcutta Math.\ Soc.}\ 106 (2014), no.\ 1, 55--64.
\MR{3380949}.  \Zbl{1318.05034}.  
\annot{} Define (semi)total signed graphs via the $\times$-line signed graph of \xref{M.\ Acharya}{2009a}{Mukti Acharya}.   Characterizes those that are compatible (\cf\ \xref{Sinha}{2005a}{Deepa Sinha}) with the canonical marking of the (semi)total signed graphs.
\dated{9 Apr 2014} 
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2015a}{}  On $\bullet$-line signed graphs $L_\bullet(S)$.  \emph{Discuss.\ Math.\ Graph Theory}\ 35 (2015), no.\ 2, 215--227.
\MR{3338747}.  \Zbl{1311.05078}.  
\annot{} Definition \cf\ \xref{B.D.~Acharya}{2010a}{B.\ Devadas Acharya}.  Notation: $S = \Sigma$.  Thm.\ 4: balance of $L_\bullet(\Sigma)$.  Thm.\ 7: sign compatibility; \cf\ \xref{Sinha}{2005a}{Deepa Sinha}.  Thm.\ 9: canonical sign compatibility; \cf\ \xref{Sinha}{2005a}{Deepa Sinha}.  Thm.\ 13 characterizes connected $\bullet$-line signed graphs.
\dated{2 Oct 2019} 
\par
\codes{\CSG:~\CLG:~\CBal,~\CBal(\CGen)}%
\itemspace

\bsgitem{2015b}{}  Unitary addition Cayley ring signed graphs $\sum_n^{\oplus*}$.  \emph{J.\ Discrete Math.\ Sci.\ Cryptography} 18 (2015), no.\ 5, 559--579.
\MR{3399713}.  
\annot{} \Cf\ \xref{Sinha and Garg}{2011e}{Deepa Sinha \& Pravin Garg}.  
\dated{7 Jan 2016} 
\par
\codes{\CSG:~\CBal,~\CClu,~\CBal(\CGen)}%
\itemspace

\bsgitem{2017a}{}  Unitary Cayley meet signed graphs.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 425--434.
\MR{3754832}.  \Zbl{1383.05160}.  
\codes{\CSG:~\CBal,~\CClu,~\CBal(\CGen)}
\itemspace

\bsgitem{2017b}{}  Negation switching invariant $3$-path signed graphs.  \emph{ J.\ Discrete Math.\ Sci.\ Cryptography} 20 (2017), no.\ 3, 703--716.
\MR{3691461}.  
\codes{\CSG:~\CLG(\CGen):~\CSw}%
\itemspace

\bsgitem{20xxa}{}  On the unitary Cayley meet signed graphs $S_n^\wedge$.  Submitted.  
\par
\codes{\CSG:~\CBal,~\CClu,~\CBal(\CGen)}%
\itemspace

\bsgitem{20xxb}{}  Negation-switching invariant $t$-path signed graphs, $t \leq 3$.  Submitted. 
\par
\codes{\CSG:~\CLG(\CGen):~\CSw}%
\itemspace

\author{Deepa Sinha, Ayushi Dhama, \& B.D.\ Acharya}
\authorspace
\bsgitem{2013a}{}  Unitary addition Cayley signed graphs.  \emph{European J.\ Pure Appl.\ Math.}\ 6 (2013), no.\ 2, 189--210.
\MR{3053442}.  \Zbl{1389.05066}.  
\annot{} 	
\Cf\ \xref{Sinha and Garg}{2011e}{Deepa Sinha \& Pravin Garg}.
\codes{\CSG:~\CBal,~\CClu,~\CBal(\CGen)}%
\itemspace

\author{Deepa Sinha \& Pravin Garg}
\authorspace
\bsgitem{2010a}{}  Consistency of semi-total signed graphs.  [Abstract.]  
In:  \emph{International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTGC-2010)} (Cochin, 2010) [Summaries], p.\ 153.  
Dept.\ of Mathematics, Cochin University of Science and Technology, 2010.
\annot{} Consistency of the canonical vertex signature of certain graphs related to the line graph and total graph of $\Sigma$; see e.g.\ \xrefd{2011f}{Deepa Sinha \& Pravin Garg}, \xrefd{2015a}{Deepa Sinha \& Pravin Garg}, \xrefd{2015b}{Deepa Sinha \& Pravin Garg}.
\dated{31 Aug 2010}
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2011a}{}  Canonical consistency of signed line structures.  \emph{Graph Theory Notes N.Y.}\ 59 (2011), 22--27.
\MR{2849400} (2012g:05098).  
\annot{} Consistency of the canonical vertex signature of two kinds of line graph: (Thm.\ 2) $\Lambda_{BC}(\Sigma)$ of \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} and (Thm.\ 8) $\Lambda_{\times}(\Sigma)$ of \xref{M.\ Acharya}{2009a}{Mukti Acharya}.
\dated{25 Mar 2011} 
\codes{\CSG:~\CLG:~\CVS:~\CBal(\CGen)}%
\itemspace

\bsgitem{2011b}{}  Balance and consistency of total signed graphs.  \emph{Indian J.\ Math.}\ 53 (2011), no.\ 1, 71--81.
\MR{2809572} (2012d:05174).  \Zbl{1238.05126}.					
\annot{$\Tau_{SG}$} Defines a vertex- and edge-signed total graph $\Tau_{SG}(\Sigma)$.  Vertex signs $\mu_1(v) := \mu_\sigma(v)$, $\mu_1(e) = \sigma(e)$.  Edge signs $\sigma_\Tau(uv) := \sigma(e_{uv})$, $\sigma_\Tau(ue) := \sigma(e)\mu_1(u)$, $\sigma_\Tau(ef) := \sigma(e)\sigma(f)$ [thus $\Tau_{SG}(\Sigma) \supseteq \Lambda_{\times}(\Sigma)$ of \xref{M.\ Acharya}{2009a}{Mukti Acharya}].  Balance and consistency characterized.  
[For spectrally suitable total graphs \cf\ \xref{Belardo--Stani\'c--Zaslavsky}{2023a}{Francesco Belardo, Zoran Stanic, \& Thomas Zaslavsky}.]
\dated{13 Oct 2009, 20 Dec 2010, 1 Apr 2022} 
\codes{\CSG,~\CVS:~\CLG(\CGen):~\CBal,~\CBal(\CGen)}%
\itemspace

\bsgitem{2011c}{}  On the regularity of some signed graph structures.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 8 (2011), no.\ 1, 63--74.
\MR{2839176} (2012f:05126).  \Zbl{1238.05127}.					
\annot{} $\Sigma$ is ``regular'' if $\Sigma^+$ and $\Sigma^-$ are regular.  For the edge signs of line graphs and total graph see \xrefd{2011b}{Deepa Sinha \& Pravin Garg}.   Characterizes $\Sigma$ such that $\Lambda_{BC}$ or $\Lambda_{\times}$ or $\Tau_{SG}$ is regular.
Dictionary:  ``signed-regular'' = regular.
\dated{25 July 2011} 
\codes{\CSG:~\CLG,~\CLG(\CGen)}%
\itemspace

\bsgitem{2011d}{}  Characterization of total signed graph and semi-total signed graphs.  \emph{Int.\ J.\ Contemp.\ Math.\ Sci.}\ 6 (2011), no.\ 5-8, 221--228.
\MR{2797063} (no rev).  \Zbl{1235.05058}.  
\annot{} Thm.\ 2.3 characterizes semi-total signed graphs.  
Thm.\ 3.2 characterizes semi-total point signed graphs.  
Thm.\ 4.4 characterizes total signed graphs.  
Each result applies a pre-existing characterization of underlying graphs.
\dated{23 Nov 2014} 
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\bsgitem{2011e}{}  On the unitary Cayley signed graphs.  \emph{Electronic J.\ Combin.}\ (2011), art.\ P229, 13 pp.
\MR{2861408} (2012k:05173).  \Zbl{1243.05110}.  
\annot{} The unitary Cayley graph $X_n = (\bbZ_n, \{ab: \exists\ (b-a)\inv\})$.  $S_n = (X_n,\sigma)$ where $\sigma(ab) = -$ iff $\not\exists\ a\inv, b\inv$.  Thm.\ 4:  $S_n$ is balanced iff $n$ is even or a prime power.  Cor.\ 5:  $S_n$ is antibalanced iff $n$ is even.  Cor.\ 7:  $\Lambda_{BC}(S_n)$ is balanced iff $n$ is a prime power.  Thm.\ 20:  Let $n$ have at most 2 distinct odd prime factors.  $S_n$ is canonically consistent iff $n$ is odd, evenly even, 2, or 6.
\dated{16 Jan 2012} 
\codes{\CSG:~\CBal,~\CLG,~\CBal(\CGen):~\CAlgeb}%
\itemspace

\bsgitem{2011f}{}  Some results on semi-total signed graphs.  \emph{Discuss.\ Math.\ Graph Theory} 31 (2011), no.\ 4, 625--638.
\MR{2952233}.  \Zbl{1255.05091}.						
\annot{} Similar to \xrefd{2011b}{Deepa Sinha \& Pravin Garg}, but for $T_2(\Sigma) := T_{SG}(\Sigma)$ without line-graph edges.
\dated{13 Oct 2009} 
\codes{\CSG,~\CVS:~\CLG(\CGen):~\CBal}%
\itemspace

\bsgitem{2013a}{}  
A characterization of canonically consistent total signed graphs.  \emph{Notes Number Theory Discrete Math.}\ 19 (2013), no.\ 3, 70--77.
\Zbl{1314.05092}.  
\annot{} \Cf\ \xref{Sinha and Garg}{2011a}{Deepa Sinha \& Pravin Garg}.  
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2014a}{}  Balance and antibalance of tensor product of two signed graphs.  \emph{Thai J.\ Math.}\ 12 ( 2013), no.\ 2, 303--311.
\MR{3217341}.  \Zbl{1307.05100}.  
\annot{} Tensor product defined by \xref{Mishra}{1974a}{V.\ Mishra}.  For connected signed graphs:  Thm.\ 2.6: $\Sigma_1 \otimes \Sigma_2$ is balanced iff $\Sigma_1$ and $\Sigma_2$ are both balanced or both antibalanced.  Thm.\ 3.1: It is antibalanced iff one is balanced and the other is antibalanced.
\dated{23 Nov 2014}
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{2015a}{}  Canonical consistency of semi-total line signed graphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 38 (2015), no.\ 5, 429--432.
\MR{3417353}.  
\annot{} \Cf\ \xref{Sinha and Garg}{2011a}{Deepa Sinha \& Pravin Garg}.  
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\bsgitem{2015b}{}  Canonical consistency of semi-total point signed graphs.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 38 (2015), no.\ 6, 497--500.
\MR{3433367}.  
\annot{} \Cf\ \xref{Sinha and Garg}{2011a}{Deepa Sinha \& Pravin Garg}.  
\codes{\CSG:~\CLG(\CGen):~\CBal(\CGen)}%
\itemspace

\author{Deepa Sinha, Pravin Garg, \& H.\ Saraswat}%  % VERIFY
\authorspace
\bsgitem{2013a}{}  On the splitting signed graphs.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 38 (2013), 103--111.  
\Zbl{1304.05067}.  
\codes{\CSG}%
\itemspace

\author{Deepa Sinha \& Sandeep Kumar}
\authorspace
\bsgitem{20xxa}{}  On the relation between index and Sombor index of signed graph.  In preparation.  \emph{}\ 
\codes{\CSG}\codesctd{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Deepa Sinha \& Anita Kumari Rao}
\authorspace
\bsgitem{2017a}{}  On co-maximal meet signed graphs of commutative rings.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 497--502.
\MR{3754840}.  \Zbl{1383.05146}.  
\codes{\CSG:~\CAlgeb}%
\itemspace

\bsgitem{2018a}{}  Co-maximal signed graphs of commutative rings.  \emph{Turkish J.\ Math.}\ 42 (2018), 1203--1220.
\MR{3804981}.  \Zbl{1424.05126}.  
\codes{\CSG:~\CAlgeb,~\CLG:~\CBal,~\CClu}%
\itemspace

\bsgitem{2018b}{}  Embedding of signed regular graphs.  \emph{Notes Number Theory Discrete Math.}\ 24 (2018), no.\ 3, 131--141.
\codes{\CSG}%
\itemspace

\bsgitem{2022a}{}  Embedding of sign-regular signed graphs and its spectral analysis.  \emph{Linear Multilinear Algebra}\ 70 (2022), no.\ 8, 1496--1512.
\MR{4420912}.  
\codes{\CSG}\codesctd{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Deepa Sinha, Anita Kumari Rao, \& Ayushi Dhama}
\authorspace
\bsgitem{2019a}{}  Spectral analysis of $t$-path signed graphs.  \emph{Linear Multilinear Algebra} 67 (2019), no.\ 9,
1879--1897.
\MR{3980671}.  \Zbl{1418.05071}.  
\annot{} Main result is Thm.\ 3.4:  $\Spec A(\Sigma)$ is sign-symmetric iff $\Sigma \simeq -\Sigma$.  [N.B. $\Sigma \simeq -\Sigma$ is unsolved.]  $(\Sigma)_t :=$ a kind of signed $t$-path graph.  Other results, e.g., Thm.\ 2.8: For signed $K_n$, $(\Sigma)_2 \simeq -\Sigma$ iff $\Sigma$ is balanced or clusterable in a limited way.
\dated{27 May 2018} 
\par
\codes{\CSG:~\CAdj:~\CEig}\codesctd{\CSG:~\CLG(\CGen),~\CClu}%
\itemspace

\author{Deepa Sinha, Anita Kumari Rao, \& Pravin Garg}
\authorspace
\bsgitem{2016a}{}  Embedding of $(i, j)$-regular signed graphs in $(i + k, j + l)$-regular signed graphs.  In:  \emph{2016 International Workshop on Computational Intelligence} (IWCI, Dhaka, 2016), pp.\ 215--218.  IEEE, 2016.
\annot{} $\Sigma$ is $(i,j)$-regular if $\Sigma^+$ is $i$-regular and $\Sigma^-$ is $j$-regular.  Embedding is as a subgraph.  The aim is to minimize the order of the supergraph.
\dated{13 Mar 2018} 
\codes{\CSG}%
\itemspace

\author{Deepa Sinha \& Anshu Sethi}
\authorspace
\bsgitem{2014a}{}  An optimal algorithm to detect balancing in common-edge sigraph.  \emph{Int.\ J.\ Computer Appl.}\ 93 (2014), no.\ 10, 19--25.
\codes{\CSG:~\CLG(\CGen):~\CBal:~\CAlgor}%
\itemspace

\bsgitem{2014b}{}  An algorithmic characterization of line signed graph.  Manuscript, 2014.
\annot{} For a signed simple graph $\Sigma$, algorithms to construct (\S3) the line graph $\Lambda(|\Sigma|)$; (\S4) $\Gamma'$ such that $|\Sigma| = \Lambda(\Gamma')$, if it exists; (\S5) the \xref{Behzad--Chartrand}{1969a}{M.\ Behzad \& G.\ Chartrand} line graph $\Lambda_{BC}(\Sigma)$; (\S6) $\Sigma'$ such that $\Sigma= \Lambda_{BC}(\Sigma')$, if it exists.  
\Cf\ \xref{M.\ Acharya and Sinha}{2005a}{Mukti Acharya \& Deepa Sinha}.
\dated{24 Dec 2014} 
\par
\codes{\CSG:~\CLG:~\CAlgor}%
\itemspace

\bsgitem{2015a}{}  An optimal algorithm to detect sign compatibility of a given sigraph.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 38 (2015), no.\ 3, 235--238.
\MR{3366153}.  
\annot{} Definition: \xref{Sinha}{2005a}{Deepa Sinha}.  The algorithm detects the forbidden subgraphs:  a path with edges $-,+,-$ and a triangle with edges $-,+,-$.
\dated{6 Jan 2016} 
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2015b}{}  An algorithmic characterization of sigraphs whose second iterated line sigraphs and common-edge sigraphs are switching equivalent..  \emph{J.\ Discrete Math.\ Sci.\ Cryptography} 18 (2015), no.\ 5, 581--603.
\MR{3399714}.  
\par
\codes{\CSG:~\CLG,~\CLG(\CGen):~\CSw:~\CAlgor}%
\itemspace

\bsgitem{2015c}{}  An algorithm to detect balancing of iterated line sigraph.  \emph{SpringerPlus} 4 (2015), art.\ 704, 19 pp.
\codes{\CSG:~\CLG:~\CBal:~\CAlgor}%
\itemspace

\bsgitem{2015d}{}  An algorithm to detect $S$-consistency in line sigraph.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 40 (2015), 61--91.
\MR{3586362}.  \Zbl{1358.05134}.  
\codes{\CSG:~\CLG:~\CAlgor}%
\itemspace

\bsgitem{2016a}{}  Encryption using network and matrices through signed graphs.  \emph{Int.\ J.\ Computer Appl.}\ 138 (2016), no.\ 4, 6--13.
\codes{\CSG:~\CAdj,~\CLG,~\CAlgor:~\CAppl}%
\itemspace

\bsgitem{2017a}{}  An algorithmic characterization of splitting signed graph.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 323--332.
\MR{3754821}.  \Zbl{1383.05147}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Deepa Sinha \& Deepakshi Sharma}
\authorspace
\bsgitem{2014a}{}  Signed graphs whose 2-path signed graphs are isomorphic to their square signed graphs.  Manuscript, 2014.
\annot{} Full version of \xrefd{2014b}{Deepa Sinha \& Deepakshi Sharma}.
\codes{\CSG:~\CLG(\CGen),~\CAlgor}%
\itemspace

\bsgitem{2014b}{}  Algorithmic characterization of signed graphs whose two path signed graphs and square signed graphs are isomorphic.  In:  \emph{2014 International Conference on Soft Computing Techniques for Engineering and Technology} (ICSCTET-2014, Bhimtal, India, 2014), 5 pp.  IEEE, [2014]. 
\annot{} Extended abstract of \xrefd{2014a}{Deepa Sinha \& Deepakshi Sharma}.
\codes{\CSG:~\CLG(\CGen),~\CAlgor}%
\itemspace

\bsgitem{2016a}{}  On square and 2-path signed graph.  \emph{J.\ Interconnection Networks} 16 (2016), no.\ 1, art.\ 1550011, 19 pp.
\codes{\CSG:~\CLG(\CGen),~\CAlgor}%
\itemspace

\bsgitem{2016b}{}  On 2-path signed graphs.  In:  \emph{2016 International Workshop on Computational Intelligence} (IWCI, Dhaka, 2016), pp.\ 218--220.  IEEE, 2016.
\annot{} Extended abstract.
\codes{\CSG:~\CLG(\CGen):~\CBal,~\CClu,~\CStr}%
\itemspace

\bsgitem{2017a}{}  Characterization of 2-path product signed graphs with Its properties.  \emph{Comput.\ Intell.\ Neurosci.}\ 2017 (2017), art.\ 1235715, 8 pp.
\codes{\CSG:~\CLG(\CGen):~\CVS,~\CSw}%
\itemspace

\bsgitem{2017b}{}  Transitivity model on signed graphs.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 455--460.
\MR{3754835}.  \Zbl{1383.05148}.  
\codes{\CSG}%
\itemspace

\bsgitem{2018a}{}  Iterated local transitivity model for signed social networks.  \emph{Appl.\ Algebra Eng.\ Commun.\ Comput.}\ 29 (2018), 149--167.
\MR{3769265}.  \Zbl{1384.05093}.  
\codes{\CSG:~\CBal,~\CClu,~\CVS}%
\itemspace

\bsgitem{2018b}{}  On the properties of square signed graph.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)}\ 41 (2018), 233--238.
\MR{3842236}.  
\codes{\CSG}%
\itemspace

\bsgitem{2019a}{}  Square signed graph.  \emph{Nat.\ Acad.\ Sci.\ Lett.\ (India)} 42 (2019), 513--518. 
\MR{4039031}.  
\codes{\CSG}%
\itemspace

\bsgitem{2020a}{}  Characterization of 2-path signed network.  \emph{Complexity} 2020 (2020), art.\ 1028941, 13 pp. 
\Zbl{1454.05119}.  
\annot{} $(\Sigma)_2 := (V,E',\sigma')$, where $uv \in E'$ iff $d(u,v)=2$, and $\sigma'(uv) = -$ iff every 2-path $uwv$ is all negative.  Thm.\ 1 characterizes $(\Sigma)_2$'s.  Thms.\ 2--4: balance, clusterability, sign-regularity of $(\Sigma)_2$.  [Some seem too complicated.  Some are not clear.]
\dated{9 Nov 2020} 
\par
\codes{\CSG:~\CVS,~\CBal,~\CClu,~\CAlgor}%
\itemspace

\author{Deepa Sinha, Deepakshi Sharma, \& Bableen Kaur}
\authorspace
\bsgitem{2017a}{}  Signed zero-divisor graph.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 517--524.
\MR{3754842}.  \Zbl{1383.05149}.  
\codes{\CSG:~\CAlgeb}%
\itemspace

\author{Deepa Sinha, Somya Upadhyaya, \& Priya Kataria}
\authorspace
\bsgitem{2013a}{}  Characterization of common-edge sigraph.  \emph{Discrete Appl.\ Math.}\ 161 (2013), no.\ 9, 1275--1285.
\MR{3030620}.  \Zbl{1277.05080}.  
\annot{} For definition \cf\ \xref{M.\ Acharya and Sinha}{2006a}{Mukti Acharya \& Deepa Sinha}.  Thm.\ 6:  $\Sigma$ is a common-edge signed graph iff $|\Sigma|$ is a common-edge graph and its edges decompose into homogeneously signed complete graphs.  \S4, ``Algorithm to output $C_E$-root sigraph of a given common-edge sigraph''.  \S5, ``Complexity of COMMON-EDGE SIGRAPH'':  It is $O(n^2\#E)$.
\dated{23 Nov 2014} 
\codes{\CSG:~\CLG(\CGen)}%
\itemspace

\author{John Sinkovic}						%SINKOVIC
\authorspace
\authref  See \hyperlink{Marina Arav}{M.~Arav}.
\itemspace

\author{Jozef \v{S}ir\'a\v{n}} 						%SIRAN
\authorx{Jozef Siran}
\authorspace
\authref  See also \hyperlink{Dan Archdeacon}{D.~Archdeacon}, \hyperlink{Pavol Gvozdjak}{P.~Gvozdjak}, \hyperlink{Cai Heng Li}{C.H.~Li}, and \hyperlink{Brendan D.\ McKay}{B.D.~McKay}.
\authrefspace

\bsgitem{1991a}{}  Characterization of signed graphs which are cellularly embeddable in no more than one surface.  \emph{Discrete Math.}\ 94 (1991), 39--44.  
\MR{1141052} (92i:05086).  \Zbl{742.05035}.					
\annot{} A signed graph orientation-embeds in only one surface iff any two circles are vertex disjoint. 
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{1991b}{}  Duke's theorem does not extend to signed graph embeddings.  \emph{Discrete Math.}\ 94 (1991), 233--238.  
\MR{1138602} (92j:05065).  \Zbl{742.05036}.					
\annot{} Richard A.\ Duke (The genus, regional number, and Betti number of a graph.  \emph{Canad.\ J.\ Math.}\ 18 (1966), 817--822.  \MR{0196731} (33 \#4917).)\ proved that the (orientable) genus range of a graph forms a contiguous set of integers.   \xref{Stahl}{1978a}{Saul Stahl} proved the analog for nonorientable embeddings.  \v{S}ir\'a\v{n}  shows this need not be the case for the demigenus range of an unbalanced signed graph.  However, any gaps consist of a single integer each.  The main examples with gaps are vertex amalgamations of balanced and uniquely embeddable unbalanced signed graphs, but a 3-connected example is $+W_6$ together with the negative diameters of the rim.  \Question\ 1 (\v{S}ir\'a\v{n}).  Do all gaps occur at the bottom of the demigenus range?  [\Question\ 2.  Can one in some way derive almost all signed graphs with gaps from balanced ones?]  
\codes{\CSG:~\CTop}%
\itemspace

\author{Jozef \v{S}ir\'a\v{n}, Jana \v{S}iagiov\'a, \& Mari\'an Olej\'ar}
\authorx{Jozef Siran, Jana Siagiova, \& Marian Olejar}
\authorspace
\bsgitem{2009a}{}  Graph coverings and graph labellings.  Special Issue on Graph Labelings.  Fifth Int.\ Workshop on Graph Labelings (IWOGL 2009) (Krishnankoil, 2009).  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 6 (2009), no.\ 1, 127--133.
\MR{2533240} (2010g:05331).  \Zbl{1210.05129}.					
\annot{} Connectivity and automorphisms of a covering graph of a gain graph (``voltage graph'').
\dated{21 July 2010} 
\codes{\CGG:~\CCov:~\CAut,~\CExp}%
\itemspace

\author{Jozef \v{S}ir\'a\v{n} \& Martin \v{S}koviera} 
\authorx{Jozef Siran \& Martin Skoviera} 
\authorspace
\bsgitem{1991a}{\dag\dag}  Characterization of the maximum genus of a signed graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 52 (1991), 124--146.  
\MR{1109428} (92b:05033).  \Zbl{742.05037}.					
\annot{} The maximum demigenus $d_M(\Sigma) =$ the largest demigenus of a closed surface in which $\Sigma$ orientation embeds.  Two formulas are proved for $d_M(\Sigma)$: one a minimum and the other a maximum of readily computable numbers.  Thus $d_M(\Sigma)$ has a ``good'' (polynomial) characterization.  Along the way, several results are proved about single-face embeddings.  \Problem\ (\S11).  Characterize those edge-2-connected $\Sigma$ such that $\Sigma$ and all $\Sigma \backslash e$ have single-face embeddings.  [A complex and lovely paper.]
\annotctd{} [Re single-face embeddings, \cf\ \xref{Bernardi and Chapuy}{2011a}{Olivier Bernardi \& Guillaume Chapuy} and \xref{Isenmann and Pecatte}{2017a}{Lucas Isenmann \& Timothee Pecatte}.]
\codes{\CSG:~\CTop}%
\itemspace

\author{Jozef \v{S}ir\'a\v{n} \& Thomas W.\ Tucker}
\authorx{Jozef Siran \& Thomas W.\ Tucker}
\authorspace
\bsgitem{2009a}{}  Symmetric maps.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 10, pp.\ 199--224.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581547} (no rev).  \Zbl{1201.05022}.  
\annot{} P.\ 203 mentions edge signs.  P.\ 218 says the Petrie dual of a rotation system on $\Sigma$ is the same rotation system on $-\Sigma$.
\dated{26 Aug 2018} 
\codes{\Csg:~\CTop,~\Csw:~\CExp}%
\itemspace

\author{[P.\ Siva Kota Reddy]}						%SIVA KOTA REDDY
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy} (under `R').
\itemspace

\author{B.\ Sivakumar}						%SIVAKUMAR
\authorspace
\authref  See also \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\authrefspace

\bsgitem{2009a}{}  Matrix units for the group algebra $kG_f = k((Z_2 \times Z_2) \wr S_f)$.  \emph{Asian-European J.\ Math.}\ 2 (2009), no.\ 2, 255--277.
\MR{2532703} (2010g:16043).  \Zbl{1198.20013}.					
\par
\codes{\Cgg:~\Cmatrd:~\CAlgeb}%
\itemspace

\author{Vaidy Sivaraman}						%SIVARAMAN
\authorspace
\authref  See also \hyperlink{John Maharry}{J.~Maharry}.
\authrefspace

\bsgitem{2012a}{}  \emph{Some Topics Concerning Graphs, Signed Graphs, and Matroids}.  Doctoral thesis, Ohio State University, 2012.
\MR{3130834} (no rev).  
\annot{} Thm.:  $D(\Sigma) := \min_X l(\Sigma^X) \leq \frac38 n$ for cubic $\Sigma$.  [\Cf\ \xref{Sehrawat and Bhattacharjya}{2019b}{Deepak Sehrawat \& Bikash Bhattacharjya}.]
\codes{\CSG:~\CSw,~\CMatrd}%
\itemspace

\bsgitem{2014a}{}  Bicircular signed-graphic matroids.  \emph{Discrete Math.}\ 328 (2014), 1--4.
\MR{3199809}.  \Zbl{1288.05035}.  
\annot{} The graphs for which $\bfF(\Gamma,\eset)$ is a frame matroid of a signed graph: iff $\bfF(\Gamma,\eset)$ is ternary, and other characterizations including forbidden subgraphs.  Successor to \xref{Matthews}{1977a}{Laurence R.\ Matthews}, and implicitly \xref{Zaslavsky}{2007a}{Thomas Zaslavsky} for group $\bbZ_2$.
\dated{1 Oct 2017, rev 11 Jun 2019}
\par
\codes{\CBic,~\CSG:~\CMatrd}%
\itemspace

\author{Vaidy Sivaraman \& Daniel Slilaty}
\authorspace
\bsgitem{2019a}{}  The graphs that have antivoltages using groups of small order.  \emph{Discrete Math.}\ 342 (2019), 2951--2965.
\MR{3996733}.  \Zbl{1419.05189}.  
\annot{} $\Gamma$ characterized for $|\fG|\leq6$ by structure, forbidden minors, and algorithmic recognition.  Successor to \xref{Zaslavsky}{2007a}{Thomas Zaslavsky}, \xref{Sivaraman}{2014a}{Vaidy Sivaraman}, and \xref{Chun, Moss, Slilaty, and Zhou}{2016a}{Deborah Chun, Tyler Moss, Daniel Slilaty, \& Xiangqian Zhou}.
\dated{11 Jun 2019}
\par
\codes{\CGG:\CBic:~\CGeom,~\CAlgor}%
\itemspace

\bsgitem{2022a}{}  The family of bicircular matroids closed under duality.  \emph{Graphs Combin.}\ 8 (2020), art.\ 24, 20 pp.
\MR{4356265}.  \Zbl{1481.05025}.  \arXiv{2012.11712}.  
\annot{} The 3-connected matroids that are bicircular and dual bicircular are the free swirls $\bfF(2C_n,\eset)$ and their minors, with exceptions of rank and dual rank $\leq5$.
\dated{29 Dec 2020} 
\codes{\CBic,~\CDual}%
\itemspace

\author{Vaidy Sivaraman \& Thomas Zaslavsky}
\authorspace
\bsgitem{20xxa}{}  The seven signed Heawood graphs.  In preparation.  
\annot{} Successor to \xref{Zaslavsky}{2012b}{Thomas Zaslavsky}, with several general theorems.  There are 7 switching isomorphism classes of signatures of the Heawood graph $H$.  $l, l_0, \chi, Q$ (inclusterability index) are computed.
General thm.:  For subcubic $|\Sigma|$, $l_0(\Sigma)=l(\Sigma)$ (for $-\Gamma$ see \xref{Choi, Nakajima, and Rim}{1989a}{Hyeong-ah Choi, Kazuo Nakajima, \& Chong S.\ Rim}).  If $\#E^-(\Sigma)=l(\Sigma)$, then $Q(\Sigma)=l(\Sigma)$.
\dated{1 Oct 2017}
\par
\codes{\CSG:~\CSw,~\CStr,~\CFr,~\CClu,~\CCol}%
\itemspace

\author{A.\ Skhre\u{\i}ver [A.\ Schrijver]}					%SKHREIVER
\authorx{A.\ Skhreiver}
\authorspace
\authref   See \hyperlink{Alexander Schrijver}{A.~Schrijver}.
\itemspace

\author{D.B.\ Skillicorn}						%SKILLICORN
\authorspace
\authref  See \hyperlink{Q.\ Zheng}{Q.~Zheng}.
\itemspace

\author{Bjarke Skjernaa}						%SKJERNAA
\authorspace
\authref  See \hyperlink{Jesper Makholm Byskov}{J.M.~Byskov}.
\itemspace

\author{Howard Skogman}						%SKOGMAN
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo} and \hyperlink{Nathan Reff}{N.~Reff}.
\itemspace

\author{Martin \v{S}koviera} 						%SKOVIERA
\authorx{Martin Skoviera}
\authorspace
\authref   See also \hyperlink{Edita Macajova}{E.~M\'a\v{c}ajov\'a}, \hyperlink{Aleksander Malnic}{A.~Malni\v{c}}, \hyperlink{Roman Nedela}{R.~Nedela}, and \hyperlink{Jozef Siran}{J.~\v{S}ir\'a\v{n}}.
\authrefspace

\bsgitem{1983a}{}  Equivalence and regularity of coverings generated by voltage graphs.  
In: Miroslav Fiedler, ed.,  \emph{Graphs and Other Combinatorial Topics} (Proc.\ Third Czechoslovak Sympos.\ on Graph Theory, Prague, 1982), pp.\ 269--272.  Teubner-Texte Math., 59.  Teubner, Leipzig, 1983.  
\MR{0737050} (85e:05064).  \Zbl{536.05019}.						
\par
\codes{\CGG:~\CTop,~\CCov,~\CSw}%
\itemspace

\bsgitem{1986a}{}  A contribution to the theory of voltage graphs.  \emph{Discrete Math.}\ 61 (1986), 281--292.  
\MR{0855333} (88a:05060).  \Zbl{594.05029}.						
\annot{} Automorphisms of covering projections of canonical covering graphs of gain graphs.
\codes{\CGG:~\CTop,~\CCov,~\CAut,~\CSw}%
\itemspace

\bsgitem{1992a}{}  Random signed graphs with an application to topological graph theory.  In: Alan Frieze and Tomasz Luczak, eds., \emph{Random Graphs, Vol.\ 2} (Proc., Pozna\'n, 1989), Ch.\ 17, pp.\  237--246.  Wiley, New York, 1992.  
\MR{1166619} (93g:05126).  \Zbl{817.05059}.						
\annot{} The model: each edge is selected with probability $p$, positive with probability $s$.  Under mild hypotheses on $p$ and $s$, $\Sigma$ is almost surely unbalanced and almost surely has a $1$-face orientation embedding.    [Related: \xref{Frank and Harary}{1979a}{Ove Frank \& Frank Harary}.]
\codes{\CSG:~\CRand,~\CEnum,~\CTop}%
\itemspace

\author{Riste \v{S}krekovski}						%SKREKOVSKI
\authorx{Riste Skrekovski}
\authorspace
\authref  See \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\author{Daniel C.\ Slilaty}						%SLILATY
\authorspace
\authref   See also \hyperlink{Lowell Abrams}{L.~Abrams}, \hyperlink{Nathan Bowler}{N.~Bowler}, \hyperlink{Arthur H.\ Busch}{A.H.~Busch}, \hyperlink{Yu Qing Chen}{Y.Q.~Chen}, \hyperlink{Deborah Chun}{D.~Chun}, \hyperlink{John Maharry}{J.~Maharry}, \hyperlink{Nancy Ann Neudauer}{N.A.~Neudauer}, \hyperlink{Hongxun Qin}{H.~Qin}, and \hyperlink{Jakayla R.\ Robbins}{J.~Robbins}.
\authrefspace

\bsgitem{2000a}{}  \emph{Orientations of Biased Graphs and Their Matroids}.  Doctoral dissertation, State University of New York at Binghamton, 2000.
\MR{2701091} (no rev).  
\annot{} Introducing orientation of biased graphs and biased signed graphs by means of proper circle orientations and their generalization, ``graphical orientation schemes''.  The definition is chosen so as to produce orientations of the bias and complete lift matroids and (though not in the thesis) to model the orientation of the bias or complete lift matroid of, respectively, an $\bbR^\times$\!- or $\bbR^+$\!-gain graph induced by its canonical bias or lift representation (\xref{Zaslavsky}{2003b}{Thomas Zaslavsky}).  Characterizations of equivalence of different orientation schemes.  The completeness question: when do graphical orientation schemes yield all orientations of the frame matroid?  Always, for additively biased (i.e., signed) graphs and for some other kinds of biased graphs.
\codes{\CGG:~\COri,~\CMatrd,~\COG,~\CSG}%
\itemspace

\bsgitem{2002a}{}  Matroid duality from topological duality in surfaces of nonnegative euler characteristic.  \emph{Combin.\ Probab.\ Computing} 11 (2002), no.\ 5, 515--528.
\MR{1930356} (2003i:05034).  \Zbl{1009.05036}.					
\annot{} Duality of matroids of biased graphs, especially signed graphs, obtained by defining gains through embedding in a surface and dualizing the graph in the surface, especially the plane, projective plane, and torus.
\dated{rev 15 Dec 2020}
\codes{\CGG,~\CSG:~\CMatrd,~\CDual,~\CTop}%
\itemspace

\bsgitem{2005a}{}  On cographic matroids and signed-graphic matroids.  \emph{Discrete Math.}\ 301 (2005), no.\ 12, 207--217.
\MR{2171313} (2007c:05049).  \Zbl{1078.05017}.					
\codes{\CSG:~\CMatrd,~\CTop}%
\itemspace

\bsgitem{2006a}{}  Bias matroids with unique graphical representations.  \emph{Discrete Math.}\ 306 (2006), no.\ 12, 1253--1256.
\MR{2245651} (2007b:05044).  \Zbl{1093.05015}.					
\annot{} When does the frame matroid $\bfF(\Omega)$ determine the biased graph $\Omega$?  Given $\Omega$ and $ \Omega_0$, without isolated vertices, loose or half edges, or balanced loops.  Assume $\Omega$ is 3-connected and contains three vertex-disjoint unbalanced circles, at most one of which is a loop.  Thm.\ 2: $\bfF(\Omega) \cong \bfF(\Omega_0)$ iff $\Omega \cong \Omega_0$.
\dated{14 Feb 2013}
\codes{\CGG:~\CMatrd:~\CStr,~\CCircles}%
\itemspace

\bsgitem{2007a}{\dag}  Projective-planar signed graphs and tangled signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 97 (2007), no.\ 5, 693--717. 
\MR{2344133} (2008j:05161).  \Zbl{1123.05046}.					
\annot{} Thm.:  The signed graphs with no two vertex-disjoint negative circles are those with a balancing vertex, or obtained from a projective-planar signed graph (\cf\ \xref{Zaslavsky}{1993a}{Thomas Zaslavsky}) or from $[-K_5]$ by $t$-summation with balanced signed graphs for $t=1,2,3$.  (Previously announced in less general form by Lov\'asz (see \xref{Seymour}{1995a}{P.D.\ Seymour}) but the proof was incorrect.)  
[\emph{Major} \Problem.  Characterize the biased graphs having no two vertex-disjoint unbalanced circles.  \xref{Lov\'asz}{1965a}{Laszlo Lovasz}, \qv, solved the contrabalanced case.]
\codes{\CSG:~\CCircles:~\CTop,~\CStr}%
\itemspace

\bsgitem{2010a}{}  Integer functions on the edges and cycle space of a graph.  \emph{Graphs Combin.}\ 20 (2010), no.\ 2, 293--299.
\MR{2606501} (2011b:05092).  \Zbl{1230.05142}.					
\annot{} Integral gains $\phi: E \to \bbZ$ induce a cycle-space homomorphism $\hat\phi: Z_1(\Gamma) \to \bbZ$.  Let $f: Z_1 (\Gamma) \to \bbZ$.  
Thm.\ 3:  $f(W) \leq k\#W$ for every walk $W$ iff $f = \hat\phi$ for some $\phi$ satisfying $\max |\phi(e)| \leq k$.  
Thm.\ 2:  For odd $k$, if also $f(W) \equiv \#W \mod 2$, there is $\phi$ which assumes only odd values; and conversely.
\dated{5 Sept 2010} 
\codes{\CGG}%
\itemspace

\bsgitem{20xxa}{}  Connectivity in signed-graphic matroids.  Submitted.  
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\author{Daniel C.\ Slilaty \& Hongxun Qin}
\authorspace
\bsgitem{2007a}{}  Decompositions of signed-graphic matroids.  \emph{Discrete Math.}\ 307 (2007), no.\ 17-18, 2187--2199.
\MR{2340600} (2008f:05032).  \Zbl{1121.05055}.  
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{2008a}{}  The signed-graphic representations of wheels and whirls.  \emph{Discrete Math.}\ 308 (2008), no.\ 10, 1816--1825. 
\MR{2394450} (2009c:05043).  \Zbl{1173.05311}.					
\annot{} All frame matroids (of biased graphs) that are wheels and whirls, characterized topologically by embeddings in the projective plane (wheels) and the cylinder (whirls).
\codes{\CGG:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{2008b}{}  Connectivity in frame matroids.  \emph{Discrete Math.}\ 308 (2008), no.\ 10, 1994--2001.  
\MR{2394467} (2009e:05139).  \Zbl{1170.05323}.					
\annot{} Graphical biconnectivity of $\Omega$ vs.\ matroid connectivity of $\bfF(\Omega)$, generalizing concepts developed by \xref{Wagner}{1985a}{Donald K.\ Wagner} for the bicircular matroid.
\par
\codes{\CGG:~\CMatrd:~\CStr}%
\itemspace

\author{Daniel C.\ Slilaty \& Thomas Zaslavsky}
\authorspace
\bsgitem{2015a}{}  Characterization of line-consistent signed graphs.  \emph{Discuss.\ Math.\ Graph Theory} 35 (2015), 589--594.  
\MR{3368992}.  \Zbl{1317.05081}.  \arXiv{1404.1651}.				
\annot{} A constructive proof of \xref{Acharya, Acharya, and Sinha's}{2009a}{Belmannu Devadas Acharya, Mukti Acharya, \& Deepa Sinha} criterion for consistency of $\Lambda(\Sigma)$. 
\dated{14 Oct 2009}
\codes{\CSG,~\CVS:~\CLG:~\CBal}%
\itemspace

\bsgitem{20xxa}{}  Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids. Submitted.  
\annot{} Cobias means choosing a linear class of bonds in $\Gamma$.  Axioimatics of the titular matroids.
\dated{16 Jun 2023} 
\codes{\CGG(\CDual):~\CMatrd:~\CStr}%
\itemspace

\author{Daniel Slilaty \& Xiangqian Zhou}
\authorspace
\bsgitem{2013a}{}  Some minor-closed classes of signed graphs.  \emph{Discrete Math.}\ 313 (2013), 313--325.
\MR{3004465}.  \Zbl{1259.05157}.  
\codes{\CSG:~\CTop}%
\itemspace

\author{N.J.A.\ Sloane} 						%SLOANE
\authorspace
\authref   See \hyperlink{P.C.\ Fishburn}{P.C.~Fishburn}, \hyperlink{R.L.\ Graham}{R.L.~Graham}, and \hyperlink{C.L.\ Mallows}{C.L.~Mallows}.
\itemspace

\author{Kaleigh Smith}						%SMITH K
\authorspace
\authref  See \hyperlink{Bruce Reed, Kaleigh Smith, \& Adrian Vetta}{B.~Reed}.
\itemspace

\author{Alex Smola}							%SMOLA
\authorspace
\authref  See \hyperlink{Shuang Hong Yang}{S.H.~Yang}.
\itemspace

\author{Chris Smyth}							%SMYTH
\authorspace
\authref   See \hyperlink{James McKee}{J.~McKee}.
\itemspace

\author{J.\ Laurie Snell} 						%SNELL
\authorspace
\authref   See \hyperlink{Joseph Berger}{J.~Berger} and \hyperlink{John G.\ Kemeny}{J.G.~Kemeny}.
\itemspace

\author{El Houssine Snoussi}						%SNOUSSI
\authorspace
\authref  See also \hyperlink{Jean-Paul Comet}{J.-P.~Comet} and \hyperlink{Denis Thieffry}{D.~Thieffry}.
\authrefspace

\bsgitem{1998a}{}  Necessary conditions for multistationarity and stable periodicity.  \emph{J.\ Biol.\ Systems} 6 (1998), no.\ 1, 3--9.
\Zbl{0982.92001}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{El Houssine Snoussi \& Rene Thomas}
\authorspace
\bsgitem{1993a}{}  Logical identification of all steady states:  The concept of feedback loop characteristic states.  \emph{Bull.\ Math.\ Biol.}\ 55 (1993), no.\ 5, 973--991.
\Zbl{0784.92002}.  
\codes{\CDyn:~\CSD}%
\itemspace

\author{Lynea Snyder}						%SNYDER
\authorspace
\authref  See \hyperlink{Yen Duong}{Y.~Duong}.
\itemspace

\author{Moo Young Sohn}						%SOHN
\authorspace
\authref  See \hyperlink{Jin Ho Kwak}{J.H.~Kwak}.
\itemspace

\author{Alan D.\ Sokal}						%SOKAL
\authorspace
\authref  See also \hyperlink{Alexander D.\ Scott}{A.D.~Scott}.
\authrefspace

\bsgitem{2005a}{}   The multivariate Tutte polynomial (alias Potts model) for graphs and matroids.  
In:  Bridget S.\ Webb, ed., \emph{Surveys in Combinatorics 2005}, pp.\ 173--226.
Cambridge Univ.\ Press, Cambridge, Eng., 2005.
\MR{2187739} (2006k:05052).  \Zbl{1110.05020}.					
\annot{} The parametrized dichromatic polynomial with parameters $d_e=1$, called the ``multivariate Tutte polynomial''.  Partly expository, partly new.  [See \xref{Zaslavsky}{1992b}{Thomas Zaslavsky}.]
\codes{\CSGw:~\CGen:~\CInvar}%
\itemspace

\author{James P.\ Solazzo}						%SOLAZZO
\authorspace
\authref  See \hyperlink{David M.\ Duncan}{D.M.~Duncan} and \hyperlink{Thomas R.\ Hoffman}{T.R.~Hoffman}.
\itemspace

\authorx{Patrick Sol\'e} 						%SOLE
\authorx{Patrick Sole}
\author{Patrick Sol\'e \& Thomas Zaslavsky}
\authorx{Patrick Sole \& Thomas Zaslavsky}
\authorspace
\bsgitem{1994a}{}  A coding approach to signed graphs.  \emph{SIAM J.\ Discrete Math.}\ 7 (1994), 544--553.  
\MR{1299082} (95k:94041).  \Zbl{811.05034}.					
\annot{} Among other things, improves some results in \xref{Akiyama, Avis, Chv\'atal, and Era}{1981a}{J.\ Akiyama, D.\ Avis, V.\ Chvatal, \& H.\ Era}.  Thm.\ 1: For a loopless graph, $D(\Gamma) \geq \frac 12 m - \sqrt{\frac 12 \ln 2}\sqrt{m(n-c(\Gamma))}$.  Thm.\ 2:  For a simple, bipartite graph, $D(\Gamma) \leq \frac 12 (m - \sqrt m)$.  
\Conjecture.  The best general asymptotic lower bound is $D(\Gamma) \geq \frac 12 m - c_1 \sqrt{mn} + o(\sqrt{mn})$ where $c_1$ is some constant between $\sqrt{\frac 12 \ln 2}$ and $\frac 12\pi$.  \Question.  What is $c_1$ for, e.g., $k$-connected graphs?  
Thm.\ 4 gives girth-based upper bounds on $D(\Gamma)$.  
\S5, ``Embedded graphs'', has bounds for several examples obtained by surface duality.  
All proofs are via covering radius of the cutset code of $\Gamma$.
\codes{\CSG:~\CFr,~\CTop}%
\annotctd{} Extends to $r=5$ the exact values of $D(K_{r,s})$ for $r \leq 4$ in \xref{Brown and Spencer}{1971a}{Thomas A.\ Brown \& Joel H.\ Spencer}.
	
[But $r=5$ has errors.  Extended correctly to all $r$ by \xref{Bowlin}{2009a}{Garry S.\ Bowlin}, \xrefd{2012a}{Garry S.\ Bowlin}.]
\dated{rev.\ 14 Feb 2011}
\codes{\CSG:~\CFr}%
\itemspace

\author{Sylvain Soliman}						%SOLIMAN
\authorspace
\authref  See \hyperlink{Francois Fages}{F.~Fages} and \hyperlink{K.\ Sriram}{K.~Sriram}.
\itemspace

\author{Louis Solomon} 						%SOLOMON
\authorspace
\authref   See \hyperlink{Peter Orlik}{P.~Orlik}.
\itemspace

\author{N.D.\ Soner}							%SONER
\authorspace
\authref  See \hyperlink{R.\ Rangarajan}{R.~Rangarajan}.
\itemspace

\authorx{Dongjin Song}						%SONG DJ
\author{Dongjin Song \& David A.\ Meyer}
\authorspace
\bsgitem{2014a}{}  A model of consistent node types in signed directed social networks.  In:  \emph{2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining} (ASONAM, Beijing, 2014), pp.\ 72--80.  IEEE, 2014.
\arXiv{1408.6822}.  
\codes{\CSD:~\CPsS}%
\itemspace

\bsgitem{2015a}{}  Recommending positive links in signed social networks by optimizing a generalized AUC.  In:  \emph{Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence} (AAAI15, Austin, Tex., U.S.A.), pp.\ 290--296.  AAAI Press, Palo Alto, Calif., 2015.  URL \url{http://www.aaai.org/ocs/index.php/AAAI/AAAI15/paper/view/9436}
\codes{\CSG:~\CAlgor}
\itemspace

\bsgitem{2015b}{}  Link sign prediction and ranking in signed directed social networks.  \emph{Social Network Anal.\ Mining} 5 (2015), art.\ 52, 24 pp.
\codes{\CSG:~\CPred:~\CPsS}
\itemspace

\author{Huimin Song}						%SONG HM
\authorspace
\authref  See \hyperlink{Xingqin Qi}{X.Q.~Qi}.
\itemspace

\author{Joungmin Song}						%SONG JM
\authorspace
\bsgitem{2017a}{}  On certain hyperplane arrangements and colored graphs.  \emph{Bull.\ Korean Math.\ Soc.}\ 54 (2017), no.\ 2, 375--382.
\MR{3632442}.  \Zbl{1373.32022}.  \arXiv{1606.07874}.				
\annot{} Regions of the [Linial-like threshold] hyperplane arrangement $\cJ_n := \{x_i+x_j=1 \text{ and } x_i=0,1\}$ are counted via graph theory related to $-K_n$.  [The hyperplanes are translates of the hyperplanes in $\cH[-K_n\full]$.  This calls for generalization via signed graphs.]
\dated{14 Apr 2017} 
\par
\codes{\Csg:~\Cpar:~\CGeom:~\CInvar}%
\itemspace

\bsgitem{2017b}{}  Enumeration of graphs and the characteristic polynomial of the hyperplane arrangements $\cJ_n$.  \emph{J.\ Korean Math.\ Soc.}\ 54 (2017), no.\ 5, 1595--1604.
\MR{3691940}.  \Zbl{06853526}.  \arXiv{1701.07313}.				
\annot{} The characteristic polynomial; \cf\ \xrefd{2017a}{Joungmin Song}.
\dated{14 Apr 2017} 
\par
\codes{\Csg:~\Cpar:~\CGeom:~\CInvar}%
\itemspace

\bsgitem{2017c}{}  Characteristic polynomial of certain hyperplane arrangements through graph theory.  Manuscript, 2017.
\arXiv{1701.07330}.								
\codes{\Csg:~\Cpar:~\CGeom:~\CInvar}%
\itemspace

\bsgitem{2018a}{}  Characteristic polynomial of the hyperplane arrangements $\cJ_n$ via finite field method.  \emph{Commun.\ Korean Math.\ Soc.}\ 33 (2018), no.\ 3, 759--765.
\MR{3846025}.  \Zbl{1401.32023}.  
\annot{} \Cf\ \xrefd{2017a}{Joungmin Song}, \xrefd{2017b}{Joungmin Song}.
\dated{1 Aug 2018} 
\codes{\Csg:~\Cpar:~\CGeom:~\CInvar}%
\itemspace

\author{Sang-Oak Song}						%SONG SO
\authorspace
\authref  See \hyperlink{Gibaek Lee}{G.~Lee}.
\itemspace

\author{Song Yi-Zhe}							%SONG YZ
\authorspace
\authref  See \hyperlink{Bai Xiao}{B.~Xiao}.
\itemspace

\author{Zi-Xia Song}							%SONG ZX
\authorspace
\authref  See \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi}.
\itemspace

\author{Eduardo D.\ Sontag}						%SONTAG
\authorspace
\authref  See also \hyperlink{David Angeli}{D.~Angeli}, \hyperlink{Boris N.\ Kholodenko}{B.N.~Kholodenko}, \hyperlink{Gheorghe Craciun}{G.~Craciun}, \hyperlink{Bhaskar DasGupta}{B.~DasGupta}, and \hyperlink{German Andres Enciso}{G.A.~Enciso}.
\authrefspace

\bsgitem{2004a}{}  Some new directions in control theory inspired by systems biology.  \emph{Systems Biol.}\ 1 (2004), no.\ 1, 9--18.
\annot{} P.\ 13 describes how a signed digraph arises from differential equations, and that it is ``monotone'' [= isotone] iff it has no negative cycles.
\dated{25 Jan 2015} 
\codes{\CBiol:~\CDyn:~\CSD:~\CExp}%
\itemspace

\bsgitem{2005a}{}  Molecular systems biology and control.  \emph{European J.\ Control} 11 (2005), no.\ 4-5, 396--435.
\MR{2201569} (no rev).  
\annot{} \S 5.1, ``Consistent graphs and monotone systems''.  
\S 5.4, ``Almost-mono\-tonicity''.  
	
Dictionary:  ``parity'' = sign, ``consistent'' = balanced, ``consistency deficit'' = frustration index, ``almost-consistency'' = small $l(\Sigma)/\#E$, ``monotone'' = isotone (monotone weakly increasing).
\dated{1 Jan 2012} 
\codes{\CSD(\Csg):~\CBal,~\CFr:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2007a}{}  Monotone and near-monotone systems.  In: I.\ Queinnec \etal, eds.,  \emph{Biology and Control Theory: Current Challenges}, pp.\ 79--122.  
Lect.\ Notes in Control and Inform.\ Sci., Vol.\ 357.  Springer-Verlag, Berlin, 2007. 
\MR{2352229} (2008k:92021).  \Cf\ \arXiv{q-bio/0612032}, \arXiv{q-bio/0612033}.  
\annot{} Conference version of \xrefd{2007b}{Eduardo D.\ Sontag}; almost the same.  
\dated{23 Jan 2015} 
\par
\codes{\CSD,~\CSG:~\CBal,~\CFr,~\CDyn,~\CBiol:~\CExp,~\CRef}%
\itemspace

\bsgitem{2007b}{}  Monotone and near-monotone biochemical networks.  \emph{Systems Synthetic Biol.}\ 1 (2007), 59--87.
\annot{} 
Dictionary:  ``graph'' = signed signed digraph; ``spin assignment'' = state = function $\zeta: V \to \signnos$; edge ``consistent'' with $\zeta$ = satisfied edge ($\sigma(e)=\zeta_i\zeta_j$); ``consistent spin assignment'' $\Sigma$ = potential function $\zeta$ (edge directions are ignored); ``monotone'' = balanced (undirected); ``consistency deficit'' = frustration index (undirected).
\par
\codes{\CSD,~\CSG:~\CBal,~\CFr,~\CDyn,~\CBiol:~\CExp,~\CRef}%
\itemspace

\author{Eduardo Sontag, Alan Veliz-Cuba, Reinhard Laubenbacher, \& Abdul Salam Jarrah}
\authorspace
\bsgitem{2008a}{\dag}  The effect of negative feedback loops on the dynamics of Boolean networks.  \emph{Biophys.\ J.}\ 95 (2008), 518--526 + suppl.\ 9 pp.
\arXiv{0707.3468}.  
\annot{} The directed frustration index $l(\vec\Gamma,\sigma)$ (called the ``PF-dist\-ance'') of a signed digraph is the smallest number of edges whose signs should be changed to eliminate all negative cycles.  This index is a measure of the number of independent negative cycles.  Then $l_{\max}(\vec\Gamma) := \max_\sigma l(\vec\Gamma,\sigma).$  P.\ 522: An algorithm ``Distance to PF'' for $l(\vec\Gamma,\sigma)$ with strongly connected $\vec\Gamma$ (sufficient, since $l$ is additive on strong components and a negative loop adds 1).  
Dictionary:  ``directed cycle'' = cycle; ``cycle'' = circle; ``odd parity'' = negative sign; ``positive feedback'' (PF) = cycle balance (no negative cycles); ``$|\vec\Gamma|$'' = $l_{\max}(\vec\Gamma)$.
\annotctd{} A unate Boolean function $f: \bbF_2^n \to \bbF_2^n$ has a (signed) interaction digraph $\cD(f)$.  A computational experiment tests the connection between $l(\cD(f))$ and the number and length of attractors (limit cycles) of $f$ in $\bbF_2^n$, which appear to be direct and inverse, respectively.
Dictionary: ``unate'' = each component function is monotone (isotone or antitone); ``monotone'' = isotone (monotone weakly increasing); ``signed dependency graph'' = (signed) interaction digraph,  ``distance-to-positive-feedback'', ``PF-distance'' = frustration index.  
\dated{16 Jan 2015}
\newline
\codes{\CSD,~\CSG:~\CFr,~\CAlgor}%
\itemspace

\author{Soorya P}							%SOORYA
\authorspace
\authref  See also \hyperlink{Shahul Hameed K}{S.~Hameed}, \hyperlink{Albin Mathew}{A.~Mathew}, and \hyperlink{Shijin T V}{T.V.~Shijin}.
\itemspace

\bsgitem{2022a}{}  \emph{A Study on Choosability in Graphs Signed Graphs and Related Parameters}.  Doctoral disserttion, Central University of Kerala, 2022, 
\codes{\CSG:~\CCol(\CGen)}%
\itemspace

\author{\'Eric Sopena}						%SOPENA
\authorx{Eric Sopena}
\authorspace
\authref  See  \hyperlink{Julien Bensmail}{J.~Bensmail}, \hyperlink{Clement Charpentier}{C.~Charpentier}, \hyperlink{Amanda Montejano}{A.~Montejano}, and \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\author{Nicola Soranzo}						%SORANZO
\authorspace
\authref  See also \hyperlink{Giovanni Iacono}{G.~Iacono}.
\authrefspace

\author{Nicola Soranzo, Fahimeh Ramezani, Giovanni Iacono, \& Claudio Altafini}
\authorspace
\bsgitem{2012a}{}  Decompositions of large-scale biological systems based on dynamical properties.  \emph{Bioinformatics} 28 (2012), no.\ 1, 76--83.
\codes{\CSD}%
\itemspace

\author{Mauricio Soto}						%SOTO
\authorspace
\authref  See \hyperlink{Eduardo G.\ Pardo}{E.G.~Pardo}.
\itemspace

\author{C.M.\ Soukoulis}						%SOUKOULIS
\authorspace
\authref  See \hyperlink{Daniel Blankschtein}{D.~Blankschtein}.
\itemspace

\author{Christophe Soul\'e}						%SOULE
\authorx{Christophe Soule}
\authorspace
\authref  See also \hyperlink{Marcelle Kaufman}{M.~Kaufman}.
\authrefspace

\bsgitem{2003a}{}  Graphic requirements for multistationarity.  \emph{ComPlexUs} 1 (2003), 123--133.
\arXiv{q-bio/0403033}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{2006a}{}  Mathematical approaches to differentiation and gene regulation.  \emph{C.R.\ Biologies} 329 (2006), 13--20.
\arXiv{q-bio/0510027}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Mona Souri}							%SOURI
\authorspace
\authref  See also \hyperlink{Saieed Akbari}{S.~Akbari}.
\authrefspace

\author{M.\ Souri, F.\ Heydari, \& M.\ Maghasedi}
\authorspace
\bsgitem{2020a}{}  Maximizing the largest eigenvalues of signed unicyclic graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ 12 (2020), no.\ 2, art.\ 2050016, 8 pp.
\MR{4099663}.  \Zbl{1456.05071}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  On the eigenvalues of some signed graphs.  \emph{Iranian J.\ Sci.\ Technol.\ Trans.\ A Sci.}\ 45 (2021), no.\ 2, 635--639.
\MR{4230645}.  
\annot{} $\Spec A(K_n(-K_{n_1,\ldots,n_k}))$.  If three $n_i$ are equal, $\Spec$ determines $K_{n_1,\ldots,n_k}$.
\dated{6 Sept 2022} 
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{N.\ Sourlas}							%SOURLAS
\authorspace
\authref  See \hyperlink{Sergio Caracciolo}{S.~Caracciolo}.
\itemspace

\authorx{B.W.\ Southern}						%SOUTHERN
\author{B.W.\ Southern, S.T.\ Chui, \& G.\ Forgacs}
\authorspace
\bsgitem{1980a}{}  Non-universality for two-dimensional frustrated lattices?  \emph{J.\ Phys.\ C} 13 (1980), L827--L830.
\annot{} Physics of signed square lattice graph, fully frustrated (all positive except for all-negative alternating vertical lines).  Reduced to the ``8-vertex'' physics model by taking alternating sites (vertices) and observing they are 4-valent and all or half positive.  
[\Cf\ \xref{Garel and J.M.\ Maillard}{1983a}{T.\ Garel \& J.M.\ Maillard}.]
\dated{16 Jun 2012} 
\codes{\CPhys:~\Csg}%
\itemspace

\author{U\'everton S.\ Souza}						%SOUZA
\authorx{Ueverton S.\ Souza}
\authorspace
\authref  See \hyperlink{Carlos V.G.C.\ Lima}{C.V.G.C.~Lima}.
\itemspace

\author{Cid C.\ de Souza}						%SOUZA	%DE SOUZA
\authorspace
\authref  See \hyperlink{Rosa M.V.\ Figueiredo}{R.M.V.~Figueiredo}.
\itemspace

\author{[Natasha D'Souza]} 						%SOUZA
\authorspace
\authref  See \hyperlink{Natasha D'Souza}{N.~D'Souza} (under `D').
\itemspace

\author{Tadeusz Soza\'nski} 						%SOZANSKI
\authorx{Tadeusz Sozanski}
\authorspace
\bsgitem{1976a}{}  Processus d'\'equilibration et sous-graphes \'equilibr\'es d'un graphe sign\'e complet.  \emph{Math.\ Sci.\ Humaines}, No.\ 55 (1976), 25--36, 83.  
\MR{0543817} (58 \#27613).  
\annot{} $\Sigma$ denotes a signed $K_n$.  The ``level of balance'' (``indice du niveau d'\'equilibre'') $\rho(\Sigma) :=$ maximum order of a balanced subgraph.  [Complement of the vertex deletion number $l_0(\Sigma)$.]  Define distance $d(\Sigma_1, \Sigma_2) := \#(E_{1+} \bigtriangleup E_{2+})$.  Say $\Sigma$ is $p$-clusterable if $\Sigma^+$ consists of $p$ disjoint cliques [its ``clusters''].  
Thm.\ 1 evaluates the frustration index of a $p$-clusterable $\Sigma$.  
Thm.\ 2 bounds $l(\Sigma)$ in terms of $n$ and $\rho(\Sigma)$.  A negation set $U$ for $\Sigma$ ``conserves'' a balanced induced subgraph if they are edge-disjoint; it is ``(strongly) conservative'' if it conserves some (resp., every) maximum-order balanced induced subgraph.  Thm.\ 3:  Every minimum negation set conserves every balanced induced subgraph of order $> \frac{2}{3}n$.  Thm.\ 4:  A minimum negation set can be ordered so that, successively negating its edges one by one, $\rho$ never decreases.
\codes{\CSG:~\CKG:~\CFr,~\CClu}%
\itemspace

\bsgitem{1980a}{}  Enumeration of weak isomorphism classes of signed graphs.  \emph{J.\ Graph Theory} 4 (1980), 127--144.  
\MR{0570348} (81g:05070).  \Zbl{434.05059}.						
\annot{} ``Weak isomorphism'' = switching isomorphism.  Principal results:  The number of  switching nonisomorphic signed $K_n$'s.  (\Cf\ \xref{Mallows and Sloane}{1975a}{C.L.\ Mallows \& N.J.A.\ Sloane}.)  The  number that are switching isomorphic to their negations.  The number of nonisomorphic (not switching nonisomorphic!) balanced signings of a given graph.  
\S2.3, ``Space of signed graphs over a fixed graph'', implicitly contains the theorem that two signed graphs are switching isomorphic iff there is an isomorphism of underlying graphs that preserves circle signs [\cf\ \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Prop.\ 3.2; \xrefd{1981b}{Thomas Zaslavsky}, Thm.\ 7]. 
\dated{rev.\ 22 Oct 2015}
\codes{\CSG,~\CKG:~\CSw:~\CEnum}%
\itemspace

\bsgitem{1982a}{}  \emph{Model rownowagi strukturalnej.  Teoria grafow oznakowanych i jej zastosowania w naukach spotecznych}.
[{The structural balance model.  The theory of signed graphs and its applications in the social sciences}.]  (In Polish.)  
Doctoral thesis, Jagellonian University, Krakow, 1982.  
\par
\codes{\CSG,~\CPsS:~\CBal,~\CFr,~\CClu,~\CAut,~\CAdj,~\CRef}%
\itemspace

\authorx{Quico Spaen}						%SPAEN
\author{Quico Spaen, Christopher Thraves Caro, \& Mark Velednitsky}
\authorspace
\bsgitem{2020a}{}  The dimension of valid distance drawings of signed graphs.  \emph{Discrete Comput.\ Geom.}\ 63 (2020), 158--168.
\MR{4045743}.  \Zbl{1429.05085}.  
\annot{} Valid distance drawing:  $V \hookrightarrow \bbR^k$ such that $(\forall v)$ all positive neighbors of $v$ are closer than all negative neighbors.  Problem:  Find $L(n) := \min k$ with a valid drawing $\forall\ \Sigma$ such that $\#V=n$.  Thm.:  $\lfloor \log_5(n-3) \rfloor + 1 \leq L(n) \leq n-2$.
\dated{28 Dec 2019} 
\codes{\CSG:~\CGeom}%
\itemspace

\author{Edward Spence}						%SPENCE
\authorspace
\authref  See \hyperlink{Willem H.\ Haemers}{W.H.~Haemers}.
\itemspace

\author{Joel Spencer} 						%SPENCER
\authorspace
\authref  See \hyperlink{Thomas A.\ Brown}{T.A.~Brown}.
\itemspace

\author{Stefano Spessato}						%SPESSATO
\authorspace
\authref  See \hyperlink{Matteo Cavaleri}{M.~Cavaleri}.
\itemspace

\author{Daniel A.\ Spielman}						%SPIELMAN
\authorspace
\authref  See \hyperlink{Afonso S.\ Bandeira}{A.S.~Bandeira} and \hyperlink{Adam W.\ Marcus}{A.W.~Marcus}.
\itemspace

\author{Joel Spencer with Laura Florescu}
\authorspace
\bsgitem{2014a}{}  \emph{Asymptopia}.  Student Math.\ Library, Vol.\ 71.  American Mathematical Society, Providence, R.I., 2014.
\MR{3185739}.  \Zbl{1331.00002}.  
\annot{} \S6,8, ``An exact formula for unicyclic graphs'':  The number of bases of $\bfL(K_n,\eset)$, the bicircular lift matroid of $K_n$.  \S6.4, ``Counting unicyclic graphs in Asymptopia'':   Asymptotics.  [N.B. $\bfL_0(K_n,\eset)$ has $n^{n-2}$ additional bases.]
\dated{3 Oct 2014} 
\codes{\Cbic:~\Cmatrd:~\CInvar}%
\itemspace

\author{Sam Spiro}							%SPIRO
\authorspace
\bsgitem{2022a}{}  The Wiener index of signed graphs.  \emph{Appl.\ Math.\ Comput.}\ 416 (2022), art.\ 126755, 10 pp.
\MR{4337738}.  
\arXiv{2106.11869}.  
\annot{} The graphs are sign-weighted, i.e., edges weighted by $\pm1\in\bbZ^+$ (the additive group).  Weiner index $W_\sigma(G) := \sum_{u,v} d_\sigma(u,v)$ where $d_\sigma(u,v)$ (``signed distance'') $:= \min_{P_{uv}} |\sum_{e\in P_{uv}} \sigma(e)|$.  Thm.\ 1.3(b):  If $W_\sigma(G)=0$, then $G$ contains an odd cycle.  
[\Problem.  This hints at generalization to a sign-weighted signed graph $(G,\sigma,\sigma_1)$ where Thm.\ 1.3(b) becomes $(G,\sigma_1)$ is not antibalanced.]
\dated{25 Nov 2021} 
\codes{\CSGw:~\CInvar}%
\itemspace

\author{K.K.\ Srimitra}						%SRIMITRA
\authorspace
\authref  See \hyperlink{Shaik Sajana}{S.~Sajana}.
\itemspace

\author{Aravind Srinivasan}						%SRINIVASAN A
\authorspace
\bsgitem{2011a}{}  Local balancing influences global structure in social networks.  \emph{Proc.\ Nat.\ Acad.\ Sci.\ (U.S.A.)} 108 (2011), no.\ 5, 1751--1752. 
\annot{} Summary of and commentary on \xref{Marvel, Kleinberg, Kleinberg, and Strogatz}{2011a}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz}.
\dated{7 Feb 2011} 
\codes{\CSG:~\CKG:~\CFr,~\CDyn}%
\itemspace

\author{Murali K.\ Srinivasan}						%SRINIVASAN MK
\authorspace
\authref  See also \hyperlink{Amitava Bhattacharya}{A.~Bhattacharya}.
\authrefspace

\bsgitem{1998a}{}  Boolean packings in Dowling geometries.  \emph{European J.\ Combin.}\ 19 (1998), 727--731.
\MR{1642742} (99i:05059).  \Zbl{990.10387}.					
\annot{} Decomposes the Dowling lattice $Q_n(\fG)$ into Boolean algebras, indexed in part by integer compositions, that are cover-preserving and centered above the middle rank.
\codes{\CGG:~\CMatrd}%
\itemspace

\author{R.\ Srinivasan}						%SRINIVASAN R
\authorspace
\authref  See \hyperlink{Vijay Kodiyalam}{V.~Kodiyalam}.
\itemspace

\author{M.A.\ Sriraj}							%SRIRAJ
\authorspace
\authref  See \hyperlink{Chandrashekar Adiga}{C.~Adiga} and \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\authorx{K.\ Sriram}							%SRIRAM K
\author{K.\ Sriram, Sylvain Soliman, \& Fran\c{c}ois Fages}
\authorx{K.\ Sriram, Sylvain Soliman, \& Francois Fages}
\authorspace
\bsgitem{2009a}{}  Dynamics of the interlocked positive feedback loops explaining the robust epigenetic switching in \emph{Candida albicans}.  \emph{J.\ Theor.\ Biol.}\ 258 (2009), 71--88.
\annot{} The effect of a pair of positive cycles sharing a single vertex, with a biological example.  \Cf\ \xref{Kim, Yoon, and Cho}{2008a}{Jeong-Rae Kim, Yeoin Yoon, \& Kwang-Hyun Cho}.
\dated{16 Jan 2015} 
\codes{\CSD:~\CBal:~\CDyn,~\CBiol}%
\itemspace

\authorx{Ajitesh Srivastava}						%SRIVASTAVA A
\author{Ajitesh Srivastava, Charalampos Chelmis, \& Viktor K.\ Prasanna}
\authorspace
\bsgitem{2016a}{}  Computing competing cascades on signed networks.  \emph{Social Network Analysis Mining} 6 (2016), art.\ 82, 13 pp.
\codes{\CSG:~\CDyn}%
\itemspace

\author{Nikhil Srivastava}						%SRIVASTAVA N
\authorspace
\authref  See \hyperlink{Adam W.\ Marcus}{A.W.~Marcus}.
\itemspace

\author{Ladislav Stacho}						%STACHO
\authorspace
\authref  See \hyperlink{Daniel Kral'}{D.~Kr\'al'}.
\itemspace

\author{Derek Stafford}						%STAFFORD
\authorspace
\authref  See \hyperlink{Derek Feng}{D.~Feng}.
\itemspace

\author{Saul Stahl} 							%STAHL
\authorspace
\bsgitem{1978a}{}  Generalized embedding schemes.  \emph{J.\ Graph Theory} 2 (1978), 41--52.  
\MR{0485488} (58 \#53180.  \Zbl{396.05013}.					
\annot{} A generalized embedding scheme for a graph is identical to a rotation system for a signing of the graph.  Thm.\ 2: Signed rotation systems describe all cellular embeddings of a graph.  Thm.\ 4: Embeddings are homeomorphic iff their signed rotation systems are switching equivalent.  Thm.\ 5: An embedding is orientable iff its signature is balanced.  Compare \xref{Ringel}{1977a}{Gerhard Ringel}.  Dictionary: $\lambda$ is the signature.  ``$\lambda$-trivial'' means balanced.
\codes{\Csg:~\CTop,~\CSw}%
\itemspace

\bsgitem{1978b}{}  The embeddings of a graph---a survey.  \emph{J.\ Graph Theory} 2 (1978), 275--298. 
\MR{0512799} (80a:05085).  \Zbl{406.05027}.
\codes{\Csg:~\CTop}%
\itemspace

\bsgitem{2005a}{}  \emph{Introduction to Topology and Geometry}.  Wiley-Interscience, Hoboken, N.J., 2005.
\MR{2102439} (2005g:57001).  \Zbl{1063.57001}.
\annot{} \S4.4, `` Voltage graphs and their coverings'':  Rotation system for surface embedding, embedded covering of embedded voltage graph, branched covering graph and embedding.
\dated{25 Apr 2014}
\par
\codes{\CGG:~\CTop:~\CExp,~\CExr}%
\itemspace

\author{Saul Stahl \& Catherine Stenson}
\authorspace
\bsgitem{2013a}{}  \emph{Introduction to Topology and Geometry}, 2nd ed.  Wiley-Interscience, Hoboken, N.J., 2013.
\MR{3235588}.  \Zbl{1286.57001}.
\annot{} See \xref{Stahl}{2005a}{Saul Stahl}.
\codes{\CGG:~\CTop:~\CExp,~\CExr}%
\itemspace

\author{David P.\ Stanford}						%STANFORD
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{Zoran Stani\'c}						%STANIC
\authorx{Zoran Stanic}
\authorspace
\authref  See also \hyperlink{Milica Andelic}{M.~An{\dj}eli\'c}, \hyperlink{Francesco Belardo}{F.~Belardo}, \hyperlink{Maurizio Brunetti}{M.~Brunetti}, \hyperlink{Gary Greaves}{G.~Greaves}, \hyperlink{Raffaella Mulas}{R.~Mulas}, \hyperlink{Leila Parsaei-Majd}{L.~Parsaei-Majd}, \hyperlink{Farzaneh Ramezani}{F.~Ramezani}, \hyperlink{Peter Rowlinson}{P.~Rowlinson}, and \hyperlink{Slobodan K.\ Simic}{S.K.~Simi\'c}.
\authrefspace

\bsgitem{2007a}{}  \emph{Neke rekonstrukcije u spektralnoj teoriji grafova i grafovi sa integralnim Q-spektrom}. [\emph{Some Reconstructions in Spectral Graph Theory and Graphs with Integral $Q$-Spectrum}.]  (In Serbian.)  Doctoral Thesis, Faculty of Math., University of Belgrade, 2007.
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2007b}{}  There are exactly 172 connected $Q$-integral graphs up to 10 vertices.  \emph{Novi Sad J.\ Math.}\ 37 (2) (2007), 193--205.
\MR{2401613} (no rev).  \Zbl{1164.05046}.					
\annot{} I.e., $\Spec \Lap(-\Gamma)$ is integral.
\dated{19 Feb 2021} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2009a}{}  On determination of caterpillars with four terminal vertices by their Laplacian spectrum.  \emph{Linear Algebra Appl.}\ 431 (2009), 2035--2048.
\MR{2567810} (2010j:05253).  \Zbl{226.05165}.  
\annot{} \S5:  $\Spec \Lap(-\Gamma)$ is mentioned.
\dated{16 Jan 2012} 
\par
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\bsgitem{2018a}{}  Perturbations in a signed graph and its index.  \emph{Discuss.\ Math.\ Graph Theory} 38 (2018), 841--852.
\MR{3811969}.  \Zbl{1391.05178}.  
\annot{} Perturbation by adding an edge or vertex or negating an edge.  Effect on $\lambda_{\max}(A(\Sigma))$.  Eg., a case of Cor.\ 8:  Negating $e$ in a nontrivial balanced block reduces $\lambda_1$.  \S4, ``Representatives of small switching equivalent signed graphs'':  Methodology for \xrefd{2018b}{Stanic}.  
Dictionary:  ``simple'' = unsigned (treated as all positive).
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2018b}{}  Signed graphs of small order.  URL \url{www.math.rs/~zstanic/siggr.htm}.
\annot{} Counts and lists of signed graphs for $n\leq8$ and cospectral ones for $n\leq7$.  Data for \xrefd{2018a}{Stanic}.
\dated{19 Feb 2021} 
\codes{\CSG:~\CSw,~\CAdj,~\CEig}
\itemspace

\bsgitem{2019a}{}  Some bounds for the largest eigenvalue of a signed graph.  \emph{Bull.\ math.\ Soc.\ sci.\ math.\ Roumanie} 62(100) (2019), no.\ 2, 183--189.
\MR{4007428}.  \Zbl{1463.05237}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2019b}{}  Bounding the largest eigenvalue of signed graphs.  \emph{Linear Algebra Appl.}\ 573 (2019), 80--89.
\MR{3933292}.  \Zbl{1411.05109}.  
\annot{} Upper bounds such as $\lambda_{\max} \leq \max_i \frac12 \big(\sqrt{5d_i^2+4(d_im_i-4t_i^-)} - d_i\big),$ where $d_i =$ degree, $m_i =$ average 2-degree, $t_i^- =$ \# negative triangles on $v_i$.
\dated{15 Feb 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2019c}{\dag}  Integral regular net-balanced signed graphs with vertex degree at most four.  \emph{Ars Math.\ Contemp.}\ 17 (2019), 103--114.
\MR{3998150}.  \Zbl{1433.05142}.  
\annot{} Important result, ``Lemma'' 2.1:  If $\Sigma$ is connected, then $\lambda_{\max}(A(\Sigma)) = \lambda_{\max}(A(|\Sigma|))$ iff $\Sigma$ is balanced.  [This strengthens \xref{Acharya's}{1980a}{B.\ Devadas Acharya} spectral characterization of balance in signed graphs.]
\annotctd{} Also, all net-regular $\Sigma$ that are 3-regular with integral $\Spec A(\Sigma)$, and some 4-regular ones.
\dated{19 Dec 2020, 21 Feb 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2019d}{\dag}  On strongly regular signed graphs.  \emph{Discrete Appl.\ Math.}\ 271 (2019), 184--190.
\MR{4030312}.  \Zbl{1428.05331}.  
Corrigendum.  \emph{Discrete Appl.\ Math.}\ 284 (2020), 640. 
\MR{4115513}.  \Zbl{1443.05185}.  
\annot{} Introduces SRSG.  The definition is combinatorial.  Assume simple, $r$-regular $|\Sigma|$.  Let $w_2^\pm(u,v) = $ \# positive $-$ \# negative 2-edge $uv$-walks.  Let $w_2^\pm(uv) = a,\ b,\ c$ according as $uv \in E^+,\ \in E^-$, or $\notin E$.  $\Sigma$ is ``strongly regular'', unless $E=\eset$ or $\Sigma = (K_n,+),\ (K_n,-)$.
\annotctd{} The balanced SRSG are the same as ordinary strongly regular graphs.  The disconnected ones are characterized in Thm.\ 4.2.  \xref{Zaslavsky's}{2010b}{Thomas Zaslavsky} ``very SRSG'' are a special case.
\annotctd{} The connection with matrices is not straightforward.  Thm.\ 4.1:  $\Sigma$ is strongly regular if $A(\Sigma)$ has two eigenvalues.
Thm.\ 5.3: An unbalanced, net-regular $(K_{r,s},\sigma)$ is an SRSG iff $\Sigma^+$, equivalently $\Sigma^-$, is the incidence graph of a symmetric block design.  
Thm.\ 5.4 combinatorially characterizes bipartite SRSG with 2, 3, and 4 eigenvalues.
\annotctd{} \Cf\ \xref{Koledin and Stani\'c}{2020a}{Tamara Koledin \& Zoran Stanic}.
\dated{4 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  Lower bounds for the least Laplacian eigenvalue of unbalanced blocks.  \emph{Linear Algebra Appl.}\ 584 (2020), 145--152.
\MR{4010177}.  \Zbl{1426.05058}.  
\annot{} $\lambda_{\min}(\Lap(\Sigma)) > 4/c^-n$ and $> \pi^2\frac{12n^2-\pi^2}{12n^4}h^-$, where $c^- =$ negative circumference, $h^- =$ packing number of negative Hamiltonian circles (if any exist).
\dated{4 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020b}{}  Spectra of signed graphs with two eigenvalues.  \emph{Appl.\ Math.\ Comput.}\ 364 (2020), art.\ 124627, 9 pp.
\MR{3996372}.  \Zbl{1433.05210}.  
\annot{} \Cf\ \xref{Ramezani}{20xxa}{Farzaneh Ramezani}.  Completes list of 3-, 4-regular such $\Sigma$, after \xref{Ghasemian and Fath-Tabar}{2017a}{E.\ Ghasemian \& G.H.\ Fath-Tabar}, \xref{Hou, Tang, and Wang}{2019a}{Yaoping Hou, Zikai Tang, \& Dijian Wang}.
\S4, ``Relations to line systems'': Line systems at angles $90^\circ$ and $\theta$ (\cf, e.g., \xref{Zaslavsky}{2012c}{Thomas Zaslavsky}).  Includes reduced line graphs $\bar\Lambda(\Sigma)$.  Thm.\ 4.3:  Connected $\bar\Lambda(\Sigma)$ has 2 eigenvalues iff $\Sigma \simeq +K_n,\ +K_{1,s},\ \pm\Delta$ where $\Delta$ is regular, or two other graphs.  
\S5, ``Computational results'':  The 9 connected, incomplete $\Sigma$ with $n\leq10$ that have 2 eigenvalues.  
\S6, ``Constructions'' of some regular examples with $n\leq24$; related (non)existence results.
\dated{4 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig,~\CLG}%
\itemspace

\bsgitem{2020c}{}  Controllability of certain real symmetric matrices with application to controllability of graphs.  \emph{Discrete Math.\ Lett.}\ 3 (2020), 9--13.
\MR{4131137}.  \Zbl{1463.05355}.  
\annot{} Cor\ 2.1 applies to signed graphs.
\dated{19 Jan 2020} 
\codes{\CAdj,~\CLap:~\CSG}%
\itemspace

\bsgitem{2020d}{}  Notes on exceptional signed graphs.  \emph{Ars Math.\ Contemp.}\ 18 (2020), 105--115.
\MR{4154730}.  \Zbl{1464.05172}.  
\annot{} Determines the maximal signed graphs with $\lambda_{\min}(A(\Sigma)) \geq -2$ that are not negated line graphs of signed graphs.
\dated{4 Oct 2019, 4 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig,~\CGeom}%
\itemspace

\bsgitem{2020e}{}  Net Laplacian controllability for joins of signed graphs.  \emph{Discrete Appl.\ Math.}\ 285 (2020), 197--203. 
\MR{4111701}.  \Zbl{1447.05099}.  
\annot{} Net Laplacian $\Lap^\pm(\Sigma) := \diag(d^\pm_\Sigma) - A(\Sigma)$.  Especially for particular signed graphs.  Application to control theory.
\codes{\CSG:~\CLap:~\CEig,~\CAppl}%
\itemspace

\bsgitem{2020f}{}  On the spectrum of the net Laplacian matrix of a signed graph.  \emph{Bull.\ Math.\ Soc.\ Sci.\ Math.\ Roumanie} 63(111) (2020), no.\ 2, 205--213.
\MR{4202394}.  
\annot{} \Cf\ \xrefd{2020e}{Zoran Stanic}.
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2020g}{}  Main eigenvalues of real symmetric matrices with application to signed graphs.  \emph{Czechoslovak Math.\ J.}\ 70 (2020), no.\ 4, 1091--1102.
\MR{4181798}.  
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020h}{}  A decomposition of signed graphs with two eigenvalues.  \emph{Filomat} 34 (2020), no.\ 6, 1949--1957.
\MR{4207727}.  
\annot{} Connected $\Sigma$ with 2 eigenvalues that admit vertex partition whose induced signed graphs also each have 2 eigenvalues.  Independently proves a result used in \xref{Huang}{2019a}{Hao Huang}.
\dated{29 Apr 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020i}{}  Oriented graphs whose skew spectral radius does not exceed 2.  \emph{Linear Algebra Appl.}\ 603 (2020), 359--367.
\MR{4117880}.  \Zbl{1446.05041}.  
\annot{} All such digraphs are determined, using signed graphs as in \xrefd{2022a}{Zoran Stanic}.  \Cf\ \xref{McKee and Smyth}{2007a}{}.
\dated{29 Apr 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  A note on a walk-based inequality for the index of a signed graph.  \emph{Special Matrices} 9 (2021), 19--21.
\MR{4203316}.  \Zbl{1457.05045}.  
\annot{} The adjacency and Laplacian indices $\lambda_1$ of $\Sigma$ satisfy, for a certain vertex $v$, $\lambda_1(n_{r,v}^+ + n_{r,v}^- + \lambda_1^{r}) \leq$ complicated bounds in terms of $r$-walks, where $n_{r,v}^\epsilon$ counts signed $r$-walks. 
\dated{12 Dec 2020} 
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\bsgitem{2021b}{}  Signed graphs with totally disconnected star complements.  \emph{Rev.\ Uni\'on Mat.\ Argentina} 62 (2021), no.\ 1, 95--104.
\MR{4281073}.  \Zbl{1473.05119}.  
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021c}{}  Connected non-complete signed graphs which have symmetric spectrum but are not sign-symmetric.  \emph{Examples Counterex.}\ 1 (2021), art.\ 100007, 3 pp.
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022a}{}  Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs.  \emph{Rev.\ Uni\'on Mat.\ Argentina} 63 (2022), no.\ 1, 41--50.
\MR{4451124}.  
\annot{} Sequel to \xrefd{20xxi}{Zoran Stanic}.
\codes{\CSG,~\Cgg:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022b}{}  Star complements in signed graphs with two symmetric eigenvalues.  \emph{Kuwait J.\ Sci.}\ 49 (2022), no.\ 2, 1--8.
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022c}{}  Notes on the polynomial reconstruction of signed graphs.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 45 (2022), 1301--1314.
\MR{4412323}.  \Zbl{1487.05134}.  
\annot{} Reconstruction of the characteristic polynomial of $A(\Sigma)$ from those of vertex-deleted subgraphs, resolved for several types of graph.
\dated{26 Apr 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022d}{}  Signed graphs with two eigenvalues and vertex degree five.  \emph{Ars Math.\ Contemp.}\ 22 (2022), art.\ P1.10, 13 pp.
\MR{4424011}.  
\annot{} Classifies 5-regular such graphs, based on weighing matrices as in \xref{Harada and Munemasa}{2012a}{Masaaki Harada \& Akihiro Munemasa}.  For $(\leq4)$-regular ones \cf\ e.g.\ \xrefd{2020b}{Zoran Stanic}.
\dated{14 Oct 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022e}{}  Some properties of the eigenvalues of the net Laplacian matrix of a signed graph.  \emph{Discuss.\ Math.\ Graph Theory} 42 (2022), no.\ 3, 893--903.
\MR{4431630}.  
\annot{} \Cf\ \xrefd{2020e}{Zoran Stanic}.
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2022f}{}  Some relations between the largest eigenvalue and the frustration index of a signed graph.  \emph{Amer.\ J.\ Combin.}\ 1 (2022), 65--72.
\codes{\CSG:~\CFr,~\CAdj:~\CEig}%
\itemspace

\bsgitem{20xxb}{}   On cospectral oriented graphs and cospectral signed graphs.  \emph{Linear Multilinear Algebra} (to appear).
\annot{} Infinite families of cospectral signed regular graphs with different underlying graphs; cospectral digraphs are derived via \xrefd{20xxi}{Zoran Stanic}.  Techniques include the signed covering graph and line graphs of signed expansions.
\annotctd{} Surprising Thm.\ 3.6:  For line graph of signed expansion of $r$-regular $\Gamma$, $\Spec \Lambda(\pm\Gamma) = \{(2)^{n(r-1)}, (-2[r-1])^n\}$, independent of $\Gamma$. 
\dated{5 Apr 2021} 
\codes{\CSG:~\CAdj:~\CEig,~\CSw,~\CCov,~\COri,~\CLG}%
\itemspace

\bsgitem{20xxi}{}  Relations between the skew spectrum of an oriented graph and the spectrum of an associated signed graph.  Submitted.  \emph{}\ 
\annot{} For each digraph $\vec{G}$ there is $\Sigma$ whose spectrum contains full information about $\Spec\vec{G}$.
\codes{\CSG,~\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Zoran Stani\'c \& Ambat Vijayakumar}
\authorspace
\bsgitem{2021a}{}  On spectral radius of signed graphs without negative even cycles.  \emph{Bull.\ Math.\ Soc.\ Sci.\ Math.\ Roumanie}\ 64 (112) (2021), no.\ 1, 89--96.
\MR{4243509}.  
\par
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Richard P.\ Stanley} 						%STANLEY
\authorspace
\authref  See also \hyperlink{Peter Doubilet}{P.~Doubilet}, \hyperlink{Lili Mu}{L.L.~Mu}, and \hyperlink{Alexander Postnikov}{A.~Postnikov}.
\authrefspace

\bsgitem{1973a}{}  Linear homogeneous diophantine equations and magic labelings of graphs.  \emph{Duke Math.\ J.}\ 40 (1973), 607--632.
\MR{0317970} (47 \#6519).  \Zbl{269.05109}.					
\annot{} P.\ 630 restates \xref{Stewart}{1966a}{B.M.\ Stewart}, Cor.\ 2.4 in a clear way and observes that, if $\Gamma$ is bipartite, then $\dim V = \#E - n + 2$.  These two statements are equivalent to \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}.
\codes{\Cpar:~\Cinc,~\Cecyc}%
\itemspace

\bsgitem{1985a}{}  Reconstruction from vertex-switching.  \emph{J.\ Combin.\ Theory Ser.\ B} 38 (1985), 132--138.  
\MR{0787322} (86f:05096).  \Zbl{572.05046}. 					
\annot{} From the $1$-vertex switching deck (the multiset of isomorphism types of signed graphs resulting by separately switching each vertex) of $\Sigma = (K_n, \sigma)$, $\Sigma$ can be reconstructed, provided that $4 \nmid n$.  The same for $i$-vertex switchings, provided that the Krawtchouk polynomial $K_i^n(x)$ has no even zeros from $0$ to $n$.  When $i = 1$, the negative-subgraph degree sequence is always reconstructible.  All done in terms of Seidel (graph) switching of unsigned simple graphs.  
[See Ellingham; Ellingham and Royle; Krasikov; Krasikov and Roditty for further developments.  \Problem\ 1.  Generalize to signings of other highly symmetric graphs.  \Problem\ 2.  Prove a similar theorem for switching of a bidirected $K_n$.]
\par
\codes{\Ckg:~\Csw,~\CTG}%
\itemspace

\bsgitem{1986a}{}  \emph{Enumerative Combinatorics, Volume I}.  Wadsworth and Brooks/Cole, Monterey, Calif., 1986.
\MR{0847717} (87j:05003).  \Zbl{608.05001}.					
\annot{} Ch.\ 3, ``Partially ordered sets'': Exercise 51, pp.\ 165 and 191, concerns the \xref{Dowling}{1973a}{Thomas A.\ Dowling}, \xrefd{1973b}{Thomas A.\ Dowling} lattices of a group and mentions Zaslavsky's generalizations [signed and biased graphs].
\par
\codes{\CGG:~\CMatrd,~\CInvar:~\CExr,~\CExp}%
\itemspace

\bsgitem{1990a}{}  (As ``R.~Stenli'')  \emph{Perechislitel'naya kombinatorika}.  ``Mir'', Moscow, 1990.
\MR{1090542} (91m:05002).  
\annot{} Russian translation of \xrefd{1986a}{Richard P.\ Stanley}.
\codes{\CGG:~\CMatrd,~\CInvar:~\CExr,~\CExp}%
\itemspace

\bsgitem{1991a}{}  A zonotope associated with graphical degree sequences.  
In: Peter Gritzmann and Bernd Sturmfels, eds., \emph{Applied Geometry and Discrete Mathematics:  The Victor Klee Festschrift}, pp.\ 555--570.  DIMACS Ser.\ Discrete Math.\ and Theor.\ Computer Sci., Vol.\ 4.  American Mathematical Soc.\ and Assoc.\ for Computing Machinery, Providence and Baltimore, 1991.  
\MR{1116376} (92k:52020).  \Zbl{737.05057}.						
\annot{} All-negative complete graphs (implicit in \S3) and signed colorings (\S4) are used to find the number of ordered degree sequences of $n$-vertex graphs and to study their convex hull.  
\codes{\CSG:~\CGeom,~\CCol}%
\itemspace

\bsgitem{1996a}{}  Hyperplane arrangements, interval orders, and trees.  \emph{Proc.\ Nat.\ Acad.\ Sci.\ USA} 93 (1996), 2620--2625.  
\MR{1379568} (97i:52013).  \Zbl{848.05005}.
\annot{} Deformed braid hyperplane arrangements, i.e., canonical affine hyperplanar representations of $\Latb(K_n,\phi)$ where $\phi(ij) = l_i \in \bbZ$ when $i < j$.  In particular, Thm.\ 3.2 for contrabalance (generic gains) [\cf\ \xref{Zaslavsky}{1982d}{Thomas Zaslavsky}] and (\S4), all $l_i = 1$.  Also (\S5), the Shi arrangement, which represents $\Latb \{0,1\}\vec K_n$. 
\dated{rev.\ 21 Jul 2022}
\par
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar:~\CExp}%
\itemspace

\bsgitem{1997a}{}  \emph{Enumerative Combinatorics, Volume 1}.  Corrected reprint.  
Cambridge Stud.\ Adv.\ Math., Vol.\ 49.  Cambridge Univ.\ Press, Cambridge, Eng., 1997.
\MR{1442260} (98a:05001).  \Zbl{970.29805}, \Zbl{945.05006}.			
\annot{} Additional exercises, some updating, some corrections to \xrefd{1986a}{Richard P.\ Stanley}.
\par
\codes{\CGG:~\CMatrd,~\CInvar:~\CExr,~\CExp}%
\itemspace

\bsgitem{1998a}{}  Hyperplane arrangements, parking functions and tree inversions.  
In: B.E.~Sagan and R.~Stanley, eds., \emph{Mathematical Essays in Honor of Gian-Carlo Rota}, Progress in Math., Vol.\ 161, pp.\ 359--375.  Birkh\"auser, Boston, 1998.
\MR{1627378} (99f:05006).  \Zbl{980.39546}.
\codes{\Cgg:~\CGeom,~\CMatrd,~\CInvar:~\CExp}%
\itemspace

\bsgitem{1999a}{}  \emph{Enumerative Combinatorics, Volume 2}.  
Cambridge Stud.\ Adv.\ Math., Vol.\ 62.  Cambridge Univ.\ Press, Cambridge, Eng., 1999.
\MR{1676282} (2000k:05026).  \Zbl{928.05001}, \Zbl{978.05002}.			
\annot{} Exercise 5.50: The Shi arrangement [the affinographic hyperplane representation of $\{0,1\}\vec K_n$ with gain group $\bbZ^+$].  Exercise 5.41(h--i): The Linial arrangement and its characteristic polynomial [$= \chi^*_{\{1\}\vec K_n}(\lambda)$].  Exercise 6.19(lll) conceals the Catalan arrangement [which represents $\{0,\pm1\}\vec K_n$].  Exercise  5.40(b): Counts two-graphs that $\not\supseteq [C_5]$.
\par
\codes{\Cgg:~\CGeom,~\Cmatrd,~\CInvar,~\CTG:~\CExr,~\CExp}%
\itemspace

\bsgitem{2012a}{}  \emph{Enumerative Combinatorics, Volume 1}.  Second ed.  
Cambridge Stud.\ Adv.\ Math., Vol.\ 49.  Cambridge Univ.\ Press, Cambridge, Eng., 2012.
\MR{2868112}.  \Zbl{1247.05003}.  
\annot{} Vastly enlarged from \xrefd{1986a}{Richard P.\ Stanley}, \xrefd{1997a}{Richard P.\ Stanley}.  Ch.\ 3, ``Partially ordered sets'': 
Exercise 115b, solution, p.\ 434, mentions \xref{Zaslavsky}{1981a}{Thomas Zaslavsky}.  
Exercise 117, solution, p.\ 435, mentions \xref{Zaslavsky}{2002a}{Thomas Zaslavsky}.
Exercise 131, pp.\ 385 and 439--440, concerns the \xref{Dowling}{1973a}{Thomas A.\ Dowling}, \xrefd{1973b}{Thomas A.\ Dowling} lattices of a group and mentions Zaslavsky's generalizations to signed and gain [and biased] graphs.  
\dated{14 Jun 2012} 
\codes{\CGG:~\CMatrd,~\CInvar:~\CExr,~\CExp}%
\itemspace

\bsgitem{2015a}{}  Valid orderings of real hyperplane arrangements.  \emph{Discrete Comput.\ Geom.}\ 53 (2015), 951--964.
\MR{3341587}.  \Zbl{1322.52018}.  \arXiv{1306.1838}.				
\annot{} Introduces ``$\psi$-graphical'' hyperplane arrangements, which are affine cross-sections of gain-graphic arrangements according to \xref{Lutz}{2019a}{Bob Lutz}.  Also see \xref{Mu--Stanley}{2015a}{Lili Mu \& Richard P.\ Stanley} and \xref{Suyama--Tsujie}{2019a}{Daisuke Suyama \& Shuhei Tsujie}.
\dated{16 Nov 2018} 
\codes{\Cgg:~\CGeom}%
\itemspace

\author{Dietrich Stauffer}						%STAUFFER
\authorspace
\authref  See \hyperlink{Guy Hed}{G.~Hed}.
\itemspace

\author{Eckhard Steffen}						%STEFFEN
\authorspace
\authref  See also \hyperlink{Chiara Cappello}{C.~Cappello}, \hyperlink{Ligang Jin}{L.-G.~Jin}, \hyperlink{Yingli Kang}{Y.-L.~Kang}, \hyperlink{You Lu}{Y.~Lu}, \hyperlink{Edita Rollova}{E.~Rollov\'a}, and \hyperlink{Michael Schubert}{M.~Schubert}.
\authrefspace

\bsgitem{2014a}{}  Circular flow numbers of (signed) regular graphs.  In:  \emph{Bordeaux Graph Workshop 2014}, pp.\ 37--38.  LaBRI, Bordeaux, 2014.  
URL \url{http://bgw.labri.fr/2014/bgw2014-booklet.pdf}
\annot{} Extended abstract.
\codes{\CSG:~\CFlows}%
\itemspace

\author{Eckhard Steffen \& Michael Schubert}
\authorspace
\bsgitem{2013a}{}  Nowhere-zero flows on signed regular graphs.  In:  Jaroslav Ne\v{s}et\v{r}il and Marco Pellegrini, eds., \emph{The Seventh European Conference on Combinatorics, Graph Theory and Applications} (EuroComb 2013, Pisa), pp.\ 621--622.  CRM Ser., Vol.\ 16.  Edizioni della Normale, Scuola Normale Superiore Pisa, Pisa, Italy, 2013.
\Zbl{1291.05084}.  
\annot{} Abstract of \xref{Schubert and Steffen}{2015a}{Michael Schubert \& Eckhard Steffen}.
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\author{Eckhard Steffen \& Alexander Vogel}
\authorspace
\bsgitem{2021a}{}  Concepts of signed graph coloring.  \emph{European J.\ Combin.}\ 91 (2021), art.\ 103226, 19 pp.
\MR{4161819}.  \Zbl{1458.05100}.  \arXiv{1909.09381}.  
\annot{} A survey of many concepts, from the direct generalization of ordinary graph coloring in \xref{Zaslavsky}{1982b}{Thomas Zaslavsky}, to variants such as circular coloring as in \xref{Kang and Steffen}{2017a}{Yingli Kang \& Eckhard Steffen}, to homomorphic ``coloring'' as expounded in \xref{Naserasr, Sopena, and Zaslavsky}{2021a}{Reza Naserasr, Eric Sopena, \& Thomas Zaslavsky}.  
\dated{5 Jan 2021} 
\par
\codes{\CSG:~\CCol:~\CInvar,~\CExp}%
\itemspace

\author{Mat\v{e}j Stehl\'{\i}k}						%STEHLIK
\authorx{Matej Stehlik}
\authorspace
\authref  See \hyperlink{Luerbio Faria}{L.~Faria}.
\itemspace

\author{Kenneth Steiglitz} 						%STEIGLITZ
\authorspace
\authref  See \hyperlink{Christos H.\ Papadimitriou}{C.H.~Papadimitriou}.
\itemspace

\author{Arthur Stein}							%STEIN A
\authorspace
\authref  See \hyperlink{Brian Healy}{B.~Healy}.
\itemspace

\author{Daniel L.\ Stein}						%STEIN DL
\authorspace
\authref  See also \hyperlink{Alberto Gandolfi}{A.~Gandolfi} and \hyperlink{Charles M.\ Newman}{C.M.~Newman}.
\authrefspace

\bsgitem{1989a}{}  Spin glasses.  \emph{Scientific American} 261 (July, 1989), no.\ 1, 52--59.
\annot{} Informally describes frustration in spin glasses in terms of randomly ferromagnetic and antiferromagnetic interactions (see \xref{Toulouse}{1977a}{Gerard Toulouse}) and gives some history and applications.
\codes{\CPhys:~\Csg:~\Cbal,~\CRand:~\CExp}%
\itemspace

\author{Benjamin Steinberg}						%STEINBERG
\authorspace
\authref  See \hyperlink{John Rhodes}{J.~Rhodes}.
\itemspace

\author{Raphael Steiner}						%STEINER
\authorspace
\bsgitem{2022a}{}  Asymptotic equivalence of Hadwiger's conjecture and its odd minor-variant.  \emph{J.\ Combin.\ Theory Ser.\ B} 155 (2022), 45--51.
\MR{4379303}.  
\arXiv{2109.02302}.  
\annot{} 
\Cf\ \xref{Geelen, Gerards, Reed, Seymour, and Vetta}{2009a}{Jim Geelen, Bert Gerards, Bruce Reed, Paul Seymour, \& Adrian Vetta}.
[\Problem.  Generalize to signed graphs.]
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\author{Douglas Steinley}						%STEINLEY
\authorspace
\authref  See \hyperlink{Michael Brusco}{M.~Brusco}.
\itemspace

\author{[R.\ Stenli (Richard P.\ Stanley)]}					%[STENLI]
\authorspace
\authref  See \hyperlink{Richard P.\ Stanley}{R.P.~Stanley}.
\itemspace

\author{Catherine Stenson}						%STENSON
\authorspace
\authref  See \hyperlink{Saul Stahl \& Catherine Stenson}{S.~Stahl}.
\itemspace

\author{Andrea Sterbini}						%STERBINI
\authorspace
\authref  See \hyperlink{Rossella Petreschi}{R.~Petreschi}.
\itemspace

\author{Dragan Stevanovi\'c}						%STEVANOVIC
\authorx{Dragan Stevanovic}
\authorspace
\authref  See also \hyperlink{Lihua Feng}{L.H.~Feng} and \hyperlink{Guihai Yu}{G.H.~Yu}.
\authrefspace

\bsgitem{2007a}{}  Research problems from the Aveiro Workshop on Graph Spectra.  \emph{Linear Algebra Appl.}\ 423 (2007), no.\ 1, 172--181.
\MR{2312333}.  
\annot{} Two problems by Krzysztof Zwierzy\'nski on the ``signless Laplacian'' matrix $\Lap(-\Gamma)$ (see \xref{Cvetkovi\'c, Rowlinson, and Simi\'c}{2007a}{Dragos Cvetkovic, Peter Rowlinson, \& Slobodan K.\ Simic} are:  
Problem AWGS.1, ``The maximum clique and the signless Laplacian''.  Compare the clique number  with the min eigenvalue.
Problem AWGS.2, ``Integral graphs''.  For which graphs are all eigenvalues (of $\Lap(-\Gamma)$, in particular) integral?
\dated{15 Sept 2010} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Brett Stevens}						%STEVENS
\authorspace
\authref  See \hyperlink{Nancy Ann Neudauer}{N.A.~Neudauer}.
\itemspace

\author{B.M.\ Stewart} 						%STEWART BM
\authorspace
\bsgitem{1966a}{}  Magic graphs.  \emph{Canad.\ J.\ Math.}\ 18 (1966), 1031--1059.
\MR{0197358} (33 \#5523).  \Zbl{149.21401} (149, p.\ 214a).					
\annot{} In $\bbR^{1+E} = \bbR\times\bbR^E$ with $x_0$ the first coordinate, let $\sigma_v(x) = \sum \{x_e : e$ is incident to $v \}$, and let $V = \{ x\in\bbR^E : \sigma_v(x) = x_0, \forall v \in V \}$.  Cor.\ 2.4 (p.\ 1059): If $\Gamma$ is connected and contains an odd circle, then $\dim V = \#E - n + 1$.  
[Restated as in \xref{Stanley}{1973a}{Richard P.\ Stanley}.  Since $V \cap \{x_0=0\} =$ null space of the incidence matrix $\incid(-\Gamma)$, this cryptically and partially anticipates the first calculation of rank$(\incid(-\Gamma))$, by \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}.]
\codes{\Cpar:~\Cinc,~\Cecyc}%
\itemspace

\author{William J.\ Stewart}						%STEWART WJ
\authorspace
\authref  See \hyperlink{Ning Liu}{N.~Liu}.
\itemspace

\author{Michael Stiebitz}						%STIEBITZ
\authorspace
\authref  See \hyperlink{Thomas Schweser}{T.~Schweser}.
\itemspace

\author{Allen H.\ Stix} 						%STIX
\authorspace
\bsgitem{1974a}{}  An improved measure of structural balance.  \emph{Human Relations} 27 (1974), 439--455.  
\codes{\CSG:~\CFr}%
\itemspace

\author{Maria Stojkow}						%STOJKOW
\authorspace
\authref  See \hyperlink{Forough Hassanibesheli}{F.~Hassanibesheli}.
\itemspace

\author{Daniel Stolarski}						%STOLARSKI
\authorspace
\authref  See \hyperlink{Jordan Carlson}{J.~Carlson}.
\itemspace

\author{Douglas Stone}						%STONE
\authorspace
\authref  See \hyperlink{William Kocay}{W.~Kocay}.
\itemspace

\author{J.\ Randolph Stonesifer} 					%STONESIFER
\authorspace
\bsgitem{1975a}{}  Logarithmic concavity for a class of geometric lattices.  \emph{J.\ Combin.\ Theory Ser.\ A} 18  (1975), 216--218.  
\MR{0357169} (50 \#9637).  \Zbl{312.05019}.
\annot{} The second kind of Whitney numbers of a Dowling lattice are binomially concave, hence strongly logarithmically concave, hence unimodal.  
[\Cf\ \xref{Damiani, D'Antona, and Regonati}{1994a}{E.\ Damiani, O.\ D'Antona, \& F.\ Regonati} and \xref{Benoumhani}{1999a}{Moussa Benoumhani}.]
[\emph{Famous Problem} (Rota).  Generalize this.]
\dated{rev.\ 30 Apr 2012}
\par
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Steven H.\ Strogatz}						%STROGATZ
\authorspace
\authref  See also \hyperlink{Seth A.\ Marvel}{S.A.~Marvel}.
\authrefspace

\bsgitem{2010a}{}  The enemy of my enemy.  \emph{New York Times}, online edition, February 14, 2010, the Opinionator blog.  URL \url{http://opinionator.blogs.nytimes.com/2010/02/14/the-enemy-of-my-enemy/}
\annot{} A gentle explanation of negatives and negation, with special reference to balance in signed graphs.
\dated{21 March 2010} 
\codes{\CSG:~\CBal:~\CExp}%
\itemspace
 
\authorx{Thomas Strohmer}						%STROHMER
\author{Thomas Strohmer \& Robert W.\ Heath Jr.}
\authorspace
\bsgitem{2003a}{}  Grassmannian frames with applications to coding and communication.  \emph{Appl.\ Comput.\ Harmonic Analysis} 14 (2003), 257--275.
\MR{1984549} (2004d:42053).  \Zbl{1028.42020}.  \arXiv{math/0301135}.			
\annot{} Notices connection with regular two-graphs via Seidel adjacency matrix (\cf\ \xref{Seidel}{1976a}{J.J.\ Seidel}), because tight Grassmannian frames are equiangular.  [Foundational, as explained in \xref{Bodmann and Paulsen}{2005a}{Bernhard G.\ Bodmann \& Vern I.\ Paulsen}, esp.\ \S4.]
\dated{6 Aug 2018} 
\codes{\Csg:~\Ckg:~\CTG:~\CAdj:~\CGeom,~\CAppl}%
\itemspace

\author{Jeffrey Stuart}						%STUART
\authorspace
\authref  See also \hyperlink{Qian Li}{Q.A.~Li}.
\authrefspace

\author{Jeffrey Stuart, Carolyn Eschenbach, \& Steve Kirkland}
\authorspace
\bsgitem{1999a}{}  Irreducible sign $k$-potent sign pattern matrices.  \emph{Linear Algebra Appl.}\ 294 (1999), 85--92.
\MR{1693935} (2000f:15017).  \Zbl{935.15008}.					
\itemspace

\author{Leanne Stuive}						%STUIVE
\authorspace
\authref  See \hyperlink{Bertrand Guenin}{B.~Guenin}.
\itemspace

\author{Bernd Sturmfels}						%STURMFELS
\authorspace
\authref  See \hyperlink{Anders Bjorner}{A.~Bj\"orner}.
\itemspace

\author{J.\ Stutz} 							%STUTZ
\authorspace
\authref   See \hyperlink{Fred Glover}{F.~Glover}.
\itemspace

\author{Li Su}							%SU L
\authorspace
\authref  See also \hyperlink{Hong-Hai Li}{H.-H.~Li}.
\authrefspace

\author{Li Su, Hong-Hai Li, \& Jing Zhang}
\authorspace
\bsgitem{2014a}{}  The minimum spectral radius of signless Laplacian of graphs with a given clique number.  \emph{Discuss.\ Math.\ Graph Theory} 34 (2014), 95--102.
\MR{3149820}.  \Zbl{1292.05180}.  
\annot{} [\Questions.  Does this apply to signed graphs, and what is the appropriate definition of a clique?  Does it apply to complex unit gain graphs (\cf\ \xref{Reff}{2012a}{Nathan Reff})?]
\dated{18 May 2018} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Ting Su}							%SU T
\authorspace
\authref  See \hyperlink{Laura Anderson}{L.~Anderson}.
\itemspace

\author{C.K.\ Subbaraya}						%SUBBARAYA
\authorspace
\authref  See \hyperlink{Chandrashekar Adiga}{C.~Adiga}.
\itemspace

\author{S.P.\ Subbiah}						%SUBBIAH
\authorspace
\bsgitem{2008a}{}  \emph{A Study of Graph Theory:  Topology, Steiner Domination and Semigraph Concepts}.  Ph.D.\ thesis, Madurai Kamaraj University, 2008. 
\annot{} Contains material summarized in \xref{Subbiah and Swaminathan}{2009a}{S.P.\ Subbiah \& V.\ Swaminathan}.
\dated{2 Aug 2010} 
\codes{\CSG}%
\itemspace

\author{S.P.\ Subbiah \& V.\ Swaminathan}
\authorspace
\bsgitem{2009a}{}  Properties of topological spaces associated with sigraphs.  In:  K.\ Somasundaram, ed., \emph{Graph Theory and its Applications} (Proc.\ ), pp.\ 233--241.  Macmillan Publishers India, Delhi, 2009.
\MR{2574613} (no rev).  
\annot{} Topologies $\tau_+, \tau_-$ on $V$ $\longleftrightarrow$ $\Sigma^\epsilon, \epsilon=+,-$ for a signed graph $\Sigma$ [not necessarily simple or finite].  $\Sigma \mapsto (\tau_\pm)$: $\tau_\epsilon = \{\text{unions of subsets of } \pi(\Sigma^\epsilon)\}$, $\pi(\Gamma) :=$ connected-component partition of $V$ in $\Gamma$.  ``Exclusive property'':  If $u,v \in$ same component of $\Sigma^\epsilon$, they are not in the same component of $\Sigma^{-\epsilon}$, for $\epsilon = \pm$.  ``Transitivity'':  Every component of $\Sigma^\pm$ is a clique.  Thm.\ 1:  Bijection between topology pairs $(\tau_+,\tau_-)$ and transitive signed graphs on a set $V$ (\xref{Subbiah}{2008a}{S.P.\ Subbiah}).  Further results [made elementary by observing that topology pairs are equivalent to partitions $\pi_+,\pi_-$ of $V$.  Exclusivity is $\pi_+ \wedge \pi_- = 0_V$ and is equivalent to simplicity of $|\Sigma|$.  Topology is an epiphenomenon].  [It is not always clear when $|\Sigma|$ is meant to be simple.]
\dated{2 Aug 2010} 
\codes{\CSG}%
\itemspace

\bsgitem{2009b}{}  Properties of topological spaces associated with sigraphs.  Int.\ Conf.\ Graph Theory Appl.\ (Coimbatore, 2008).
\emph{Electronic Notes Discrete Math.}\ 33 (2009), 59--66. 
\MR{2574613}.  
\annot{} Shorter version of \xrefd{2009a}{S.P.\ Subbiah \& V.\ Swaminathan}.
\dated{2 Aug 2010} 
\codes{\CSG}%
\itemspace

\author{Anand Subramanian}						%SUBRAMANIAN
\authorspace
\authref  See \hyperlink{Eduardo Queiroga}{E.~Queiroga}.
\itemspace

\author{M.S.\ Subramanya}						%SUBRAMANYA
\authorspace
\authref  See also \hyperlink{R.\ Rangarajan}{R.~Rangarajan}, \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}, and \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\authrefspace

\author{M.S.\ Subramanya \& P.\ Siva Kota Reddy}
\authorspace
\bsgitem{2008a}{}  On balance and clusters in graph structures.  \emph{Ultra Sci.}\ 20(1)M (2008), 159--162.
\annot{} A ``graph structure'' (due to E.\ Sampathkumar in 2005) is $G := (V,\cR)$ where $\cR = \{R_1,\ldots,R_k)$, $k\geq2$, $R_i \subseteq \cP^{(2)}(V)$, and the $R_i$ are disjoint.  Let $\cS \subseteq \cR$ and $\|\cS\| := \bigcup \{R : R \in \cS\}$.  Define $\|G\| := (V,\|\cS\|)$ and let $\Sigma(\cS)$ be the signed $\|G\|$ with negative edge set $\|\cS\|$.  [$\Sigma(\cS)$ is not defined but is implicit.]  
$G$ is ``$\cS$-balanced'' if $\Sigma(\cS)$ is balanced, and ``$\cS$-clusterable'' if $\Sigma(\cS)$ is clusterable.  
Prop.\ 3 [hard to interpret] seems to be \xref{Harary's}{1953a}{Frank Harary} theorem for $\Sigma(\cS)$.  Thm.\ 4: $G$ is $\cS$-balanced for all $\cS$ iff it is $\{R_i\}$-balanced for all $i$.  
Thm.\ 7 is \xref{Davis's}{1967a}{James A.\ Davis} characterization of clusterability applied to $\cS$-clusterability.  Thm.\ 8 has three conditions equivalent to $\cS$-clusterability, assuming $\bigcup_1^k R_i = \cP^{(2)}(V)$ and no $R_i=\eset$.  [$k=2, \#\cS=1$ is signed $K_n$.]  
Thm.\ 9: $G$ is $\cS$-clusterable for all $\cS$ iff it is $\{R_i\}$-clusterable [the paper says ``balanced''] for all $i$.  
\dated{1 Aug 2009} 
\codes{\Csg,~\CSG(\CGen),~\Cgg:~\CBal,~\CSw,~\CClu}%
\itemspace

\bsgitem{2009a}{}  Triangular line signed graph of a signed graph.  \emph{Adv.\ Appl.\ Discrete Math.}\ 4 (2009), no.\ 1, 17--23.
\MR{2555622} (2010m:05136).  \Zbl{1176.05036}.					
\annot{} Definitions as at \xref{Sampathkumar, Reddy, and Subramanya}{2008a}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}, \xrefd{2010c}{E.\ Sampathkumar, P.\ Siva Kota Reddy, \& M.S.\ Subramanya}.  Let $T(\Gamma) := (E, E_T)$ where $E_T := \{ef : e,f \in C_3 \text{ in } \Gamma\}$.  The triangular line signed graph is $T(\Sigma) := (T(|\Sigma|), \sigma^c)$.  
Solved:  $T(\Sigma) \simeq \Lambda_{\times}(\Sigma)$, $T^k(\Sigma) \simeq T^2(\Sigma)$.
[$\Lambda_{\times}$ as in \xref{M.\ Acharya}{2009a}{Mukti Acharya}.]
\dated{3 Aug 2009}
\codes{\CSG:~\CBal,~\CSw,~\CLG(\CGen),~\CLG}%
\itemspace

\author{Benjamin Sudakov}						%SUDAKOV
\authorspace
\authref  See \hyperlink{Igor Balla}{I.~Balla} and \hyperlink{Gregory Gutin}{G.~Gutin}.
\itemspace

\author{Naduvath K.\ Sudev}						%SUDEV
\authorx{Sudev Naduvath}
\authorspace
\authref  See also \hyperlink{Javeria Amreen}{J.~Amreen}, \hyperlink{Achu Aniyan}{A.~Aniyan}, \hyperlink{P.K.\ Ashraf}{P.K.~Ashraf}, and \hyperlink{K.A.\ Germina}{K.A.~Germina}.
\authrefspace

\author{N.K.\ Sudev, P.K.\ Ashraf, \& K.A.\ Germina}
\authorspace
\bsgitem{2019a}{}  Some new results on integer additive set-valued signed graphs.  \emph{TWMS J.\ App.\ Eng.\ Math.}\ 9 (2019), no.\ 3, 638--645.
\arXiv{1609.00295}.	\HAL{hal-02263299}.	  
\codes{\CSG:~\CLab}%
\itemspace

\author{N.K.\ Sudev, K.P.\ Chithra, \& K.A.\ Germina}
\authorspace
\bsgitem{2017a}{}  Switched signed graphs of integer additive set-valued signed graphs.  \emph{Discrete Math.\ Algorithms Appl.}\ 9 (2017), no.\ 4, art.\ 1750043, 10 pp.
\MR{3686920}.  \Zbl{1373.05169}.  
\codes{\CSG:~\CLab,~\CSw}%
\itemspace

\author{N.K.\ Sudev \& K.A.\ Germina}
\authorspace
\bsgitem{2015a}{}  A study on integer additive set-valuations of signed graphs.  \emph{Carpathian Math.\ Publ.}\ 7 (2015), no.\ 2, 236--246.
\MR{3457909}.  \Zbl{1329.05141}.  \arXiv{1511.00678}.  \HAL{hal-02099245}.		
\codes{\CSG:~\CLab}%
\itemspace

\authorx{Sudev Naduvath \& Germina Augustine}
\author{Sudev Naduvath \& Germina Augustine [K.A.\ Germina]}
\authorspace
\bsgitem{2018a}{}  \emph{An Introduction to Sumset Valued Graphs (Theory and Applications)}, Lambert Academic Publishing, 2018.
\annot{} \S1.3, ``Signed graphs'': expository.  Fix $X \subseteq \bbZ$.  
\S6.2, ``Sumset labeled signed graphs'':  A sumset labelling of $\Sigma$ is an injective $f: V \to \cP(X)\setminus\{\eset\}$ such that $\sigma(uv) = (-1)^{\#(f(u)+f(v))}$, $\forall uv \in E$.  
It is strong if $\forall uv\in E$, $f(u)+f(v)$ has no duplicate sums.  Thm.\ 6.2.4 [6.2.5]:  Let $f$ be strong and $\#(f(u)+f(v)) = k$, $\forall uv$.  Then $\Sigma$ is balanced [clusterable] iff $|\Sigma|$ is bipartite or $\sqrt{k} \in 2\bbZ$ [$k$ is odd].  More results for strong and weak $f$.
\S6.3, ``Arithmetic sumset labeled signed graphs'':  Here $f(v)$ is an arithmetic progression.
\dated{23 Oct 2019} 
\codes{\CSG:~\CLab:~\CBal,~\CClu}%
\itemspace

\author{Sudev Naduvath \& Johan Kok}
\authorspace
\bsgitem{2020a}{}  On certain topological indices of signed graphs.  \emph{J.\ Math.\ Comput.\ Sci.}\ 10 (2020), no.\ 2, 248--261.
\arXiv{2002.10240}.  
\annot{} Definitions and simple properties of positive, negative, mixed, and net Zagreb, Schultz, Gutman indices, obtained from positive, negative, and net degrees.
\dated{29 Dec 2020} 
\codes{\CSG:~\CInvar}%
\itemspace

\author{J.\ Christabel Sudha}							%SUDHA
\authorspace
\authref  See \hyperlink{C.\ Jayasekaran}{C.~Jayasekaran}.
\itemspace

\author{N.\ Sudharsanam}						%SUDHARSANAM
\authorspace
\authref  See \hyperlink{R.\ Balakrishnan}{R.~Balakrishnan}.
\itemspace

\author{R.\ Sujatha}							%SUJATHA
\authorspace
\authref  See \hyperlink{R.\ Sundareswaran}{R.~Sundareswaran}.
\itemspace

\author{G.\ Sumathy}							%SUMATHY
\authorspace
\authref  See also \hyperlink{C.\ Jayasekaran}{C.~Jayasekaran}.
\authrefspace

\author{G.\ Sumathy \& C.\ Jayasekaran}
\authorspace
\bsgitem{2018a}{}  A characterization of bicyclic but not two-cyclic graphs with a cut vertex as self vertex switching.  \emph{Utilitas Math.}\ 107 (2018), 87--101.
\MR{3793045}.  \Zbl{1395.05176}.  
\codes{\Ctg:~\CSw}%
\itemspace

\author{Mei-Yu Sun}							%SUN MY
\authorspace
\authref  See \hyperlink{Fu-Tao Hu}{F.-T.~Hu}.
\itemspace

\author{Qiang Sun}							%SUN Q
\authorspace
\authref  See \hyperlink{Hongyan Cai}{H.-Y.~Cai}, \hyperlink{Reza Naserasr}{R.~Naserasr}, and \hyperlink{Chao Wen}{C.~Wen}.
\itemspace

\author{Shiwen Sun}							%SUN SW
\authorspace
\authref  See \hyperlink{Shasha Feng}{S.-S.~Feng}.
\itemspace

\author{Wanting Sun}						%SUN WT
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.-C.~Li}.
\itemspace

\authorx{Zhenyu Sun}						%SUN ZY
\author{Zhenyu Sun, Hongwei Zhang, \& Frank L.\ Lewis}
\authorspace
\bsgitem{2021a}{}  Output sign-consensus of heterogeneous multiagent systems over fixed and switching signed graphs: an observer-based approach.  \emph{Int.\ J.\ Robust Nonlinear Control} 31 (2021), no.\ 12, 5849--5864.
\MR{4329717} (no rev).  
\annot{} \Cf\ \xref{Jiang and Zhang}{2020a}{Huidi Jiang \& Hongwei Zhang}.
\codes{\CSD}%
\itemspace

\author{Zhi Ren Sun}							%SUN ZR
\authorspace
\authref  See \hyperlink{Xiao Xin Zhu}{X.X.~Zhu}.
\itemspace

\author{Zhongyao Sun}						%SUN ZY
\authorspace
\bsgitem{2015a}{}  \emph{Analysis and Logical Modeling of Biological Signaling Transduction Networks}.  Doctoral dissertation, Pennsylvania State University, 2015.
\annot{} Ch.\ 5, ``Determining the attractors of a boolean network using an elementary signaling mode approach'', employs signed digraphs.	
\par
\codes{\CSD:~\CDyn}%
\itemspace

\authorx{R.\ Sundareswaran}						%SUNDARESWARAN
\author{R.\ Sundareswaran, R.\ Sujatha, \& Goksen Bacak-Turan}
\authorspace
\bsgitem{2020a}{}  A study on vulnerability parameters of signed fuzzy graphs.  In:  Cengiz Kahraman \etal, eds., \emph{Intelligent and Fuzzy Techniques in Big Data Analytics and Decision Making} (INFUS 2019 Conf., Istanbul), pp.\ 24--32.  Adv.\ Intell.\ Syst.\ Comput., Vol.\ 1029.  Springer Nature, 2020.
\codes{\CSG:~\CInvar}%
\itemspace

\author{V.S.\ Sunder}						%SUNDER
\authorspace
\authref  See \hyperlink{Vijay Kodiyalam}{V.~Kodiyalam}.
\itemspace

\author{Borworn Suntornpoch}						%SUNTORNPOCH
\authorspace
\authref  See \hyperlink{Yotsanan Meemark}{Y.~Meemark}.
\itemspace

\author{Didi Surian}							%SURIAN
\authorspace
\authref  See \hyperlink{David Lo}{D.~Lo}.
\itemspace

\author{Daisuke Suyama}						%SUYAMA
\authorspace
\authref  See also \hyperlink{Takuro Abe}{T.~Abe}.
\authrefspace

\author{Daisuke Suyama, Michele Torielli, \& Shuhei Tsujie}
\authorspace
\bsgitem{2019a}{}  Signed graphs and the freeness of the Weyl subarrangements of type $B_l$.  \emph{Discrete Math.}\ 342 (2019), no.\ 1, 233--249.
\MR{3886267}.  \Zbl{1400.05106}.  \arXiv{1707.01967}.  
\codes{\CSG:~\CGeom:~\CAlgeb}%
\itemspace

\author{Daisuke Suyama \& Shuhei Tsujie}
\authorspace
\bsgitem{2019a}{}  Vertex-weighted graphs and freeness of $\psi$-graphical arrangements.  \emph{Discrete Comput.\ Geom.}\ 61 (2019), no.\ 1, 185--197.
\MR{3925550}.  \Zbl{1412.52016}.  \arXiv{1511.04853}.				
\annot{} \Cf\ \xref{Stanley}{2015a}{Richard P.\ Stanley}.
\dated{16 Nov 2018} 
\codes{\Cgg:~\CGeom}%
\itemspace

\author{Masuo Suzuki}						%SUZUKI M
\authorspace
\bsgitem{1991a}{}  Lee-Yang complex-field systems and frustrated Ising models.  \emph{J.\ Phys.\ Soc.\ Japan} 60 (1991), no.\ 2, 441--449.
\MR{1104390} (92h:82034) (\qv).  
\annot{} \S2, ``Equivalence of Villain's frustrated system to Lee-Yang's complex-field systems'':  (2.3) summarizes \xref{Villain's}{1977a}{Jacques Villain} ``fully frustrated'' signed-graphic Ising model.
\dated{17 Jun 2012} 
\codes{\CPhys:~\CSG:~\CExp}%
\itemspace

\authorx{Sei Suzuki}							%SUZUKI S
\author{Sei Suzuki, Jun-ichi Inoue, \& Bikas K.\ Chakrabarti}
\authorspace
\bsgitem{2013a}{}  \emph{Quantum Ising Phases and Transitions in Transverse Ising Models}, second edition.  Lect.\ Notes Physics, Vol.\ 862.  Springer, Berlin, 2013.
\MR{3222781}.  \Zbl{1268.82001}.						
\annot{} \Cf\ first edition: \xref{Chakrabarti, Dutta, and Sen}{1996a}{Bikas K.\ Chakrabarti, Amit Dutta, \& Parongama Sen}.
\par
\codes{\CPhys:~\CSG,~\Cwg:~\CFr}%
\itemspace

\author{V.\ Swaminathan}						%SWAMINATHAN
\authorspace
\authref  See \hyperlink{S.P.\ Subbiah}{S.P.~Subbiah}.
\itemspace

\author{Chaitanya Swamy}						%SWAMY C
\authorspace
\bsgitem{2004a}{}  Correlation clustering: Maximizing agreements via semidefinite programming.  In:  \emph{Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA, New Orleans, 2004), pp.\ 526--527.  Assoc.\ for Computing Machinery, New York, and SIAM, Philadelphia, 2004.
\MR{2291092}.  
\par
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}
\itemspace

\author{Shivakumar Swamy C.S.}					%SWAMY CSS
\authorspace
\authref  See \hyperlink{Chandrashekar Adiga}{C.~Adiga}.
\itemspace

\author{Ed Swartz}							%SWARTZ
\authorspace
\authref  See \hyperlink{Patricia Hersh}{P.~Hersh}.
\itemspace

\author{Eric Swartz}							%SWARTZ
\authorspace
\authref  See \hyperlink{Alex Schaefer}{A.~Schaefer}.
\itemspace

\author{Swathyprabhu Mj [Swathy Prabhu]}				%SWATHY PRABHU
\authorx{Swathyprabhu Mj}						%SWATHYPRABHU
\authorspace
\authref  See \hyperlink{Sandip Das}{S.~Das}.
\itemspace

\author{Robert H.\ Swendsen}						%SWENDSEN
\authorspace
\authref  See \hyperlink{Jian-Sheng Wang}{J.-S.~Wang}.
\itemspace

\author{Katia Sycara}						%SYCARA
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Itiro Sy\^ozi}							%SYOZI
\authorx{Itiro Syozi}
\authorspace
\authref  See also \hyperlink{Yasuhiro Kasai}{Y.~Kasai}.
\authrefspace

\bsgitem{1950a}{}  The statistics of honeycomb and triangular lattice.  II.  \emph{Progress Theor.\ Phys.}\ 5 (1950), 341--351.
\MR{0039629} (12, 576g).  
\annot{} Physics of the all-negative (``antiferromagnetic'') toroidal honeycomb (\S7) and triangular (\S9) lattices.  The former is similar to all-positive (``ferromagnetic'') [because balanced] while the latter is not [because unbalanced].  
[Also see \xref{Houtappel}{1950a}{R.M.F.\ Houtappel}, \xrefd{1950b}{R.M.F.\ Houtappel}, \xref{Newell}{1950a}{G.F.\ Newell}, \xref{Wannier}{1950a}{G.H.\ Wannier}.]
\dated{21 Jun 2012} 
\codes{\CPhys,~\Csg:~\CPar:~\CFr}%
\itemspace

\author{Edward Szczerbickl}						%SZCZERBICKI
\authorspace
\bsgitem{1996a}{}  Signed directed graphs and reasoning for agents and multi-agent systems.  \emph{Int.\ J.\ Systems Sci.}\ 27 (1996), no.\ 10, 1009--1015.
\Zbl{860.90071}.  
\annot{} A state (``pattern'') is $s: V \to \{+,-,0\}$.  An arc $uv$ is ``consistent'' if $s(v) = \sigma(uv)s(u)$.  In a ``solution'' $\zeta$ all arcs are consistent.  A state is propagated by $s'(v) = \sigma(uv)s(u)$.  Rules for simplification to get a signed digraph with equivalent solutions.  [The model is incomplete.  The role of propagation is unclear.]
\dated{24 Nov 2012} 
\codes{\CSD:~\CAppl,~\Cbal}%
\itemspace

\author{Janusz Szczypula} 						%SZCZYPULA
\authorspace
\authref   See \hyperlink{Patrick Doreian}{P.~Doreian}.
\itemspace

\author{Andrzej Szepietowski}						%SZEPIETOWSKI
\authorspace
\authref  See also \hyperlink{Janusz Dybizbanski}{J.~Dybizba\'nski}.
\authrefspace

\bsgitem{2019a}{}  Negative closed walks in signed graphs: A note.  Manuscript, 2019.
\arXiv{1910.06032}.  
\annot{} Corrects an error in manuscript of \xref{Naserasr, Sopena, and Zaslavsky}{2021a}{Reza Naserasr, Eric Sopena, \& Thomas Zaslavsky}.
\dated{6 Jul 2022} 
\codes{\CSG}%
\itemspace

\author{Stefan Szeider}						%SZEIDER
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon}.
\itemspace

\author{Endre Szemer\'edi}						%SZEMEREDI
\authorx{Endre Szemeredi}
\authorspace
\authref  See \hyperlink{Bela Bollobas}{B.~Bollob\'as}.
\itemspace

\author{Zolt\'an Szigeti}						%SZIGETI
\authorspace
\authref  See \hyperlink{A.A.\ Ageev}{A.A.~Ageev}.
\itemspace   

\author{Ferenc Sz\"oll\H{o}si}						%SZOLLOSI
\authorx{Ferenc Szollosi}
\authorspace
\authref  See also \hyperlink{Gary Greaves}{G.~Greaves}.
\authrefspace

\author{Ferenc Sz\"oll\H{o}si \& Patric R.J.\ \"Osterg\r{a}rd}
\authorspace
\bsgitem{2018a}{}  Enumeration of Seidel matrices.  \emph{European J.\ Combin.}\ 69 (2018), 169--184.
\MR{3738150}.  \Zbl{1376.05026}.  
\annot{} Seidel matrix = $A(K_n,\sigma)$.  Spectrum \etal\ for $n\leq 13$.  Classification of those with 3 distinct eigenvalues for $n\leq 23$.  Application to equiangular lines.
\dated{22 Dec 2017} 
\codes{\Csg:~\CKG:~\CAdj,~\CGeom}%
\itemspace

\author{Jayme L.\ Szwarcfiter}						%SZWARCFITER
\authorspace
\authref  See \hyperlink{Carlos V.G.C.\ Lima}{C.V.G.C.~Lima}.
\itemspace


\label{LETTER T} \hypertarget{LETTER T}{}	% LETTER  T


\authorx{Bosiljka Tadi\'c}						%TADIC
\authorx{Bosiljka Tadic}
\author{Bosiljka Tadi\'c, Krzysztof Malarz, \& Krzysztof Ku{\l}akowski}
\authorx{Bosiljka Tadic, Krzysztof Malarz, \& Krzysztof Kulakowski}
\authorspace
\bsgitem{2005a}{}  Magnetization reversal in spin patterns with complex geometry.  \emph{Phys.\ Rev.\ Lett.}\ 94 (2005), art.\ 137204, 4 pp.
\codes{\Csg:~\Cpar:~\CFr}%
\itemspace

\author{B.\ Taglienti}							%TAGLIENTI
\authorspace
\authref  See \hyperlink{M.\ Falcioni}{M.~Falcioni}.
\itemspace

\author{Artin Tajdini}							%TAJDINI
\authorspace
\authref  See \hyperlink{Saieed Akbari}{S.~Akbari}.
\itemspace

\author{Martin Tak\'a{\u c}}						%TAKAC
\authorx{Martin Takac}
\authorspace
\bsgitem{1997a}{}  Fixed point classification method for qualitative simulation.  In: Ernesto Coasta and Amilcar Cardoso, eds., \emph{Progress in Artificial Intelligence} (8th Portuguese Conf., EPIA-97, Coimbra, Portugal, 1997), pp.\ 255--266.  \LNCS 1323.  Springer, Berlin, 1997.
\MR{1703015} (no rev).  \Zbl{1044.68883}.  
\codes{\CSD:~\CQM:~\CSta}%
\itemspace

\author{K\'aroly Tak\'acs}						%TAKACS
\authorx{Karoly Takacs}
\authorspace
\authref  See \hyperlink{Simone Righi}{S.~Righi}.
\itemspace

\author{Shingo Takahashi}						%TAKAHASHI
\authorspace
\authref  See \hyperlink{Takehiro Inohara}{T.~Inohara}.
\itemspace

\author{Akimichi Takemura}						%TAKEMURA
\authorspace
\authref  See \hyperlink{Hidehiko Kamiya}{H.~Kamiya}.
\itemspace

\author{Lynn Takeshita}						%TAKESHITA
\authorspace
\bsgitem{20xxa}{}  Coloring signed graphs.  Manuscript, [n.d.]
\codes{\CSG:~\CCol:~\CExp}%
\itemspace

\author{Michel Talagrand}						%TALAGRAND
\authorspace
\bsgitem{1998a}{}  Huge random structures and mean field models for spin glasses.  Proc.\ Int.\ Congress of Mathematicians, Vol.\ I (Berlin, 1998).  \emph{Documenta Math.}, Extra Vol.\ ICM 1998 (1998), Vol.\ I, pp.\ 507--536.
\MR{1648045} (2000c:60164).  \Zbl{902.60089}.
\codes{\Csg:~\CGen:~\Cfr:`\CExp}%
\itemspace

\author{Irving Tallman} 						%TALLMAN
\authorspace
\bsgitem{1967a}{}  The balance principle and normative discrepancy.  \emph{Human Relations} 20 (1967), 341--355.
\codes{\CPsS:~\CECol}%
\itemspace

\author{Ilan Talmud}							%TALMUD
\authorspace
\authref  See \hyperlink{Zeev Maoz}{Z.~Maoz}.
\itemspace

\authorx{Bit-Shun Tam}						%TAM BS
\author{Bit-Shun Tam, Yi-Zheng Fan, \& Jun Zhou}
\authorspace
\authref  See also \hyperlink{Ting-Chung Chang}{T.-J.~Chang} [T.-C.~Chang], \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}, \hyperlink{Hong-Hai Li}{H.-H.~Li}, and \hyperlink{Qi Wu}{Q.~Wu}.
\authrefspace

\bsgitem{2008a}{}  Unoriented Laplacian maximizing graphs are degree maximal.  \emph{Linear Algebra Appl.}\ 429 (2008), 735--758.
\MR{2428127} (2009c:05143).  \Zbl{1149.05034}.					
\annot{} The matrix is $\Lap(-\Gamma)$.  ``Maximizing'' graphs are those whose degree sequences are maximal in the majorization ordering.  
[For majorization also see \xref{Liu, Liu, and You}{2013a}{Bolian Liu, Muhuo Liu, \& Zhifu You}.]
\dated{23 Mar 2009}
\codes{\CPar:~\CLap}%
\itemspace

\author{Bit-Shun Tam \& Shu-Hui Wu}
\authorspace
\bsgitem{2010a}{}  On the reduced signless Laplacian spectrum of a degree maximal graph.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 7, 1734--1756.
\MR{2592914} (2011c:15041).  \Zbl{1230.05202}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{A.\ Tamilselvi}						%TAMILSELVI
\authorspace
\authref  See also \hyperlink{M.\ Parvathi}{M.~Parvathi}.
\authrefspace

\bsgitem{2010a}{}  Robinson-Schensted correspondence for the $G$-vertex colored partition algebra.  \emph{Asian-European J.\  Math.}\ 3 (2010), no.\ 2, 369--385.
\MR{2669040} (2011j:16057).  \Zbl{1230.05010}.					
\codes{\Cgg:~\CAlgeb,~\Cmatrd}%
\itemspace

\author{Arie Tamir} 							%TAMIR
\authorspace
\authref   See also \hyperlink{Dorit S.\ Hochbaum}{D.~Hochbaum}.
\authrefspace

\bsgitem{1976a}{}  On totally unimodular matrices.  \emph{Networks} 6 (1976), 373--382.  
\MR{0472865} (57 \#12553).  \Zbl{356.15020}.
\codes{\CSD:~\CBal}%
\itemspace

\author{Christino Tamon}						%TAMON
\authorspace
\authref  See \hyperlink{John Brown}{J.~Brown} and \hyperlink{Devlin Mallory, Abigail Raz, Christino Tamon, \& Thomas Zaslavsky}{D.~Mallory}.
\itemspace

\author{Akihisa Tamura}						%TAMURA A
\authorspace
\authref  See also \hyperlink{Yoshiko T.\ Ikebe}{Y.T.~Ikebe} and \hyperlink{Daishin Nakamura}{D.~Nakamura}.
\authrefspace

\bsgitem{1997a}{}  The generalized stable set problem for perfect bidirected graphs.  \emph{J.\ Operations Res.\ Soc.\ Japan} 40 (1997), 401--414.
\MR{1476832} (99e:05063).  \Zbl{894.90156}.					
\annot{} Problem: maximize an integral weight function over the bidirected stable set polytope (\cf\ \xref{Johnson and Padberg}{1982a}{Ellis L.\ Johnson \& Manfred W.\ Padberg}).  \S3 concerns the effect on perfection of deleting all incoming edges at a vertex.  \S4 reduces the ``generalized stable set problem'' for bidirected graphs to the maximum weighted stable set problem for ordinary graphs, whence the problem for perfect bidirected graphs is solvable in polynomial time.
\par
\codes{\Csg:~\COri:~\CInc,~\CGeom,~\CSw,~\CAlgor}%
\itemspace

\bsgitem{2000a}{}  Perfect $(0, \pm1)$-matrices and perfect bidirected graphs.  Combinatorics and Optimization (Okinawa, 1996).  \emph{Theor.\ Comput.\ Sci.}\ 235 (2000), no.\ 2, 339--356.
\MR{1756130} (2001i:15019).  \Zbl{938.68061}.					
\annot{} The stable set problem associated with bidirected graphs.  
\par
\codes{\Csg:~\COri:~\CGeom,~\CAlgor}
\itemspace

\author{Takeyuki Tamura}						%TAMURA T
\authorspace
\authref  See \hyperlink{Tatsuya Akutsu}{T.~Akutsu}.
\itemspace

\author{Jinsong Tan}							%TAN JS
\authorspace
\bsgitem{2008a}{}  A note on the inapproximability of correlation clustering.  \emph{Inform.\ Processing Lett.}\ 108 (2008), 331--335.
\MR{2456610} (2009k:05167).  \Zbl{1189.05158}.  \arXiv{0704.2092}.			
\codes{\Csg:~\CClu:~\CAlgor}%
\itemspace

\author{Shang Wang Tan}						%TAN SW
\authorspace
\authref  See also \hyperlink{Lin Feng}{L.~Feng}, \hyperlink{Xiao Li Wu}{X.L.~Wu}, and \hyperlink{De Long Zhang}{D.L.~Zhang}.
\authrefspace

\bsgitem{2010a}{}  On the Laplacian spectral radius of weighted trees with a positive weight set.  \emph{Discrete Math.}\ 310 (2010), no.\ 5, 1026--1036.
\MR{2575820} (2011e:05156).  \Zbl{1230.05147}.  
\annot{} The results on $\Lap(\Gamma,w)$ with edge weights $w: E \to \bbR_{>0}$ are deduced from results on $\Lap(-\Gamma,w)$.  
	
[\Problem.  Show the same reasoning applies to all signatures of $\Gamma$.] 
\dated{20 Jan 2012} 
\codes{\Cpar:~\CWG:~\CEig}%
\itemspace

\bsgitem{2010b}{}  On the weighted trees with given degree sequence and positive weight set.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 2, 380--389.
\MR{2645091} (2011e:05157).  \Zbl{1209.05054}.  
\annot{} Similar to \xrefd{2010a}{Shang Wang Tan}.
\dated{20 Jan 2012} 
\codes{\Cpar:~\CWG:~\CEig}%
\itemspace

\author{Shang-wang Tan, Ji-ming Guo, \& Jian Qi}	
\authorspace
\bsgitem{2003a}{}  The spectral radius of Laplacian matrices and quasi-Laplacian matrices of graphs.  \emph{Gongcheng Shuxue Xuebao} [\emph{Chinese J.\ Engineering Math.}] 20 (2003), no.\ 6, 69--74.
\MR{2031534} (2004k:05137).  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shang-Wang Tan \& Jing-Jing Jiang}
\authorspace
\bsgitem{2011a}{}  On the Laplacian spectral radius of weighted trees with fixed diameter and weight set.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 2, 173--192.
\MR{2773649} (2012a:05202).  \Zbl{1226.05169}.					
\annot{} The ``(signless) Laplacian'' of a graph with positive edge weights, $(\Gamma,w)$ where $w: E \to \bbR_{>0}$, is $\Lap(-\Gamma,w) := D(\Gamma,w) + A(\Gamma,w)$ (called $R$).  The spectral radius is that of $\Lap(-\Gamma,w)$.  
	
[\Problem.  Generalize to all weighted signed graphs.]
\dated{11 Jan 2011, 21 Jan 2012} 
\codes{\Cpar:~\CWG,~\CEig}%
\itemspace

\author{Shang Wang Tan \& Xing Ke Wang}
\authorspace
\bsgitem{2009a}{}  On the largest eigenvalue of signless Laplacian matrix of a graph.  \emph{J.\ Math.\ Res.\ Exposition} 29 (2009), no.\ 3, 381--390.
\MR{2510212} (2010h:05183).  \Zbl{1212.05164}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xuezhong Tan}						%TAN XZ
\authorspace
\authref  See also \hyperlink{Mu Huo Liu}{M.H.~Liu}.
\authrefspace

\author{Xuezhong Tan \& Bolian Liu}
\authorspace
\bsgitem{2006a}{}  On the spectrum of the quasi-Laplacian matrix of a graph.  \emph{Australasian J.\ Combin.}\ 34 (2006), 49--55.
\MR{2195309} (2006i:05106).  \Zbl{1102.05039}.  
\dated{25 Oct 2014} 
\par
\codes{\CPar:~\CEig,~\Cecyc}%
\itemspace


\author{Ying-Ying Tan}						%TAN YY
\authorspace
\authref  See also \hyperlink{Yi-Zheng Fan}{Y.-Z.~Fan}.
\authrefspace

\author{Ying Ying Tan \& Yi Zheng Fan}
\authorspace
\bsgitem{2008a}{}  On edge singularity and eigenvectors of mixed graphs.  \emph{Acta Math.\ Sinica (Engl.\ Ser.)}\ 24 (2008), no.\ 1, 139--146.
\MR{2384238} (2008k:05134).  \Zbl{1143.05058}.					
\annot{} Relations between least Laplacian eigenvalue, its eigenvector, and $l(\Sigma)$.  Properties of the eigenvector when $l(\Sigma) = 1$, e.g., $\lambda_{\min} \leq (4/n)l(\Sigma)$.
Dictionary:  ``mixed graph'' = signed graph, ``edge singularity'' = frustration index $l(\Sigma)$.  
[Generalized in \xref{Bapat, Kalita, and Pati}{2012a}{R.B.\ Bapat, D.\ Kalita, \& S.\ Pati}.]
\dated{28 Oct 2011, 20 Jan 2012} 
\codes{\Csg:~\CFr,~\CEig}%
\itemspace

\author{Adrian Tanasa}						%TANASA
\authorspace
\authref  See \hyperlink{Thomas Krajewski}{T.~Krajewski}.
\itemspace

\author{B.Z.\ Tang}							%TANG BZ
\authorspace
\authref  See \hyperlink{Y.\ Chen}{Y.~Chen}.
\itemspace

\author{Jiliang Tang}							%TANG JL
\authorspace
\authref  See also \hyperlink{Ghazaleh Beigi}{G.~Beigi}.
\authrefspace

\author{Jiliang Tang, Yi Chang, Charu Aggarwal, \& Huan Liu}
\authorspace
\bsgitem{2015a}{}  Negative link prediction in social media.  In:  \emph{Proceedings of the Eighth ACM International Conference on Web Search and Data Mining} (WSDM'15, Shanghai, 2015), pp.\ 87--96.  ACM, New York, 2015.
\arXiv{1412.2723}.  
\codes{\CSG:~\CPred:~\CAlgor}%
\itemspace

\bsgitem{2016a}{}  A survey of signed network mining in social media.  \emph{ACM Computing Surveys} 49 (2016), no.\ 3, art.\ 42, 39 pp.
\arXiv{1511.07569}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Wen Tang}							%TANG W
\authorspace
\authref  See \hyperlink{Er Ling Wei}{E.L.~Wei}.
\itemspace

\author{Wenliang Tang}						%TANG WL
\authorspace
\authref  See \hyperlink{Er Ling Wei}{E.L.~Wei}
\itemspace

\author{Zikai Tang}							%TANG ZK
\authorspace
\authref  See \hyperlink{Yaoping Hou}{Y.-P.~Hou}.
\itemspace

\author{Shin-ichi Tanigawa}						%TANIGAWA
\authorspace
\authref  See also \hyperlink{Rintaro Ikeshita}{R.~Ikeshita} and \hyperlink{Tibor Jordan}{T.~Jord\'an}.
\authrefspace

\bsgitem{2015a}{}  Matroids of gain graphs in applied discrete geometry.  \emph{Thans.\ Amer.\ Math.\ Soc.}\ 367 (2015), no.\ 12, 8597--8641.
\MR{3403067}.  \Zbl{1325.05048}.  \arXiv{1207.3601}.				
\codes{\CGG:~\CMatrd:~\CGen}%
\itemspace

\author{Tetsuji Taniguchi}						%TANIGUCHI
\authorspace
\authref  See \hyperlink{Taeyoung Chung}{T.Y.~Chung}, \hyperlink{Alexander L.\ Gavrilyuk}{A.L.~Gavrilyuk}, \hyperlink{Gary Greaves}{G.~Greaves}, \hyperlink{Hye Jin Jang}{Hye Jin Jang}, and \hyperlink{Akihiro Munemasa, Yoshio Sano, \& Tetsuji Taniguchi}{A.~Munemasa}.
\itemspace

\author{Percy H.\ Tannenbaum} 					%TANNENBAUM
\authorspace
\authref   See \hyperlink{Charles E.\ Osgood}{C.E.~Osgood}.
\itemspace

\author{\'Eva Tardos}							%TARDOS
\authorx{Eva Tardos}	
\authorspace
\authref  See also \hyperlink{Andrew V.\ Goldberg}{A.V.~Goldberg}.
\authrefspace

\author{\`Eva Tardos \& Kevin D.\ Wayne}
\authorx{Eva Tardos \& Kevin D.\ Wayne}
\authorspace
\bsgitem{1998a}{}  Simple generalized maximum flow algorithms.  
In: Robert E.\ Bixby, E.~Andrew Boyd, and Roger Z.\ R\'{\i}os-Mercado, eds., \emph{Integer Programming and Combinatorial Optimization} (6th Int.\ IPCO Conf., Houston, 1998, Proc.), pp.\ 310--324.  \LNCS 1412.  Springer, Berlin, 1998.
\MR{1726354} (2000i:90111).  \Zbl{911.90156}.						
\annot{} Max flow in a network with positive rational gains.  Multiple sources and sinks are allowed.  ``Relabeling'' is switching the gains.  Useful references to previous work.
\codes{\CGN:~\CSw,~\CAlgor,~\CRef}%
\itemspace

\author{Robert E.\ Tarjan}						%TARJAN RE
\authorspace
\authref  See \hyperlink{Andrew V.\ Goldberg}{A.V.~Goldberg}.
\itemspace

\author{Christos Tatakis}						%TATAKIS
\authorspace
\authref  See \hyperlink{Enrique Reyes}{E.~Reyes}.
\itemspace

\author{U.\ Tatt [W.T.\ Tutte]}						%TATT
\authorspace
\authref  See \hyperlink{W.T.\ Tutte}{W.T.~Tutte}.
\itemspace

\author{Behruz Tayfeh-Rezaie}						%TAYFEH-REZAIE
\authorspace
\authref  See \hyperlink{F.\ Ayoobi}{F.~Ayoobi} and \hyperlink{Leila Parsaei-Majd}{L.~Parsaei-Majd}.
\itemspace

\author{D.E.\ Taylor} 							%TAYLOR DE
\authorspace
\authref   See also \hyperlink{J.J.\ Seidel}{J.J.~Seidel}.
\authrefspace

\bsgitem{1977a}{} Regular 2-graphs.  \emph{Proc.\ Lond.\ Math.\ Soc.}\ (3) 35 (1977), 257--274.  
\MR{0476587} (57 \#16147).  \Zbl{362.05065}.
\annot{} Introducing two-graphs and regular two-graphs (defined by G.~Higman, unpublished).  [See \xref{Seidel}{1976a}{J.J.\ Seidel} etc.\ for more.]  
A ``two-graph'' is the class $\cC_{3}^-$ of negative triangles of a signed complete graph $(K_n,\sigma)$.  (See \S2, p.\ 258, where the group is $\bbZ_2 \cong \{+,-\}$ and the definition is in terms of the $2$-coboundary operator.)  Two-graphs and switching classes of signed complete graphs are equivalent concepts (stated in terms of Seidel switching in \S2, p.\ 260).
A two-graph is ``regular'' if every edge lies in the same number of negative triangles.  Thm.:  $\cC_{3}^-$ is regular iff $A(K_n,\sigma)$ has at most two eigenvalues.  Various parameters of regular two-graphs are calculated.
\codes{\CTG:~\CEig.~\CGeom}%
\itemspace

\author{Graeme Taylor}						%TAYLOR G
\authorspace
\bsgitem{2010a}{}  \emph{Cyclotomic Matrices and Graphs.}  Doctoral dissertation, University of Edinburgh, 2010.
\annot{} See \xrefd{2011a}{Graeme Taylor}.	
\codes{\CSG}%
\itemspace

\bsgitem{2011a}{}  Cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields.  \emph{J.\ Algebra} 331 (2011), no.\ 1, 523--545.
\MR{2774674} (2012b:\-15058).  \Zbl{1238.05166}.  \arXiv{1011.2737}.			
\codes{\CSG}%
\itemspace

\author{Herbert Taylor}						%TAYLOR H
\authorspace
\authref  See \hyperlink{Paul Erdos}{P.~Erd\H{o}s}.
\itemspace

\author{Howard F.\ Taylor} 						%TAYLOR HF
\authorspace
\bsgitem{1970a}{} \emph{Balance in Small Groups}.  Van Nostrand Reinhold, New York, 1970.
\annot{} A thorough and pleasantly written survey of psychological theories of balance, including formalizations by signed graphs (Chs.\ 3 and 6), experimental tests and critical evaluation of the formalisms, and so forth.  Ch.\ 2, ``Substantive models of balance'', takes the perspective of social psychology.  \S2.2, ``Varieties of balance theory'', reviews the theories of \xref{Heider}{1946a}{Fritz Heider} (the source of \xref{Harary's}{1953a}{Frank Harary} invention of signed graphs), \xref{Osgood and Tannenbaum}{1955a}{Charles E.\ Osgood \& Percy H.\ Tannenbaum}, and others.  \S2.2e, ``The Rosenberg-Abelson modifications'', discusses their introduction of the ``cost'' of change of relations, which led them (\xref{Abelson and Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg}) to propose the frustration index as a measure of imbalance.
\par
\codes{\CPsS:~\CSG,~\CWG:~\CExp,~\CRef}%
\annotctd{} Ch.\ 3, ``Formal models of balance'', reviews various graph-theoretic models: signed and weighted signed, different ways to weigh imbalance, etc., the relationship to theories in social psychology being constantly kept in mind.  \S3.1, ``Graph theory and balance theory'', presents the basics of balance, measures of degree of balance by circles (\xref{Cartwright and Harary}{1956a}{Dorwin Cartwright \& Frank Harary}), circles with strengths of edges (\xref{Morrissette}{1958a}{Julian O.\ Morrissette}), local balance and $N$-balance (\xref{Harary}{1955a}{Frank Harary})), edge deletion and negation (\xref{Abelson and Rosenberg}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg}, \xref{Harary}{1959b}{Frank Harary}), vertex frustration number (\xref{Harary}{1959b}{Frank Harary}).  \S3.2, ``Evaluation of formalizations: strong points'', and \S3.3, ``Evaluation of formalizations: weak points'', judged from the applied standpoint.  \S3.3a, ``Discrepancies between cycles or subsets of cycles'', suggests that differing degrees of imbalance among certain different subsets of the vertices may be significant [Is this reasonable?] and proposes measures, e.g., a variance measure (p.\ 71), of this ``discrepancy''. 
\codes{\CPsS:~\CSG,~\CWG:~\CBal,~\CFr:~\CExp}%
\annotctd{} Ch.\ 6, ``Issues involving formalization'', goes into more detail.  \S6.1, ``Indices of balance'', compares five indices, in particular \xref{Phillips'}{1967a}{J.L.\ Phillips} eigenvalue index (also in \xref{Abelson}{1967a}{Robert P.\ Abelson}) with examples to show that the index differentiates among different balanced signings of the same graph.  \S6.2, ``Extrabalance properties'', discusses \xref{Davis's}{1967a}{James A.\ Davis} clustering (\S6.2b) and indices of clustering (\S6.2c).  \S6.3, ``The problem of cycle length and non-local cycles''.  Are long circles less important?  Do circles at a distance from an actor (that is, a vertex) have less effect on the actor in balancing processes?
\codes{\CPsS:~\CSG:~\CFr,~\CAdj:~\CExp}%
\itemspace

\author{Siamak Tazari}						%TAZARI
\authorspace
\bsgitem{2010a}{}  Faster approximation schemes and parameterized algorithms on $H$-minor-free
and odd-minor-free graphs.  In:  Petr Hlin\v{e}n\'y and Anton\'{\i}n Ku\v{c}era, eds., \emph{Mathematical Foundations of Computer Science 2010} (35th Int.\ Sympos., Brno), pp.\ 641--652.  \LNCS 6281.  Springer, Berlin, 2010.
\MR{2727265 (2012g:68143)}.  \Zbl{1287.68183}.  
\annot{} Extended abstract of \xrefd{2012a}{Siamak Tazari}.
\codes{\Csg:~\Cpar:~\Cfr:~\CAlgor}%
\itemspace

\bsgitem{2012a}{}  Faster approximation schemes and parameterized algorithms on (odd-)$H$-minor-free graphs.  \emph{Theor.\ Computer Sci.}\ 417 (2012), 95--107.
\MR{2885892} (2012m:\-68486).  \Zbl{1241.05027}.  \arXiv{1004.3392} (prelim version).	
\codes{\Csg:~\Cpar:~\Cfr:~\CAlgor}%
\itemspace

\author{Mina Teicher}						%TEICHER
\authorspace
\authref  See \hyperlink{Meirav Amram, Robert Shwartz, \& Mina Teicher}{M.~Amram}.
\itemspace

\author{Roberto Tempo}						%TEMPO
\authorspace
\authref  See \hyperlink{Anton Proskurnikov}{A.V.~Proskurnikov}.
\itemspace

\author{Jeffrey C.Y.\ Teo}						%TEO JCY
\authorspace
\authref  See \hyperlink{AtMa P.O.\ Chan}{A.P.O.~Chan}.
\itemspace

\author{Hiroaki Terao}						%TERAO
\authorspace
\authref  See \hyperlink{Hidehiko Kamiya}{H.~Kamiya}.
\itemspace

\author{Hidetaka Terasaka} 						%TERASAKA
\authorspace
\authref  See \hyperlink{Shin'ichi Kinoshita}{S.~Kinoshita}.
\itemspace

\author{Lesley G.\ Terris}						%TERRIS
\authorspace
\authref  See \hyperlink{Zeev Maoz}{Z.~Maoz}.
\itemspace

\authorx{Evimaria Terzi}						%TERZI
\author{Evimaria Terzi \& Marco Winkler}
\authorspace
\bsgitem{2011a}{}  A spectral algorithm for computing social balance.  In:  Alan Frieze, Paul Horn, and Pawe{\l} Pra{\l}at, eds., \emph{Algorithms and Models for the Web Graph} (WAW 2011, 8th Int.\ Workshop, Atlanta, 2011), pp.\ 1--13.  \LNCS 6732.  Springer, Berlin, 2011.
\MR{2842308} (2012i:68186).  \Zbl{1327.68035}.  
\annot{} $\hat b :=$ fraction of balanced triangles, calculated via cubes of adjacency eigenvalues.
\dated{15 Jul 2019} 
\codes{\CSG:~\CFr:~\CAdj}%
\itemspace

\author{Jelena Te\v{s}i\'c}						%TESIC
\authorx{Jelena Tesic}
\authorspace
\authref  See \hyperlink{Ghadeer Alabandi}{G.~Alabandi}.
\itemspace

\author{Ambuj Tewari}						%TEWARI
\authorspace
\authref  See \hyperlink{Kai-Yang Chiang}{K.-Y.~Chiang}.
\itemspace

\author{Dirk Oliver Theis}						%THEIS
\authorspace
\authref  See \hyperlink{Nancy E.\ Clarke}{N.E.~Clarke}.
\itemspace

\author{Michel Thellier}						%THELLIER
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{Denis Thieffry}						%THIEFFRY
\authorspace
\authref  See also \hyperlink{Jean-Paul Comet}{J.-P.~Comet}, \hyperlink{Aurelien Naldi}{A.~Naldi}, \hyperlink{Elisabeth Remy}{\'E.~Remy}, and \hyperlink{Rene Thomas}{R.\ Thomas}.
\authrefspace

\bsgitem{2007a}{}  Dynamical roles of biological regulatory circuits.  \emph{Briefings Bioinformatics}\ 8 (2007), no.\ 4, 220--225.
\annot{} Survey of positive and negative cycles in biological regulation.
\dated{25 Jan 2015} 
\codes{\CSD:~\CDyn:~\CExp}%
\itemspace

\author{D.\ Thieffry, E.H.\ Snoussi, J.\ Richelle, \& R.\ Thomas}
\authorspace
\bsgitem{1995a}{}  Positive loops and differentiation.  \emph{J.\ Biol.\ Systems} 3 (1995), no.\ 2, 157--466.
\codes{\CSD}%
\itemspace

\author{Dimitrios M.\ Thilikos}						%THILIKOS
\authorspace
\authref  See \hyperlink{Christos Giatsidis}{C.~Giatsidis}.
\itemspace

\author{Morwen B.\ Thistlethwaite}					%THISTLETHWAITE
\authorspace
\bsgitem{1988a}{}  On the Kauffman polynomial of an adequate link.  \emph{Invent.\ Math.}\ 93 (1988), 285--296.  
\MR{0948102} (89g:57009).  \Zbl{645.57007}.						
\annot{} A $1$-variable Tutte-style polynomial $\Gamma_\Sigma$ of a sign-colored graph.  Fix an edge ordering.  For each spanning tree $T$ and edge $e$, let $\mu_T(e) = -A^{3\tau_T(e)\sigma(e)}$ if $e$ is active with respect to $T$, $A^{\tau_T(e)\sigma(e)}$ if it is inactive, where $\tau_T(e) = +1$ if $e \in T$, $-1$ if $e \notin T$.  Then $\Gamma_\Sigma(A) = \sum_T \prod_{e \in T} \mu_T(e)$.  
[In the notation of \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}, $\Gamma_\Sigma(A) = Q_\Sigma$ with $a_\epsilon = A^{-\epsilon}, b_\epsilon = A^\epsilon$ for $\epsilon = \pm1$ and $u = v = -(A^2 + A^{-2})$.]  
\S\S3 and 4 show $\Gamma_\Sigma$ is independent of the ordering.  Other sections derive consequences for knot theory.  
[This marks the invention of a Tutte-style polynomial of a colored, or parametrized or weighted, graph or matroid, developed in \xref{Kauffman}{1989a}{Louis H.\ Kauffman} and successors.]
\codes{\CSGc:~\CKnot:~\CInvar}%
\itemspace

\author{Apostolos Thoma}						%THOMA
\authorspace
\authref  See \hyperlink{Enrique Reyes}{E.~Reyes}.
\itemspace

\author{A.D.\ Thomas} 						%THOMAS AD
\authorspace
\authref  See \hyperlink{F.W.\ Clarke}{F.W.~Clarke}.
\itemspace

\authorx{Creighton K.\ Thomas}						%THOMAS CK
\author{Creighton K.\ Thomas, David A.\ Huse, \& A.\ Alan Middleton}
\authorspace
\bsgitem{2011a}{}  Zero- and low-temperature behavior of the two-dimensional $\pm J$ ising spin glass.  \emph{Phys.\ Rev.\ Lett.}\ 107 (2011), art.\ 047203, 4 pp.
\arXiv{1103.1946}.  
\annot{} A droplet model of a signed square lattice shows long-range correlations (spin-glass behavior) in the ground state.	
\dated{3 Jan 2015} 
\par
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\author{Creighton K.\ Thomas \& A.\ Alan Middleton}
\authorspace
\bsgitem{2009a}{}  Exact algorithm for sampling the two-dimensional Ising spin glass.  \emph{Phys.\ Rev.\ E} 80 (2009), art.\ 046708, 16 pp.
\annot{} Both pure signed graphs ($\pm J$ model) and randomly weighted ones (Gaussian model), using the Kasteyn and Temperley--Fisher decoration and Pfaffian method.
	
\dated{10 Jan 2015} 
\par
\codes{\CSG,~\CWG:~\CState(\Cfr),~\CPhys,~\CAlgor}%
\itemspace

\bsgitem{2013a}{}  Numerically exact correlations and sampling in the two-dimensional Ising spin glass.  \emph{Phys.\ Rev.\ E} 87 (2013), art.\ 043303, 16 pp.
\arXiv{1301.1252}.  
\annot{} Both pure signed graphs ($\pm J$ model) and randomly weighted ones (Gaussian model), using the Kasteyn and Temperley--Fisher decoration and Pfaffian method.
	
\dated{10 Jan 2015} 
\par
\codes{\CSG,~\CWG:~\CState(\Cfr),~\CPhys,~\CAlgor}%
\itemspace

\author{Ren\'e Thomas}						%THOMAS R
\authorx{Rene Thomas}
\authorspace
\authref  See also \hyperlink{Jacques Demongeot}{J.~Demongeot}, \hyperlink{Marcelle Kaufman}{M.~Kaufman}, \hyperlink{J.\ Leclercq}{J.~Leclercq}, \hyperlink{El Houssine Snoussi}{E.H.~Snoussi}, and \hyperlink{Denis Thieffry}{D.~Thieffry}.
\authrefspace

\bsgitem{1973a}{}  Boolean formalization of genetic control circuits.  \emph{J.\ Theor.\ Biol.}\ 42 (1973), no.\ 3, 563--585.  
Errata.  \emph{Ibid.}\ 44 (1974), no.\ 2, 44.
\annot{} A main progenitor of a large field of inquiry about biological and chemical regulatory systems with positive and negative feedback.  [See, e.g., \hyperlink{Julio Aracena}{J.~Aracena}, \hyperlink{Elisabeth Remy}{\'E.~Remy}, \hyperlink{Adrien Richard}{A.~Richard}, \hyperlink{Heike Siebert}{H.~Siebert}, R.~Thomas, and their many coauthors.]  The diagrams show the embryonic appearance of signed digraphs.
\dated{25 Apr 2014} 
\codes{\CBiol:~\Csd:~\CDyn}%
\itemspace

\bsgitem{1978a}{}  Logical analysis of systems comprising feedback loops.  \emph{J.\ Theor.\ Biol.}\ 73 (1978), no.\ 4, 631--656.
\codes{\Csd:~\CDyn,~\CBiol}%
\itemspace

\bsgitem{1979a}{}  The dynamic behavior of boolean systems comprising feedback loops.  
In:  Ren\'e Thomas, ed., \emph{Kinetic Logic:  A Boolean Approach to the Analysis of Complex Regulatory Systems} (Proc.\ EMBO Course, Brussels, 1977), Ch.\ VII, pp.\ 127--142.  Lect.\ Notes in Biomath., Vol.\ 29.  Springer-Verlag, Berlin, 1979. 
\annot{} Describes dynamics of very simple signed digraphs with up to two cycles.  E.g.:  One positive cycle leads to one of two steady states.  One negative cycle implies cycling states.  With two cycles having one common vertex, both positive are like one positive cycle.  Both negative allow for multiple cyclic states.  One of each sign allow both a steady state and cyclic states.
\dated{4 Aug 2018} 
\codes{\CSD:~\CDyn,~\CChem}%
\itemspace

\bsgitem{1981a}{}  On the relation between the logical structure of systems and their ability to generate multiple steady states and sustained oscillations.  
In:  J.\ Della Dora, Jacques Demongeot, \& B.\ Lacolle, eds., \emph{Numerical Methods in the Study of Critical Phenomena} (Proc.\ Colloq., Carry-le-Rouet, 1980), pp.\ 180--193.  Springer Ser.\ Synergetics, Vol.\ 9.  Springer, Berlin, 1981.
\MR{0660499} (83g:92037).  \Zbl{0489.92025}.					
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{1994a}{}  The role of feedback circuits: Positive feedback circuits are a necessary condition for positive real eigenvalues of the Jacobian matrix.  \emph{Berichte Bunsenges.\ phys.\ Chem.}\ 98 (1994), no.\ 9, 1148--1151.
\codes{\CSD:~\CDyn}%
\itemspace

\bsgitem{1996a}{}  Analyse et synth\`ese de syst\`emes \'a dynamique chaotique en termes de circuits de r\'etroaction (feedback circuits).  (In French.)  \emph{Acad.\ Roy.\ Belg.\ Bull.\ Cl.\ Sci.}\ (6) 7 (1996), no.\ 1-6, 101--124 (1997).
\MR{1475761} (98h:58121).  \Zbl{1194.94211}.  
\codes{\Csd:~\CDyn}%
\itemspace

\bsgitem{2006a}{}  Nullclines and nullcline intersections.  \emph{Int.\ J.\ Bifurcation Chaos} 16 (2006), no.\ 10, 3023--3033.
\MR{2283557} (2007g:34092).  \Zbl{1146.34308}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Ren\'e Thomas \& Richard D'Ari}
\authorx{Rene Thomas \& Richard D'Ari}
\authorspace
\bsgitem{1990a}{}  \emph{Biological Feedback}.  CRC Press, Boca Raton, 1990.
\Zbl{743.92003}.  
\par
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Ren\'e Thomas \& Marcelle Kaufman}
\authorx{Rene Thomas \& Marcelle Kaufman}
\authorspace
\bsgitem{2005a}{}  Frontier diagrams:  Partition of phase space according to the signs of eigenvalues or sign patterns of the circuits.  \emph{Int.\ J.\ Bifurcation Chaos} 15 (2005), no.\ 10, 3051--3074.
\MR{2192633} (2006m:37037).  \Zbl{1093.37502}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{R.\ Thomas \& J.\ Richelle}
\authorspace
\bsgitem{1986a}{}  Boolean and continuous analyses of systems containing feedback loops.  IV.\ Positive feedback and multistationarity.  \emph{Acad.\ Roy.\ Belg.\ Bull.\ Cl.\ Sci.}\ (5) 72 (1986), no.\ 11, 435--453. 
\MR{0637745} (84h:92020a).  
\codes{\Csd:~\CDyn}%
\itemspace

\bsgitem{1988a}{}  Positive feedback loops and multistationarity.  \emph{Discrete Appl.\ Math.}\ 19 (1988), 381--386.
\MR{0936224} (89g:92007).  \Zbl{639.92003}.					
\codes{\Csd:~\CDyn}%
\itemspace

\author{Ren\'e Thomas, Denis Thieffry, \& Marcelle Kaufman}
\authorx{Rene Thomas, Denis Thieffry, \& Marcelle Kaufman}
\authorspace
\bsgitem{1995a}{}  Dynamical behaviour of biological regulatory networks---I.  Biological role of feedback loops and practical use of the concept of the loop-characteristic state.  \emph{Bull.\ Math.\ Biol.}\ 57 (1995), no.\ 2, 247--276.
\Zbl{821.92009}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Robin Thomas}						%THOMAS R
\authorspace
\authref  See also \hyperlink{William McCuaig}{W.~McCuaig} and \hyperlink{Neil Robertson}{N.~Robertson}.
\authrefspace

\author{Robin Thomas \& Peter Whalen}
\authorspace
\bsgitem{2016a}{}  Odd $K_{3,3}$ subdivisions in bipartite graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 118 (2016), 76--87.
\MR{3471845}.  \Zbl{1332.05116}.  
\annot{} An ``odd $K_{3,3}$'' is an all-negative subdivision of $-K_{3,3}$, treated as unsigned.
\codes{\Csg:~\CPar:~\CStr}%
\itemspace

\author{Andrew Thomason}						%THOMASON
\authorspace
\bsgitem{1988a}{}  A graph property not satisfying a ``zero-one law''.  \emph{European J.\ Combin.}\ 9 (1988), 517--521.
\MR{0970386} (90e:05051).  \Zbl{675.05057}.						
\annot{} The property is the existence of an Eulerian cut.  The asymptotic probability is $.57\ldots$.  [\Problem.  Generalize to gain graphs with finite gain group, esp.\ to signed graphs.  The property is that of being switchable so that the identity-gain edges form an Eulerian subgraph.  (This has various meanings.)  Variation:  The property is that of having a maximal balanced subgraph that is Eulerian.  One expects the asymptotic probabilities to be the same for both problems and to depend only on the group's order.]
\codes{\Cpar:~\CRand}%
\itemspace

\author{Carsten Thomassen} 						%THOMASSEN
\authorspace
\authref  See also \hyperlink{P.D.\ Seymour}{P.D.~Seymour}.
\authrefspace

\bsgitem{1985a}{}  Even cycles in directed graphs.  \emph{European J.\ Combin.}\ 6 (1985), 85--89.  
\MR{0793491} (86i:05098).  \Zbl{606.05039}.					
\annot{} It is an NP-complete problem to decide whether a given signed digraph has a positive but not all-positive cycle, even if there are only $2$ negative arcs.  This follows from Lemma 3 of Steven Fortune, John Hopcroft, and James Wyllie, The directed subgraph homeomorphism problem (see \emph{Theor.\ Computer Sci.}\ 10 (1980), 111--121.  \MR{0551599} (81e:68079).  \Zbl{419.05028}.) by the simple argument in the proof of Prop.\ 2.1 here.  
\annotctd{} To decide whether a specified arc of a digraph lies in an even cycle, or in an odd cycle, are NP-complete problems (Prop.\ 2.1).  To decide existence of an even cycle [hence, by the negative subdivision trick, of a positive cycle in a signed digraph] is difficult [but is solvable in polynomial time; see \xref{Robertson, Seymour, and Thomas}{1999a}{Neil Robertson, P.D.\ Seymour, \& Robin Thomas}], although existence of an odd cycle [resp., of a negative cycle] is easy, by a trick here attributed to Edmonds (unpublished).  Prop.\ 2.2: Deciding existence of a positive cycle in a signed digraph is polynomial-time solvable if $\#E^-$ is bounded.  Thm.\ 3.2: If the outdegrees of a digraph are all $> \log_2 n$, then \emph{every signing} has a positive cycle, and this bound is best possible; restricting to the all-negative signature, the lower bound might (it's not known) go down by a factor of up to $2$, but certainly (Thm.\ 3.1) a constant minimum on outdegree does not imply existence of an even cycle.  [See \xrefd{1992a}{Carsten Thomassen} for the effect of connectivity.]
\codes{\CSD,~\CPar:~\CBal,~\CAlgor}%
\itemspace

\bsgitem{1986a}{}  Sign-nonsingular matrices and even cycles in directed graphs.  \emph{Linear Algebra Appl.}\ 75 (1986), 27--41.  
\MR{0825397} (87k:05120).  \Zbl{589.05050}.						
Erratum.  \emph{Ibid.}\ 240 (1996), 238.  
\MR{1387301}.  
\codes{\CQM,~\Csd:~\Cpar:~\CSol,~\Cbal,~\CAlgor}%
\itemspace

\bsgitem{1988a}{}  Paths, circuits and subdivisions.  In: Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Selected Topics in Graph Theory 3}, Ch.\ 5, pp.\ 97--131.  Academic Press, London, 1988.
\MR{1205394} (93h:05003) (book).  \Zbl{659.05062}.					
\annot{} \S8:  ``Even directed circuits and sign-nonsingular matrices.''
\par
\codes{\CSD,~\CQM:~\CBal,~\CSol:~\CExp}%
\annotctd{} \S\S8--10 treat even cycles in digraphs. 
\codes{\CSD:~\CBal:~\CExp}%
\annotctd{} [\emph{General} \Problem.  Generalize even-cycle and odd-cycle results to positive and negative cycles in signed digraphs, the unsigned results corresponding to all-negative signatures.]
\itemspace

\bsgitem{1988b}{}  On the presence of disjoint subgraphs of a specified type.  \emph{J.\ Graph Theory} 12 (1988), 101--111.
\MR{0928740} (89e:05174).  \Zbl{662.05032}.
\annot{} There is an algorithm for detecting a balanced circle in a $\bbZ_m$-gain graph.  Balance of such a gain graph is characterized.
	
\codes{\Cgg:~\CBal,~\CCircles:~\CAlgor}%
\itemspace

\bsgitem{1989a}{}  When the sign pattern of a square matrix determines uniquely the sign pattern of its inverse.  \emph{Linear Algebra Appl.}\ 119 (1989), 27--34.  
\MR{1005232} (90f:05099).  \Zbl{673.05067}.						
\codes{\CQM,~\CSD:~\CSol,~\CAdj}%
\itemspace

\bsgitem{1990a}{}  Embeddings of graphs with no short noncontractible cycles.  \emph{J.\ Combin.\ Theory Ser.\ B} 48 (1990), 155--177.
\MR{1046752} (91b:05069).  \Zbl{704.05011}.						
\annot{} \S5 describes the ``fundamental cycle method'', a simple algorithm for a shortest unbalanced circle in a biased graph (Thm.\ 5.1).  Thus the method finds a shortest noncontractible circle (Thm.\ 5.2).  A noteworthy linear class: the surface-separating (``$\Pi$-separating'') circles (p.\ 166).
Dictionary:  ``$3$-path-condition'' on a class $F$ of circles = property that $F^c$ is a linear class.  ``M\"obius cycle'' = negative circle in the signature induced by a nonorientable embedding (only on p.\ 166).
\par
\codes{\Cgg,~\Csg:~\CCircles:~\CAlgor,~\CTop}%
\itemspace

\bsgitem{1992a}{}  The even cycle problem for directed graphs.  \emph{J.\ Amer.\ Math.\ Soc.}\ 5 (1992), 217--229.  
\MR{1135027} (93b:05064).  \Zbl{760.05051}. 						
\annot{} A digraph that is strongly connected and has all in- and out-degrees $\geq 3$ contains an even cycle.
\codes{\Csd:~\Cpar:~\CCycles}%
\itemspace

\bsgitem{1993a}{}  The even cycle problem for planar digraphs.  
\emph{J.\ Algorithms} 15 (1993), 61--75.  
\MR{1218331} (94d:05077).  \Zbl{784.68045}.						
\annot{} A polynomial-time algorithm for deciding the existence of an even cycle in a planar digraph.
\codes{\Csd:~\Cpar:~\CCycles:~\CAlgor}%
\itemspace

\bsgitem{1994a}{}  Embeddings of graphs.  Graphs and Combinatorics (Qawra, 1990).  \emph{Discrete Math.}\ 124 (1994),217--228.
\MR{1258855} (95f:05035).  \Zbl{797.05035}.
\annot{} P.~225 and Thm.\ 6.3: the ``$3$-path-condition'' and shortest unbalanced circle algorithm from \xrefd{1990a}{Carsten Thomassen}.  Examples mentioned (under other names) are parity bias (all-negative signs) [underlying the even-circle matroid of \xref{Tutte}{1981a}{W.T.\ Tutte} and \xref{Doob}{1973a}{Michael Doob} via \xref{Zaslavsky}{1989a}{Thomas Zaslavsky}], poise bias [underlying a matroid of \xref{Matthews}{1978c}{Laurence R.\ Matthews}], and noncontractible or orientation-reversing embedded circles [for the latter see esp.\ \xref{Lins}{1985a}{Sostenes Lins} and \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}].
\codes{\Cgg,~\Cpar:~\CExp}%
\itemspace

\bsgitem{2001a}{}  The Erd\"os--P\'osa property for odd cycles in graphs of large connectivity.  Paul Erd\"os and His Mathematics (Budapest, 1999).  \emph{Combinatorica} 21 (2001), no.\ 2, 321--333.
\MR{1832455} (2002c:05108).  \Zbl{989.05062}.						
\annot{} Given $k$, there exists $K$ such that every sufficiently connected graph has $k$ vertex-disjoint odd circles or $K$ vertices whose deletion leaves a bipartite graph.  
[\Problem.  Given $k$, there exists $K$ such that every sufficiently connected signed graph has $k$ vertex-disjoint negative circles or $K$ vertices whose deletion leaves a balanced graph.]
\dated{rev.\ 26 Dec 2012}
\codes{\Cpar:~\CFr:~\CCircles}%
\itemspace

\bsgitem{2001b}{}  Totally odd $K_4$-subdivisions in 4-chromatic graphs.  \emph{Combinatorica} 21 (2001), no.\ 3, 417--443.
\MR{1848060} (2002e:05058).  \Zbl{1012.05064}.					
\annot{} Re-proves \xref{Zang}{1998b}{Wenan Zang} (\xref{Toft's}{1975a}{Bjarne Toft} conjecture).  [\Question.  What is the signed-graph generalization?]
\dated{rev.\ 26 Dec 2012, 29 Oct 2017}
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\author{G.L.\ Thompson} 						%THOMPSON
\authorspace
\authref  See \hyperlink{V.\ Balachandran}{V.~Balachandran}.
\itemspace

\authorx{Christopher Thraves}						%THRAVES
\author{Christopher Thraves Caro [Christopher Thraves]}
\authorspace
\authref  See \hyperlink{Felipe Benitez}{F.~Benitez}, \hyperlink{Anne-Marie Kermarrec}{A.-M.~Kermarrec}, \hyperlink{Eduardo G.\ Pardo}{E.G.~Pardo}, and \hyperlink{Quico Spaen}{Q.~Spaen}.
\itemspace

\author{Florence Thuderoz}						%THUDEROZ
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{Fenglei Tian}							%TIAN FL
\authorspace
\authref  See also \hyperlink{Xiaobin Ma}{X.B.~Ma}, \hyperlink{Sai Wang}{S.~Wang}, \hyperlink{Feng Xu}{F.~Xu}, and \hyperlink{Min Zhu}{M.~Zhu}.
\authrefspace

\author{Fenglei Tian, Li Chen, \& Rui Chu}	
\authorspace
\bsgitem{2018a}{}  Rank of the Hermitian-adjacency matrix of a mixed graph in terms of matching number.  \emph{Ars Combin.}\ 37 (2018), 221--232.
\MR{3790972}.  \Zbl{06890323}.  
\annot{} Generalizes \xref{Ma--Wong--Tian}{2016a}{Xiaobin Ma, Dein Wong, \& Fenglei Tian} to $2\mu(\Gamma) - 2\xi(\Gamma) \leq \rk A(\Phi) \leq 2\mu(\Gamma) + \xi(\Gamma)$ for gain graph $\Phi$ with gains in $\{\pm1,\pm\mathrm{i}\}$.  Characterizes equalities.  
[A different matching formula is in \xref{Chen--Huang--Li}{2018a}{Chen Chen, Jin Huang, \& Shuchao Li}.]
\dated{10 May 2019, rev 16 Oct 2020} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Fenglei Tian, Xiaoming Li, \& Jianling Rou}
\authorspace
\bsgitem{2014a}{}  A note on the signless Laplacian and distance signless Laplacian eigenvalues of graphs.  \emph{J.\ Math.\ Res.\ Appl.}\ 34 (2014), no.\ 6, 647--654.
\MR{3288067}.  \Zbl{1324.05120}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Fenglei Tian, Dengyin Wang, \& Min Zhu}
\authorspace
\bsgitem{2016a}{}  A characterization of signed planar graphs with rank at most 4.  \emph{Linear Multilinear Algebra} 64 (2016), no.\ 5, 807--817.
\MR{3479381}.  \Zbl{1335.05079}.  
\annot{} Characterized:  All signed graphs with $\rk A(\Sigma)=2,3$ and signed planar graphs with $\rk A(\Sigma)=4$.
\dated{22 Jan 2016} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{Fenglei Tian \& Dein Wong}
\authorspace
\bsgitem{2017a}{}  Relation between the skew energy of an oriented graph and its matching number.  \emph{Discrete Appl.\ Math.}\ 222 (2017), 179--184.
\Zbl{1396.05078}.  
\annot{} Given a digraph.  In an oriented circle $C$, an edge has sign $+1$ or $-1$ depending on whether it is consistent or inconsistent with $C$.  An even circle is ``evenly/oddly oriented" if its sign product is $+1$ or $-1$.
\dated{15 Aug 2022} 
\codes{\Csg}%

\bsgitem{2018a}{}  Nullity of Hermitian-adjacency matrices of mixed graphs.  \emph{J.\ Math.\ Res.\ Appl.}\ 38 (2018), no.\ 1, 23--33.
\MR{3752028}.  \Zbl{1413.05249}.  
\annot{} For $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$ ($\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges), characterizes the spectrally unique ones with $\rk A(\Phi) = 3$.
\dated{15 Dec 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Gui-Xian Tian}						%TIAN GX
\authorspace
\authref  See also \hyperlink{Shu-Yu Cui}{S.-Y.~Cui}.
\authrefspace

\author{Gui-Xian Tian, Ting-Zhu Huang, \& Bo Zhou}
\authorspace
\bsgitem{2009a}{}  A note on sum of powers of the Laplacian eigenvalues of bipartite graphs.  \emph{Linear Algebra Appl.}\ 430 (2009), no.\ 8-9, 2503--2510.
\MR{2508309} (2010e:05191).  \Zbl{1165.05020}.  
\annot{} A lower bound on $\sum_i \lambda_i(\Lap(\Gamma))^\alpha$, over nonzero eigenvalues, for bipartite $\Gamma$ and $\alpha \in \bbR^\times$.  
[\Question.  Is there a nonbipartite generalization involving $\Lap(-\Gamma)$?]
\dated{23 Jan 2012} 
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\authorx{Xiao-Jun Tian}						%TIAN XJ
\author{Xiao-Jun Tian, Xiao-Peng Zhang, Feng Liu, \& Wei Wang}
\authorspace
\bsgitem{2009a}{}  Interlinking positive and negative feedback loops creates a tunable motif in gene regulatory networks.  \emph{Phys.\ Rev.\ E} 80 (2009), no.\ 1, 011926.
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\author{Yi Tian}							%TIAN Y
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.C.~Li}.
\itemspace

\author{Jonathan Tidor}						%TIDOR
\authorspace
\authref  See \hyperlink{Zilin Jiang}{Z.-L.~Jiang}.
\itemspace

\author{R.M.\ Tifenbach}						%TIFENBACH
\authorspace
\bsgitem{2011a}{}  Strongly self-dual graphs.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 12, 3151--3167.
\MR{2831603} (2012h:05202).  \Zbl{1226.05170}.  
\annot{} \Cf\ \xref{Tifenbach and Kirkland}{2009a}{R.M.\ Tifenbach \& S.J.\ Kirkland}.  
An $h$-graph $\Gamma$ is ``self-dual'' if it has inverse $\Sigma$ and $\Gamma\cong|\Sigma|$, ``strongly self-dual'' if $\Gamma=\Sigma$.  
Thm.\ 3.2 is \xref{Tifenbach and Kirkland}{2009a}{R.M.\ Tifenbach \& S.J.\ Kirkland} Thm.\ 2.5 with strong self-duality instead of duality.  \S4, ``Constructions of strongly self-dual graphs''.  
\S5, ``Eigenvalues of self-dual $h$-graphs'':  For an eigenvalue $\lambda$ of a self-dual $h$-graph, $-\lambda$ and $\pm1/\lambda$ are eigenvalues.  $\lambda=\pm1$ if rational.  $\pm1$ has multiplicity $\equiv m (\mod 2)$.  Thm.\ 5.5:  Let $k^-:=$ \# of vertices switched in changing $\Sigma$ to $\Gamma^+$; then $\lambda=\pm1$ has multiplicity $\geq |m-2k^-|$.  Examples.
\dated{4 May 2017} 
\codes{\Csg:~\CAdj,~\CEig}%
\itemspace

\author{R.M.\ Tifenbach \& S.J.\ Kirkland}
\authorspace
\bsgitem{2009a}{}  Directed intervals and the dual of a graph.  \emph{Linear Algebra Appl.}\ 431 (2009), nos.\ 5-7, 792--807.
\MR{2535551} (2010m:05185).  \Zbl{1226.05171}.  
\annot{} Inspired by \xref{Godsil}{1985a}{C.D.\ Godsil} \etal\ \  Graphs are simple.  An ``$h$-graph'' is bipartite with left set $\{u_1,\ldots,u_m\}$, right set $\{v_1,\ldots,v_m\}$, and a unique perfect matching $M=\{u_iv_i\}_i$.  
Thm.\ 1.1: $\exists$ labelling so every edge $u_iv_j$ has $i\leq j$ (\xref{Simion and Cao}{1989a}{R.\ Simion \& D.-S.\ Cao}).  
Thus, $\exists$ vertex labelling and partial order $P_\Gamma$ on $\{v_1,\ldots,v_p\}$ so every edge $u_iv_j$ has $i\leq j$; this gives an acyclic digraph.  
If $A(\Gamma)\inv=A(\Sigma)$ for some $\Sigma$, then $\Gamma^+:=|\Sigma|$ (``dual'' of $\Gamma$) is an $h$-graph, $\Sigma$ is balanced, and every covering edge of $P_{\Gamma^+}$ is negative in $\Sigma$ (Thm.\ 2.3).  Thm.\ 2.4: $P_\Gamma \cong P_{\Gamma^+}$.  Thm.\ 2.5:  Intervals of $P_\Gamma$ have duals; intervals respect duality.  Thm.\ 2.6:  $\Gamma^+$ exists iff $P_\Gamma$ has bipartite Hasse diagram and all intervals have duals.  [The former is due to balance and the negative covering edges (i.e., antibalance).]  \S\S3--4:  Examples.
\dated{4 May 2017} 
\codes{\Csg:~\CAdj}%
\annotctd{}  [\Problem.  Generalize this and other $h$-graph research to bipartite signed graphs with unique perfect matching, so having a signed-graphic inverse is natural.  What do balance and the negative covering edges (antibalance) of the inverse digraph become?]
\dated{4 May 2017} 
\codes{\CSG:~\CAdj}%
\itemspace

\author{Shailesh K.\ Tipnis}						%TIPNIS
\authorspace
\authref  See \hyperlink{Arthur H.\ Busch}{A.H.~Busch}, \hyperlink{Ajit A.\ Diwan}{A.A.~Diwan}, and \hyperlink{Heather Jordon}{H.~Jordon}.
\itemspace

\author{R.L.\ Tobin} 							%TOBIN
\authorspace
\bsgitem{1975a}{} Minimal complete matchings and negative cycles.  \emph{Networks} 5 (1975), 371--387.  
\MR{0395786} (52 \#16578).  \Zbl{348.90151}.
\codes{\Csg:~\Cvs}
\itemspace

\author{Bjarne Toft}							%TOFT
\authorspace
\authref   See also \hyperlink{Tommy R.\ Jensen}{T.R.~Jensen} and \hyperlink{Uffe Krusenstjerna-Hastrom}{U.~Krusenstjerna-Hastr{\o}m}.
\authrefspace

\bsgitem{1975a}{}  Problem 10.  In:  M.\ Fiedler, ed., \emph{Recent Advances in Graph Theory} (Proc.\ Second Czechoslovak Symp., Prague, 1974), pp.\ 543--544.  Academia Praha, 1975.
\MR{0363962} (51 \#217) (book).  \Zbl{316.00007} (book).				
\annot{} Proposes that for every 4-chromatic graph $\Gamma$, $-\Gamma$ contains a subdivision of $-K_4$ (that means every $K_4$ edge subdivides into an odd path).  
[Proved by \xref{Zang}{1998b}{Wenan Zang}, \xref{Thomassen}{2001b}{Carsten Thomassen}.  \Cf\ \xref{Krusenstjerna-Hastr{\o}m and Toft}{1980a}{Uffe Krusenstjerna-Hastrom \& Bjarne Toft} and \xref{Jensen and Shepherd}{1995a}{T.R.\ Jensen \& F.B.\ Shepherd}.]
\dated{29 Oct 2017} 
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\author{Sivan Toledo}						%TOLEDO
\authorspace
\authref  See \hyperlink{Erik G.\ Boman}{E.G.~Boman} and \hyperlink{Doron Chen}{D.~Chen}.
\itemspace

\author{Ioan Tomescu} 						%TOMESCU
\authorspace
\authref   See also \hyperlink{Dragos Popescu}{D.R.~Popescu}.
\authrefspace

\bsgitem{1973a}{}  Note sur une caracterisation des graphes dont le degr\'e de des\'equilibre est maximal.  \emph{Math.\ Sci.\ Humaines}, No.\ 42 (1973), 37--40.  
\MR{0366757} (51 \#3003).  \Zbl{266.05115}.					
\annot{} Independent proof of \xref{Petersdorf's}{1966a}{M.\ Petersdorf} Satz 1.  Also, treats similarly a variation on the frustration index.
	
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{1974a}{}  La r\'eduction minimale d'un graphe \`a une r\'eunion de cliques.  \emph{Discrete Math.}\ 10 (1974), 173--179.  
\MR{0363992} (51 \#247).  \Zbl{288.05127}.					
\annot{} The fewest sign changes needed to make $(K_n,\sigma)$ clusterable is $\leq \lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor$.
\dated{6 Jan 2017}
\codes{\CSG:~\CBal,~\CClu}%
\itemspace


\bsgitem{1976a}{} Sur le nombre des cycles n\'egatifs d'un graphe complet sign\'e.  \emph{Math.\ Sci.\ Humaines}, No.\ 53 (1976), 63--67.  
\MR{0457285} (56 \#15493).  \Zbl{327.05119}.					
\annot{} Consider $(K_n,\sigma)$ with $\#E^-=p$.  The parity of the number of negative triangles = that of $np$.  The number of negative $t$-gons, for $t \geq 4$, is even [strengthened in \xref{Popescu}{1991a}{Dragos Popescu}, \xrefd{1996a}{Dragos Popescu}].  
[\xref {Kittipassorn \& M\'esz\'aros}{2015a}{Teeradej Kittipassorn \& Gabor Meszaros} performs a detailed study of the number of negative triangles.]
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{1978a}{} Problem 2.  In:  A.~Hajnal and Vera T.\ S\'os, eds., \emph{Combinatorics} (Proc.\ Fifth Hungarian Colloq., Keszthely, 1976), Vol.\ II, p.\ 1217.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 18.  J\'anos Bolyai Math.\ Soc., Budapest, and North-Holland, Amsterdam, 1978.
\MR{0519295} (80a:05002b) (book).  \Zbl{378.00007}.
\codes{\CSG:~\CBal}%
\itemspace

\author{Mark Tomforde}						%TOMFORDE
\authorspace
\authref  See \hyperlink{Bernhard G.\ Bodmann}{B.G.~Bodmann}.
\itemspace

\authorx{Joanna Tomkowicz}						%TOMKOWICZ
\author{Joanna Tomkowicz \& Krzysztof Ku{\l}akowski}
\authorx{Joanna Tomkowicz \& Krzysztof Kulakowski}
\authorspace
\bsgitem{2010a}{}  Scaling of connected spin avalanches in growing networks.  \emph{Phys.\ Rev.\ E} 81 (2010), art.\ 052101, 4 pp.
\arXiv{0904.2697}.  
\codes{\Cpar:~\CState(\Cfr)}%
\itemspace

\author{C.B.\ Tompkins} 						%TOMPKINS
\authorspace
\authref  See \hyperlink{I.\ Heller}{I.~Heller}.
\itemspace

\author{Arnaud Tonnelier}						%TONNELIER
\authorspace
\authref  See \hyperlink{Jacques Demongeot}{J.~Demongeot}.
\itemspace

\author{Hatice Topcu}						%TOPCU
\authorspace
\authref  See \hyperlink{Willem H.\ Haemers}{W.H.~Haemers}.
\itemspace

\authorx{J.\ Topp}							%TOPP
\author{J.\ Topp \& W.\ Ulatowski}
\authorspace
\bsgitem{1987a}{}  On functions which sum to zero on semicycles.  \emph{Zastosowanie Mat.} (\emph{Applicationes Math}.) 19 (1987), 611--617.  
\MR{0951376} (89i:05138).  \Zbl{719.05044}.					
\annot{} An additive real gain graph is balanced iff every circle in a circle basis is balanced, iff the gains are induced by a vertex labelling [in effect, switch to $0$], iff every two paths with the same endpoints have the same gains.  A digraph is gradable (\xref{Harary, Norman, and Cartwright}{1965a}{Frank Harary, Robert Z.\ Norman, \& Dorwin Cartwright}; also see \xref{Marcu}{1980a}{Danut Marcu}) iff $\phi_1$ is balanced, where for each arc $e$, $\phi_1(e) = 1 \in \bbZ$ (Thm.\ 3).  The Windy Postman Problem (Thms.\ 4, 5).
\codes{\CGG,~\CGD:~\CBal}%
\itemspace

\author{Aleksandar Torga\v{s}ev}					%TORGASEV
\authorx{Aleksandar Torgasev}
\authorspace
\authref  See also \hyperlink{Dragos M.\ Cvetkovic}{D.M.~Cvetkovi\'c}. 
\authrefspace

\bsgitem{1982a}{}  The spectrum of line graphs of some infinite graphs.  \emph{Publ.\ Inst.\ Math.\ (Beograd) (N.S.)} 31(45) (1982), 209--222.  
\MR{0710960} (85d:05175).  \Zbl{526.05039}.
\annot{} An infinite analog of \xref{Doob's}{1973a}{Michael Doob} characterization via the even-cycle matroid of when a line graph has $-2$ as an eigenvalue.  [\Problem.  Generalize to line graphs of infinite signed graphs.]
	
\codes{\Cpar:~\CEig(\CLG)}%
\itemspace

\bsgitem{1983a}{}  A note on infinite generalized line graphs.  In:  D.~Cvetkovi\'c \etal, eds., \emph{Graph Theory} (Proc.\ Fourth Yugoslav Seminar,  Novi Sad, 1983), pp.\ 291--297.  Univ.\ Novom Sadu, Inst.\ Mat., Novi Sad, 1984.  
\MR{0751456} (85i:05168).  \Zbl{541.05042}.					
\annot{} An infinite graph is a generalized line graph iff its least ``limit'' eigenvalue $\geq -2$.  [\Problem.  Generalize to line graphs of infinite signed graphs.]
\codes{\Cpar:~\CEig(\CLG)}%
\itemspace

\author{Michele Torielli}						%TORIELLI
\authorspace
\authref  See also \hyperlink{Weili Guo}{W.-L.~Guo} and \hyperlink{Daisuke Suyama}{D.~Suyama}.
\authrefspace

\author{Michele Torielli \& Shuhei Tsujie}
\authorspace
\bsgitem{2020a}{}  Freeness of hyperplane arrangements between boolean arrangements and Weyl arrangements of type $B_l$.  \emph{Electronic J.\ Combin.}\ 27 (2020), no.\ 3, art.\ P3.10, 15 pp.
\MR{4245123}.  \Zbl{1444.52011}.  \arXiv{1807.02432}.  
\codes{\CSG:~\CGeom:~\CAlgeb}%
\itemspace

\author{Juan R.\ Torregrosa}						%TORREGROSA
\authorspace
\authref  See \hyperlink{C.\ Mendes Araujo}{C.~Mendes Ara\'ujo}.
\itemspace

\author{[N\'uria Ballber Torres]}						%TORRES
\authorspace
\authref  See \hyperlink{Nuria Ballber Torres}{N.~Ballber~Torres} (under `B').
\itemspace

\author{Dejan V.\ To\v{s}i\'c}						%TOSIC
\authorx{Dejan V.\ Tosic}
\authorspace
\authref  See \hyperlink{Milica Andelic}{M.~An\'deli\'c}.
\itemspace

\author{G\'erard Toulouse} 						%TOULOUSE
\authorx{Gerard Toulouse}
\authorspace
\authref  See also \hyperlink{B.\ Derrida}{B.~Derrida} and \hyperlink{Jean Vannimenus}{J.~Vannimenus}.
\authrefspace

\bsgitem{1977a}{}  Theory of the frustration effect in spin glasses:\ I.  \emph{Commun.\ Phys.}\ 2 (1977), 115--119.
Repr.\ in M.~M\'ezard, G.~Parisi, and M.A.~Virasoro, \emph{Spin Glass Theory and Beyond}, pp.\ 99--103.  World Scientific Lect.\ Notes in Physics, Vol.\ 9.  World Scientific, Singapore, 1987.
\annot{} Introduces the notion of imbalance (``frustration'') of a signed graph to account for inherent disorder in an Ising model (here synonymous with a signed graph, usually a lattice graph).  (Positive and negative edges are called   ``ferromagnetic and antiferromagnetic bonds''.)  Observes that switching the edge signs from all positive (the model of D.D.\ Mattis, \emph{Phys.\ Lett.}\ 56A (1976), 421--?) makes no essential difference.  In a planar lattice [or any plane graph] frustration of face boundaries (``plaquettes'') can be thought of as curvature, i.e., failure of flatness.  Proposes two kinds of asymptotic behavior of frustration as a circle encloses more plaquettes.  The planar-duality approach for finding the states with minimum frustration (i.e., switchings with fewest negative edges); the number of such states is the ``ground-state degeneracy'' and is important.  Ideas are sketched; no proofs.
\annotctd{} [A foundational paper.  See \xref{Wannier}{1950a}{G.H.\ Wannier} and, e.g., \xref{Villain \etal{}}{1977a}{Jacques Villain} \etal, \xref{Hoever, Wolff, and Zittartz}{1981a}{P.\ Hoever, W.F.\ Wolff, \& J.\ Zittartz}, \xref{Barahona, Maynard, Rammal, and Uhry}{1982a}{F.\ Barahona, R.\ Maynard, R.\ Rammal, \& J.P.\ Uhry}, \xref{van Hemmen}{1983a}{J.L.\ van Hemmen}, \xref{Wolff and Zittartz}{1983a}{W.F.\ Wolff \& J.\ Zittartz}, \xref{M\'ezard, Parisi, and Virasoro}{1987a}{Marc Mezard, Giorgio Parisi, \& Miguel Angel Virasoro}, \xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz}, \xref{Schw\"arzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh}, \etal]
\codes{\CSG:~\CPhys,~\CSw,~\CBal}%
\itemspace

\bsgitem{1979a}{}  Symmetry and topology concepts for spin glasses and other glasses.  
Non-perturbative Aspects in Quantum Field Theory (Proc.\ Les Houches Winter Adv.\ Study Inst., 1978).  \emph{Phys.\ Rep.}\ 49 (1979), no.\ 2, 267--272.  
\MR{0518399} (82j:82063).  
\annot{} Mainly for signed lattice graphs, with spins $s(v) \in S^{n-1}$ having symmetry group $\SO(n)$; $n=1$ (Ising model) gives $\SO\signnos$; $n=2$ is planar spins; $n=3$ is Heisenberg spins.  Two symmetry groups:  $\bbZ_2^{\#V}$ acts on $\Sigma$ (the ``microscopic level''); $\SO(n)$ or $\O(n)$ acts on states $s$ (the ``macroscopic level'').  
[An edge is satisfied if $s(w) = \sigma(vw)s(v)$, otherwise frustrated.]  A ``ground state'' (where the most edges are satisfied) has a topology of frustrated plaquettes [negative girth circles], whose nature, depending on the lattice dimension, is described intuitively.  Regions (``packets'') of relatively fixed spins can be identified.  Topology of frustrated plaquettes leads to the homotopy groups of $\O(n)$.  The effect on thermodynamic phases is discussed.
Dictionary:  ``Local transformation'' = switching.
\dated{20 Aug 2012} 
\codes{\CPhys,~\CSG:~\CFr,~\CSw:~\CExp,~\CRef}%
\itemspace

\bsgitem{1981a}{}  Spin glasses with special emphasis on frustration effects.  In:  Claudio Castellani \etal, eds., \emph{Disordered Systems and Localization} (Rome, 1981), pp.\ 166--173.  Lect.\ Notes in Phys., Vol.\ 149.  Springer, Berlin, 1981.
\annot{} \S3, ``Frustration'': in signed graphs [after normalization to bond strength 1].  ``Frustration function'' of circles [$= \sigma(C)$] determines physical properties because they are ``gauge [= switching] invariant'', if no external magnetic field.  \S3.i, ``Periodic frustrated models'' [= toroidally embedded graphs].  \S3.ii, ``Fully frustrated models'', where every ``plaquette'' [girth circle] is negative: overblocking effect, i.e., positive density of plaquettes with more than one negative edge.  [A mathematically interesting concept, not understood today.]  \S3.iii, ``Systems with finite residual entropy'': e.g., antiferromagnetic [all-negative] Potts models.  \S3.iv, ``Approach to spin glasses, by dilution of periodic frustrated systems'' [embedding an unbalanced toroidal graph in a larger balanced graph?].  \S3.v, ``Connections with gauge theories; topological defects and their hydrodynamics'': \cf, e.g., \xrefd{1979a}{Gerard Toulouse}.  \S3.vi, ``Random frustration (edge weights $\pm1$) models, in various space dimensions'':  comparing random signs $\pm1$ with Gaussian random edge weights (centered at 0, hence with signs and magnitudes).  For signed $K_n$'s (``Sherrington--Kirkpatrick model''), ``in the thermodynamic limit [both] have the same physics.''
\dated{20 Aug 2012} 
\codes{\CPhys:~\Csg,~\CFr,~\CSw:~\CExp,~\CRef}%
\itemspace

\author{G\'erard Toulouse \& Jean Vannimenus} 
\authorx{Gerard Toulouse \& Jean Vannimenus} 
\authorspace
\bsgitem{1977a}{}  La frustration:  un monde sem\'e de contradictions.  \emph{La Recherche}, No.\ 83, Vol.\ 8 (Nov., 1977), 980--981.
\annot{} Popular exposition of the elements of frustration in relation to the Ising model [evidently based on \xref{Toulouse}{1977a}{Gerard Toulouse}].  Briefly mentions the social psychology application.  
\codes{\CPhys:~\CSG,~\CBal:~\CExp}\codesctd{\CSG:~\CPsS:~\CExp}%
\itemspace

\bsgitem{1980a}{}  On the connection between spin glasses and gauge field theories.  \emph{Phys.\ Rep.}\ 67 (1980), no.\ 1, 47--54.
\MR{0600878} (no rev).  
\annot{} Annealed and quenched models on a square lattice are compared.  Annealed:  edge weights $J_{ij}$ (``bond strengths'') are random variables; this is randomly weighted, randomly signed graphs.  Quenched, edge weights $=\pm J$; this is signed graphs.  The annealed model ``grossly underestimates frustration effects.''  Proposed corrective:  introduce Lagrange multipliers for the plaquettes.  This leads to unexplored theory.  App.\ (c), ``The frustration model'':  randomly signed graphs, especially regular graphs; compared to models with Gaussian random edge weights and signs.
\dated{20 Aug 2012} 
\codes{\CPhys:~\Csg,~\CFr}\codesctd{\CPhys:~\Csg,~\CFr:~\CExp}%
\itemspace

\authorx{L.\ Tournier}							%TOURNIER
\author{L.\ Tournier \& M.\ Chaves}
\authorspace
\bsgitem{2009a}{}  Uncovering operational interactions in genetic networks using asynchronous Boolean dynamics.  \emph{J.\ Theor.\ Biol.}\ 260 (2009), 196--209.
\MR{2973075} (no rev).  \Zbl{1402.92207}.  \HAL{hal-00554625}.  
\codes{\CSD:~\CDyn,~\CBiol}%
\itemspace

\authorx{Vincent Antonio Traag}					%TRAAG
\author{V.A.\ Traag \& Jeroen Bruggeman}
\authorspace
\bsgitem{2009a}{}  Community detection in networks with positive and negative links.  \emph{Phys.\ Rev.\ E} 80 (2009), art.\ 036115, 6 pp.
\arXiv{0811.2329}.  
\annot{} Generalizes a Potts model for positive links to signed graphs.  Method is more general than the clustering model for signed graphs.  
[Applied in \xref{Yoshikawa, Iino, and Iyetomi}{2012a}{Takeo Yoshikawa, Takashi Iino, \& Hiroshi Iyetomi}.]
\codes{\CSG:~\CClu,~\CPsS}%
\itemspace

\author{Vincent Antonio Traag, Paul Van Dooren, \& Patrick De Leenheer}
\authorspace
\bsgitem{2013a}{}  Dynamical models explaining social balance and evolution of cooperation.  \emph{PLOS One} 8 (2014), no.\ 4, art.\ e60063, 7 pp.\ + 4 supplements.
\arXiv{1207.6588}.  
\par
\codes{\CSG,~\CWG:~\CBal:~\CKG:~\CDyn}%
\itemspace

\author{Lorenzo Traldi}						%TRALDI
\authorspace 
\authref  See also \hyperlink{Joanna A.\ Ellis-Monaghan}{J.~Ellis-Monaghan}.
\authrefspace

\bsgitem{1989a}{}  A dichromatic polynomial for weighted graphs and link polynomials.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 106 (1989), 279--286.  
\MR{0955462} (90a:57013).  \Zbl{713.57003}.					
\annot{} Generalizing \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} Tutte polynomial of a sign-colored graph, Traldi's ``weighted dichromatic polynomial'' $Q(\Gamma; t,z)$ is \xref{Zaslavsky's}{1992b}{Thomas Zaslavsky} $Q_\Gamma(1,w; t,z)$, in which the deletion-contraction parameters $a_e = 1$ and $b_e = w(e)$, the weight of $e$.  Thm.\ 2 gives the Tutte-style spanning-tree expansion.  
Thm.\ 4: Kauffman's Tutte polynomial $Q[\Sigma](A,B,d) = d\inv A^{\#E^+} B^{\#E^-} Q_{|\Sigma|}(1,w; d,d)$ for connected $\Sigma$, with $w(e)$ $= (AB\inv)^{\sigma(e)}$.  
[See \xref{Kauffman}{1989a}{Louis H.\ Kauffman} for other generalizations.  Traldi gives perhaps too much credit to \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn}.]  
\annotctd{} P.\ 284: Invariance under Reidemeister moves of type II constrains the weighted dichromatic polynomial to, in essence, equal Kauffman's.  Thus no generalization is evident in connection with general link diagrams.  There is an interesting application to special link diagrams.
\par
\codes{\CSGc:~\CGen:~\CInvar,~\CKnot}%
\itemspace

\bsgitem{2004a}{}  A subset expansion of the coloured Tutte polynomial.  \emph{Combin.\ Probab.\ Comput.}\ 13 (2004), no.\ 2, 269--275.
\MR{2047240} (2004k:05095).  \Zbl{1049.05024}.					
\annot{} The corank-nullity expansion of the usual Tutte polynomial generalizes to colored Tutte polynomials in the universal sense of \xref{Bollob\'as and Riordan}{1999a}{Bela Bollobas \& Oliver Riordan}.  
	
\codes{\CSGc:~\CGen:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2005a}{}  Parallel connections and coloured Tutte polynomials.  \emph{Discrete Math.}\ 290 (2005), no.\ 2--3, 291--299.
\MR{2123398} (2005j:05033).  \Zbl{1069.05021}.  
\annot{} The Tutte polynomial of a parallel connection of colored graphs or matroids.
\codes{\CSGc:~\CGen:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{2006a}{}  On the colored Tutte polynomial of a graph of bounded treewidth.  \emph{Discrete Appl.\ Math.}\ 154 (2006), no.\ 6, 1032--1036.
\MR{2212555} (2006j:05199).  \Zbl{1091.05027}.  
\annot{} Polynomial-time computability for colored graphs of bounded tree width.  [Also see \xref{Makowsky}{2005a}{J.A.\ Makowsky}.]
\par
\codes{\CSGc:~\CGen:~\CInvar:~\CAlgor,~\CKnot}%
\itemspace

\bsgitem{2015a}{}  The transition matroid of a 4-regular graph: An introduction.  \emph{European J.\ Combin.}\ 50 (2015), 180--207.
\MR{3361421}.  \Zbl{1319.05034}.  \arXiv{1307.8097}.				
\annot{} \S8, ``Topological Tutte polynomials'', defines the \xref{Bollob\'as--Riordan}{2002a}{Bela Bollobas \& Oliver Riordan} ribbon polynomial via edge signs, then via transition circuits.
\dated{3 Nov 2015} 
\codes{\CSG:~\CTop:~\CInvar}%
\itemspace

\author{Tan Nhat Tran}						%TRAN TN
\authorspace
\bsgitem{2019a}{}  Characteristic quasi-polynomials of ideals and signed graphs of classical root systems.  \emph{European J.\ Combin.}\ 79 (2019), 179--192.
\MR{3926514}.  \Zbl{1412.52018}.  \arXiv{1805.00179}.				
\codes{\CSG:~\CGeom,~\CInvar,~\CAlgeb}%
\itemspace

\authorx{Tuan Tran}							%TRAN T
\author{Tuan Tran \& G\"unter M.\ Ziegler}
\authorx{Tuan Tran \& Gunter M.\ Ziegler}
\authorspace
\bsgitem{2014a}{}  Extremal edge polytopes.  \emph{Electronic J.\ Combin.}\ 21 (2014), no.\ 2, art.\ P2.57, 16 pp.
\MR{3244823}.  \Zbl{1300.05145}.  \arXiv{1307.6708}.				
\annot{} Edge polytope $P_{-\Gamma}$ (\cf\ \xref{Ohsugi and Hibi}{1998a}{Hidefumi Ohsugi \& Takayuki Hibi}).  
[This is the antibalanced case.  \Problem.  Generalize to signed graphs, including balanced graphs.]
\codes{\Csg:~\CPar:~\CGeom}%
\itemspace

\author{Ben Tremblay}						%TREMBLAY
\authorspace
\authref  See \hyperlink{Gary MacGillivray}{G.~MacGillivray}.
\itemspace

\author{Mari\'an Trenkler} 						%TRENKLER
\authorspace
\authref  See \hyperlink{Samuel Jezny}{S.~Jezn\'y}.
\itemspace

\author{Vilmar Trevisan}						%TREVISAN
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo}.
\itemspace

\author{Irene Triantafillou}						%TRIANTAFILLOU
\authorspace
\bsgitem{2022a}{}  Spectra of signed graphs.  In:  Nicholas J.\ Daras \& Themistocles M.\ Rassias, eds.,  \emph{Approximation and Computation in Science and Engineering}, pp.\ 861--873.  Springer Optim.\ Appl., Vol.\ 180.  Springer Nature, Cham, 2022.
\MR{4436994}.  
\annot{} Surveys adjacency and Laplacian spectra.  For ``sign-symmetric'' \cf\ \xref{Ghorbani--Haemers--Maimani--Majd}{2020a}{Ebrahim Ghorbani, Willem H.\ Haemers, Hamid Reza Maimani, \& Leila Parsaei Majd}.
\dated{23 May 2022} 
\par
\codes{\CSG:~\CExp:~\CBal,~\CSw,~\CAdj,~\CLap:~\CEig;~\CRef}%
\itemspace

\author{Nenad Trinajsti\'c} 						%TRINAJSTIC
\authorx{Nenad Trinajstic}
\authorspace
\authref  See also \hyperlink{Ante Graovac}{A.~Graovac}.
\authrefspace

\bsgitem{1977a}{}  Computing the characteristic polynomial of a conjugated system using the Sachs theorem.  \emph{Croatica Chemica Acta} 49 (1977), No. 4, 593--633.
\annot{} ``Extension of the Sachs formula to M\"obius systems'', pp.\ 608--613.  ``M\"obius system'' = signed graph.
\dated{17 Mar 2020} 
\par
\codes{\CChem:~\CSG:~\CAdj}%
\itemspace

\bsgitem{1983a}{}  \emph{Chemical Graph Theory}.  2 vols.  CRC Press, Boca Raton, Florida, 1983.
\MR{0772570} (86g:92044a), \MR{0772571} (86g:92044b).  
\annot{} Vol.\ I:  Ch.\ 3, \S~VI: `` M\"obius graphs.'' Ch.\ 5, \S~VI: ``Extension of Sachs formula to M\"obius systems.'' \S~VII:  ``The characteristic polynomial of a M\"obius cycle.'' Ch.\ 6, \S~VIII: ``Eigenvalues of M\"obius annulenes.''
\par
\codes{\CSG:~\CChem,~\CEig:~\CExp}%
\itemspace

\bsgitem{1992a}{}  \emph{Chemical Graph Theory}.   Second ed.  CRC Press, Boca Raton, Florida, 1992.
\MR{1169298} (93g:92034).  
\annot{} Ch.\ 3, \S~V.B: ``M\"obius graphs.''  Ch.\ 4, \S~I: ``The adjacency matrix'': see  pp.\ 42--43.  Ch.\ 5: ``The characteristic polynomial of a graph'', \S~II.B: ``The extension of the Sachs formula to M\"obius systems''; \S~III.D: ``M\"obius cycles''.  Ch.\ 6, \S~VIII: ``Eigenvalues of M\"obius annulenes'' (i.e., unbalanced circles); \S~IX: ``A classification scheme for moncyclic systems'' (i.e., characteristic polynomials of circles).  
\codes{\CSG:~\CEig,~\CChem}%
\annotctd{} Ch.\ 7: ``Topological resonance energy,'' \S~V.C: ``M\"obius annulenes''; \S~V.G: ``Aromaticity in the lowest excited state of annulenes''.   
\par
\codes{\CChem;~\Csg:~\Cbal}%
\itemspace

\author{Anastasia Trofimova}						%TROFIMOVA
\authorspace
\authref  See \hyperlink{Yurii Burman}{Y.~Burman}.
\itemspace

\author{Nicolas Trotignon}						%TROTIGNON
\authorspace
\authref  See also \hyperlink{Pierre Aboulker}{P.~Aboulker}.
\authrefspace

\author{Nicolas Trotignon \& Kristina Vu\v{s}kovi\'c}
\authorx{Nicolas Trotignon \& Kristina Vuskovic}
\authorspace
\bsgitem{2010a}{}  A structure theorem for graphs with no cycle with a unique chord and its consequences.  \emph{J.\ Graph Theory} 63 (2010), no.\ 1, 31--67.
\MR{2590324} (2011g:05260).  \Zbl{1186.05104}.  \arXiv{1309.0979}.			
\annot{} Pp.\ 35--36:  Brief description of graphs having special kinds of signatures.  \Cf\ \xref{Conforti, Cornu\'ejols, and Vu\v{s}kovii\'c}{2006a}{Michele Conforti, Gerard Cornuejols, \& Kristina Vuskoviic} \etal
\dated{19 Jan 2015} 
\codes{\CSGw,~\Csg:~\CBal(\CGen):~\CExp}%
\itemspace

\author{L.E.\ Trotter, Jr.}						%TROTTER LE
\authorspace
\authref  See \hyperlink{E.C.\ Sewell}{E.C.~Sewell}.
\itemspace

\author{Klaus Truemper} 						%TRUEMPER
\authorspace
\authref   See also \xref{Conforti, Cornu\'ejols, and Truemper}{1994a}{Michele Conforti, Gerard Cornuejols, \& Klaus Truemper} and \xref{Gerards, Lov\'asz, \etal{}}{1990a}{A.M.H.\ Gerards, L.\ Lovasz, A.\ Schrijver, P.D.\ Seymour, C.-S.\ Shi, \& K.\ Truemper}, \xref{Tseng and Truemper}{1986a}{F.T.\ Tseng \& K.\ Truemper}.
\authrefspace

\bsgitem{1976a}{} An efficient scaling procedure for gain networks.  \emph{Networks} 6 (1976), 151--159.  
\MR{0452603} (56 \#10882).  \Zbl{331.90027}.  
\codes{\Cgg:~\CGN,~\Csg:~\CBal,~\CSw}%
\itemspace

\bsgitem{1977a}{}  On max flows with gains and pure min-cost flows.  \emph{SIAM J.\ Appl.\ Math.}\ 32 (1977), 450--456.  
\MR{0432208} (55 \#5197).  \Zbl{352.90069}.  
\par
\codes{\CGG,~\COG,~\CGN,~\CBal}%
\itemspace

\bsgitem{1977b}{}  Unimodular matrices of flow problems with additional constraints.  \emph{Networks} 7 (1977), 343--358.  
\MR{0503664} (58 \#20352).  \Zbl{373.90023}.  
\codes{\Csg:~\CInc:~\CBal}%
\itemspace

\bsgitem{1978a}{}  Optimal flows in nonlinear gain networks.  \emph{Networks} 8 (1978), 17--36.  
\MR{0465133} (57 \#5041).  \Zbl{381.90039}.  
\codes{\CGN}%
\itemspace

\bsgitem{1982a}{\dag\dag} Alpha-balanced graphs and matrices and $\GF(3)$-representability of matroids.  \emph{J.\ Combin.\ Theory Ser.\ B}  32 (1982), 112--139. 
\MR{0657681} (83i:05025).  \Zbl{478.05026}.					
\annot{} A $0,\pm1$-matrix is called ``balanced'' if it contains no submatrix that is the incidence matrix of a negative circle.  More generally, $\alpha$-balance of a $0,\pm1$-matrix corresponds to prescribing the signs of holes in a signed graph.  Main theorem characterizes the sets of holes (chordless circles) in a graph that can be the balanced holes in some signing.  [See \xref{Conforti and Kapoor}{1998a}{Michele Conforti \& Ajai Kapoor} for a new proof and discussion of applications.]
\par
\codes{\Csg:~\CBal,~\CInc}%
\itemspace

\bsgitem{1992a}{}  \emph{Matroid Decomposition}.  Academic Press, San Diego, 1992.  
\MR{1170126} (93h:\-05046).  \Zbl{760.05001}.					
\annot{} \S12.1: ``Overview.''  \S12.2: ``Characterization of alpha-balanced graphs,'' exposition of \xrefd{1982a}{Klaus Truemper}.
\codes{\Csg:~\CBal,~\CSw}%
\itemspace

\bsgitem{1992b}{}  A decomposition theory for matroids.  VII.  Analysis of minimal violation matrices.  \emph{J.\ Combin.\ Theory Ser.\ B} 55 (1992), 302--335.
\MR{1168967} (93e:05021).  \Zbl{809.05024}.  
\annot{} According to \xref{Cornu\'ejols}{2001a}{Gerard Cornuejols}, this paper contains the following theorem: A bipartite graph is ``balanceable'' (has a $\pm1$-weighting $\pmod{4}$ in which all polygons have sum $0 \pmod{4}$) iff it does not contain an induced subgraph that is a subdivided odd wheel or a theta graph with nodes in opposite color classes.  
[The weights are not gains because they are not oriented.  However, this has major applications to signed hypergraphs; \cf\ \xref{Rusnak}{2010a}{Lucas Rusnak}.]
[\Problem.  Generalize to arbitrary graphs.]  
\annotctd{} [In a bipartite graph the sum around a polygon has to be $0$ or $2 \pmod{4}$ and therefore belongs to a group $\cong \bbZ_2$ so can be considered a sign.  However, it may not be possible to relabel the edges from $\bbZ_2$ so as to get the same polygon sums.  I.e., the polygon signing may not be derivable from a signed graph.]	
\codes{\CSGw:~\Cbal}%
\itemspace

\author{Th\'eophile Trunck}						%TRUNCK
\authorx{Theophile Trunck}
\authorspace
\authref  See \hyperlink{Pierre Aboulker}{P.~Aboulker}.
\itemspace

\author{Anke Truss}							%TRUSS
\authorspace
\authref  See \hyperlink{Sebastian Bocker}{S.~B\"ocker}.
\itemspace

\author{Marcello Truzzi} 						%TRUZZI
\authorspace
\authref  See \hyperlink{Frank Harary}{F.~Harary}.
\itemspace

\author{S.V.\ Tsaranov} 						%TSARANOV
\authorspace
\authref  See also \hyperlink{F.C.\ Bussemaker}{F.C.~Bussemaker}, \hyperlink{Peter J.\ Cameron}{P.J.~Cameron}, and \hyperlink{J.J.\ Seidel}{J.J.~Seidel}.
\authrefspace

\bsgitem{1992a}{}  On spectra of trees and related two-graphs.  
In: Jaroslav Ne\v{s}et\v{r}il and Miroslav Fiedler, eds., \emph{Fourth Czechoslovak Symposium on Combinatorics, Graphs and Complexity} (Prachatice, 1990), pp.\ 337--340.  Ann.\ Discrete Math., Vol.\ 51.  North-Holland, Amsterdam, 1992. 
\MR{1206289} (no rev).  \Zbl{776.05077}.						
\annot{} A two-graph whose points are the edges of a tree $T$ and whose triples are the nonseparating triples of edges of $T$ (from \xref{Seidel and Tsaranov}{1990a}{J.J.\ Seidel \& S.V.\ Tsaranov} via \xref{Cameron}{1994a}{Peter J.\ Cameron}).  An associated signed complete graph $\Sigma_T$ on vertex set $E(T)$ is obtained by orienting $T$ arbitrarily, then taking $\sigma_T(ef) = +$ or $-$ depending on whether $e$ and $f$ are similarly or oppositely oriented in the path of $T$ that contains both.  Reorienting edges corresponds to switching $\Sigma_T$.  Thm.: Letting $n = \#V(T)$, the matrices $3I_n + A(\Sigma_T)$ and $2I_{n+1} - A(T)$ have the same numbers of zero and negative eigenvalues.
\codes{\CTG:~\CEig,~\CGeom}%
\itemspace

\bsgitem{1993a}{}  Trees, two-graphs, and related groups.  
In: D.~Jungnickel and S.A.~Vanstone, eds., \emph{Coding Theory, Design Theory, Group Theory} (Proc.\ Marshall Hall Conf., Burlington, Vt., 1990), pp.\ 275--281.  Wiley, New York, 1993.  
\MR{1227141} (94j:05062).  
\annot{} New proof of theorem on the group (\xref{Seidel and Tsaranov}{1990a}{J.J.\ Seidel \& S.V.\ Tsaranov}) of the two-graph (\xref{Tsaranov}{1992a}{S.V.\ Tsaranov}) of a tree.
\codes{\CTG:~\CEig,~\CGeom}%
\itemspace

\author{Michael J.\ Tsatsomeros}					%TSATSOMEROS
\authorspace
\authref   See \hyperlink{M.\ Cavers, S.M.\ Cioaba, S.\ Fallat, D.A.\ Gregory, W.H.\ Haemers, S.J.\ Kirkland, J.J.\ McDonald, \& M.\ Tsatsomeros}{M.~Cavers}, \hyperlink{Charles R.\ Johnson, Michael Neumann, \& Michael J.\ Tsatsomeros}{C.R.~Johnson}, \hyperlink{Steve Kirkland, J.J.\ McDonald, \& M.J.\ Tsatsomeros}{S.~Kirkland}, and \hyperlink{D.D.\ Olesky, M.J.\ Tsatsomeros, \& P.\ van den Driessche}{D.D.~Olesky}.
\itemspace

\author{Dennis Tseng}						%TSENG D
\authorspace
\authref  See \hyperlink{Victor Reiner}{V.~Reiner}.
\itemspace

\authorx{Fan T.\ Tseng}						%TSENG FT
\author{F.T.\ Tseng \& K.\ Truemper}
\authorspace
\bsgitem{1986a}{}  A decomposition of the matroids with the max-flow min-cut property.  \emph{Discrete Appl.\ Math.}\ 15 (1986), 329--364.
\MR{0865011} (88b:05046).  \Zbl{679.90052}.					
Addendum.  \emph{Discrete Appl.\ Math.}\ 20 (1988), 87--88.
\MR{0936900} (89b:05070).  \Zbl{818.05031}.					
\annot{} A special case is decomposition of $\bfL_0(\Sigma)$. 
\dated{18 Jan 2021} 
\codes{\Csg}%
\itemspace

\authorx{Charalampos E.\ Tsourakakis}					%TSOURAKAKIS
\author{Charalampos E.\ Tsourakakis, Michael Mitzenmacher, Jaros{\l}aw B{\l}asiok, Ben Lawson, Preetum Nakkiran, \& Vasileios Nakos}
\authorspace
\bsgitem{2020a}{}  Predicting positive and negative links with noisy queries: theory \& practice.  \emph{Internet Math.}\ (2020), 17 pp.
\MR{4196797}.  \arXiv{1709.07308v3}.  
\codes{\CSG:~\CClu}%
\itemspace

\author{Shuhei Tsujie}						%TSUJIE
\authorspace
\authref  See \hyperlink{Takuro Abe}{T.~Abe}, \hyperlink{Norihiro Nakashima}{N.~Nakashima}, \hyperlink{Daisuke Suyama}{D.~Suyama}, and \hyperlink{Michele Torielli}{M.~Torielli}.
\itemspace

\authorx{D.\ Tsvetkovich}						%TSVETKOVICH
\author{D.\ Tsvetkovich, M.\ Dub, \& Kh.\ Zakhs}
\authorspace
\bsgitem{1984a}{}  \emph{Spektry grafov.  Teoriya i primenenie.}  
(In Russian.)  Transl.\ V.V.\ Strok, ed.\ V.S.\ Korolyuk.  Preface by Strok and Korolyuk.  Naukova Dumka, Kiev, 1984.
\MR{0746475} (85c:05025).  
\annot{} Russian ed.\ of \xref{Cvetkovi\'c, Doob, and Sachs}{1980a}{Dragos M.\ Cvetkovic, Michael Doob, \& Horst Sachs}.
\par
\codes{\CSD,~\Cpar,~\CTG:~\CSw,~\CAdj,~\CEig,~\CGeom:~\CExp,~\CExr,~\CRef}%
\itemspace

\author{Jianhua Tu}							%TU JH
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.-H.~Yu}.
\itemspace

\author{Thomas W.\ Tucker} 						%TUCKER
\authorspace
\authref  See also \hyperlink{Jonathan L.\ Gross}{J.L.~Gross} and \hyperlink{Jozef Siran}{J.~\v{S}ir\'a\v{n}}.
\authrefspace

\bsgitem{1983a}{}  Finite groups acting on surfaces and the genus of a group.  \emph{J.\ Combin.\ Theory Ser.\ B} 34 (1983), no.\ 1, 82--98.
\MR{0701174} (85b:20055).  \Zbl{521.05027}.					
\codes{\CGG:~\CTop}%
\itemspace

\bsgitem{2009a}{}  The genus of a group.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 11, pp.\ 225--244.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581548} (no rev).  \Zbl{1225.05130}.  
\annot{} \S3, ``Quotient embeddings and voltage graphs''.
\dated{12 Jun 2013} 
\par
\codes{\CTop:~\CGG,~\Csg,~\CCov:~\CExp}%
\itemspace

\author{Francesco Tudisco}						%TUDISCO
\authorspace
\authref  See \hyperlink{Pedro Mercado}{P.~Mercado}.
\itemspace

\author{Vanda Tulli}							%TULLI
\authorspace
\authref  See \hyperlink{A.\ Bellacicco}{A.~Bellacicco}.
\itemspace

\author{Pavel Tumarkin}						%TUMARKIN
\authorspace
\authref  See \hyperlink{Mathieu Dutour Sikiric}{M.D.~Sikiri\'c}.
\itemspace

\author{Hande Tun\c{c}el}						%TUNCEL
\authorspace
\authref  See \hyperlink{Fatihcan M.\ Atay}{F.M.~Atay}.
\itemspace

\author{Edward C.\ Turner} 						%TURNER
\authorspace
\authref  See \hyperlink{Richard Z.\ Goldstein}{R.Z.~Goldstein}.
\itemspace

\author{Krzysztof Turowski}						%TUROWSKI
\authorspace
\authref  See \hyperlink{Robert Janczewski}{R.~Janczewski}.
\itemspace

\author{Daniel Turz\'{\i}k} 						%TURZIK
\authorx{Daniel Turzik}
\authorspace
\authref  See \hyperlink{Svatopluk Poljak}{S.~Poljak}.
\itemspace

\author{W.T.\ Tutte} 							%TUTTE
\authorspace
\bsgitem{1967a}{\dag}  Antisymmetrical digraphs.  \emph{Canad.\ J.\ Math.}\ 19 (1967), 1101--1117.
\MR{0214512} (35 \#5362).  \Zbl{161.20905} (161, p.\ 209e).
\annot{} Integral $(u,u)$-flows on a signed graph with edge capacities, presented in the language of integral $(\tilde u,\tilde u^*)$-flows on a digraph with edge capacities, with an orientation-reversing, fixed-point free, capacity-preserving involution ${}^*$.  [Such a digraph is the double covering digraph of a bidirected graph, thus the capacities and flows are equivalent to $(u,u)$-flows on a capacitated signed graph.]  Analog of the Min-Flow Max-Cut Theorem (see 3.3).  Structure of flows.  Application to undirected graph factors.
[\Problem.  Convert the entire paper to the language of signed graphs.  Express the structure of $(u,u)$-flows in terms of  signed-graphic objects such as unbalanced unicyclic subgraphs.  Extract the implicit matroid theory, including the structure of cocircuits (\cf\ \xref{Chen and Wang}{2009a}{Beifang Chen \& Jue Wang}).]
\dated{9 Sept 2010, 12 Jan 2012}
\codes{\Csg:~\Cori,~\Ccov:~\CFlows}%
\itemspace

\bsgitem{1981a}{\dag} On chain-groups and the factors of graphs.  
In:  L.~Lov\'asz and Vera T.\ S\'os, eds., \emph{Algebraic Methods in Graph Theory} (Proc.\ Colloq., Szeged, 1978), Vol.\ 2, pp.\ 793--818.  
Colloq.\ Math.\ Soc.\ J\'anos Bolyai, 25.  J\'anos Bolyai Math.\ Soc., Budapest, and North-Holland, Amsterdam, 1981.  
\MR{0642073} (83b:05104).  \Zbl{473.05023}.
\annot{} The chain-group approach to the dual even-cycle matroid, $\bfF(-\Gamma)^*$.  Developed entirely in terms of the group $\Delta(\Gamma)$ [topologically, $B^1(\Gamma,\bbZ)$] of integral $1$-coboundaries.  
Assuming $\Gamma$ connected:  ``Dendroids of $\Delta(\Gamma)$'' = bases of $\bfF(-\Gamma)$;  Thms.\ 8.6--7 give their structure in the bipartite and nonbipartite cases.  
Support of an elementary coboundary = circuit of $\bfF(-\Gamma)^*$; this is a bond of $\Gamma$ if $\Gamma$ is bipartite (Thm.\ 7.5) and a minimal balancing set otherwise (Thm.\ 7.6).  
Thm.\ 7.8: Any coboundary times some power of 2 is a sum of primitive coboundaries.  [\Problem.  Explain how this is related to total dyadicity of the incidence matrix.]  
``Rank of $\Delta(\Gamma)$'' = $\rk \bfF(-\Gamma)$; its value is given at the end of \S8.  
\S9 develops a relationship between ``homomorphisms'' of $\Delta(\Gamma)$ (linear functionals) and graph factors.  
\S10:  The dual chain group; characterization of circuits of $\rk \bfF(-\Gamma)$.  
[It is amazing what can be done with nothing but integral 1-coboundaries.  \Problem\ 1.  Extend Tutte's theory of integral chain groups to all signed graphs.  \xref{Grossman, Kulkarni, and Schochetman}{1994a}{Jerrold W.\ Grossman, Devadatta M.\ Kulkarni, \& Irwin E.\ Schochetman} have a development over a field but this is very different, even aside from their opposite viewpoint that goes from matroids to vector spaces.  \Problem\ 2.  Extend to signed hypergraphs, where each hyperedge has a function $\tau_e : V(e) \to \{+,-\}$ (not distinguished from $-\tau_e$; as with bidirected graphs, choosing one of them corresponds to orienting $e$).]
\annotctd{} [Tutte knew and lectured on $\bfF(-\Gamma)^*$ and/or $\bfF(-\Gamma)$ before anyone (\xref{Doob}{1973a}{Michael Doob}, \xref{Sim\~oes-Pereira}{1973a}{J.M.S.\ Simoes-Pereira}) published it.---information from Neil Robertson.]
\codes{\Csg:~\CECyc,~\CDual,~\Cinc}%
\itemspace

\bsgitem{1984a}{}  \emph{Graph Theory.}  Encycl.\ Math.\ Appl., Vol.\ 21.  Addison-Wesley, Menlo Park, Calif., 1984.  
\MR{0746795} (87c:05001).  \Zbl{554.05001}.  					
Repr.\ Cambridge Univ.\ Press, Cambridge, Eng., 2001.  
\MR{1813436} (2001j:05002).  \Zbl{964.05001}.  
\annot{} Note VIII.12.1, ``Unoriented coboundaries'', mentions the work of \xrefd{1981a}{W.T.\ Tutte}.  
\dated{9 May 2014} 
\codes{\Cpar:~\Cmatrd}%
\itemspace

\bsgitem{1988a}{}  (as ``U.\ Tatt'')  \emph{Teoriya grafov.}  Transl.\ G.P.\ Gavrilov.  ``Mir'', Moscow, 1988.
\MR{0977974} (89i:05093).  
\annot{} Russian trans.\ of \xrefd{1984a}{W.T.\ Tutte}.
\codes{\Cpar:~\Cmatrd}%
\itemspace

\author{Kaya Tutuncuoglu}						%TUTUNCUOGLU
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Zsolt Tuza}							%TUZA
\authorspace
\authref   See \hyperlink{Svatopluk Poljak}{S.~Poljak}.
\itemspace

\author{Ilya Tyomkin}							%TYOMKIN
\authorspace
\authref  See \hyperlink{Amos Beimel}{A.~Beimel}.
\itemspace


\label{LETTER U} \hypertarget{LETTER U}{}	% LETTER  U


\author{Frank Uhlig}							%UHLIG
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{J.P.\ Uhry} 							%UHRY
\authorspace
\authref  See \hyperlink{Francisco Barahona}{F.~Barahona} and \hyperlink{I.\ Bieche}{I.~Bieche}.
\itemspace

\author{W{\l}odzimierz Ulatowski} 					%ULATOWSKI
\authorx{Wlodzimierz Ulatowski}
\authorspace
\authref  See also \hyperlink{J.\ Topp}{J.~Topp}.
\authrefspace

\bsgitem{1991a}{}  On Kirchhoff's voltage law in $Z_n$.  \emph{Discuss.\ Math.}\ 11 (1991), 35--50.  
\MR{1178357} (93g:05121).  \Zbl{757.05058}.  					
\annot{} Examines injective, nowhere zero, balanced gains (called ``graceful labellings'') from $Z_{m+1}$, $m = \#E$, on arbitrarily oriented circles and variously oriented paths.  [\Question.  Does this work generalize to bidirected circles and paths?] 
\codes{\CGD:~\Cbal:~\CCircles,~\CPaths}%
\itemspace

\author{[N.B.\ Ul'janov]}						%ULJANOV
\authorspace
\authref   See \hyperlink{N.B.\ Ul'yanov}{N.B.~Ul'yanov}.
\itemspace

\author{N.B.\ Ul'yanov}						%ULYANOV
\authorspace
\authref   See \hyperlink{D.O.\ Logofet}{D.O.~Logofet}.
\itemspace

\author{Somya Upadhyaya}						%UPADHYAYA
\authorspace
\authref  See \hyperlink{Deepa Sinha}{D.~Sinha}.
\itemspace


\label{LETTER V} \hypertarget{LETTER V}{}	% LETTER  V


\author{Gurunath Rao Vaidya}						%VAIDYA GR
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\authorx{Samir K.\ Vaidya}						%VAIDYA SK
\author{Samir K.\ Vaidya \& Kalpesh M.\ Popat}
\authorspace
\bsgitem{2019a}{}  Some new results on Seidel equienergetic graphs.  \emph{Kyungpook Math.\ J.}\ 59 (2019), no.\ 2, 335--340.
\MR{3987713}.  \Zbl{1439.05145}.  
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{J.F.\ Vald\'es}						%VALDES
\authorx{J.F.\ Valdes}
\authorspace
\authref  See also \hyperlink{W.\ Lebrecht}{W.~Lebrecht} and \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\authrefspace

\author{J.F.\ Vald\'es, J.\ Cartes, \& E.E.\ Vogel}
\authorx{J.F.\ Valdes, J.\ Cartes, \& E.E.\ Vogel}
\authorspace
\bsgitem{2000a}{}  Polyhedra as $\pm J$ closed Ising lattices.  \emph{Rev.\ Mexicana F{\'\i}s.}\ 46 (2000), no.\ 4, 348--356.
\MR{1783780} (2001g:82035).  \Zbl{1291.82030}.  
\annot{} Physics and signed graph theory on a signed polyhedral graph, esp.\ properties of ground states as functions of $x := \#E^+/\#E$.  Effects of vertex and face degrees.
\dated{17 Jun 2012, 9 Jan 2015} 
\par
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\author{J.F.\ Vald\'es, J.\ Cartes, E.E.\ Vogel, S.\ Kobe, \& T.\ Klotz}
\authorx{J.F.\ Valdes, J.\ Cartes, E.E.\ Vogel, S.\ Kobe, \& T.\ Klotz}
\authorspace
\bsgitem{1998a}{}  Relationship between the structure of the ground level and frustration in $\pm J$ Ising lattices.  \emph{Physica A} 257 (1998), 557--562.
\annot{} Maps the ground states of a $6\times6$ toroidal square lattice with various signatures.
	
Dictionary:  \cf\ \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}.
\dated{2 Jan 2015} 
\codes{\CSG:~\CState(\Cfr),~\CSw,~\CPhys}%
\itemspace

\author{J.F.\ Vald\'es, W.\ Lebrecht, \& E.E.\ Vogel}
\authorx{J.F.\ Valdes, W.\ Lebrecht, \& E.E.\ Vogel}
\authorspace
\bsgitem{2007a}{}  $\pm J$ Ising model on Dice lattices.  \emph{Physica A} 385 (2007), 551--557.
\annot{} Randomly signed dice lattice (planar, with rhombic faces) with specified $x := \#E^+/\#E$: frustration index, distribution of frustrated plaquettes (rhombi), \etal, as functions of $x$.  This lattice is interesting because the average degree (``coordination number'') is not integral; \cf\ \xref{Lebrecht, Vogel, and Vald\'es}{2004a}{W.\ Lebrecht, E.E.\ Vogel, \& J.F.\ Valdes} \etal\ \ 
	
\dated{3 Jan 2015} 
\par
\codes{\CSG,~\CPhys:~\CFr~\CState,~\CAlgor}%
\itemspace

\bsgitem{2012a}{} $\pm J$ Ising model on homogeneous Archimedean lattices.  \emph{Physica A} 391 (2012), 2585--2599.
\MR{2882041}.  
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CFr,~\CState}%
\itemspace

\authorx{Carlos E.\ Valencia}						%VALENCIA
\author{Carlos E.\ Valencia \& Rafael H.\ Villarreal}
\authorspace
\bsgitem{2006a}{}  Explicit representations of the edge cone of a graph.  \emph{Int.\ J.\ Contemp.\ Math.\ Sci.}\ 1 (2006), no.\ 2, 53--66.
\MR{2288999} (2007m:05164).  \Zbl{1118.05068}.  
\annot{} As intersection of half-spaces.
\dated{7 Jul 2022} 
\codes{\Csg:~\CGeom,~\CAlgeb}%
\itemspace

\author{Miguel A.\ Valencia Bucio}					%VALENCIA-BUCIO
\authorspace
\authref  See also \hyperlink{Jose Martinez-Bernal}{J.~Mart\'{i}nez-Bernal}.
\authrefspace

\author{Miguel \'Angel Valencia Bucio}
\authorspace
\bsgitem{2019a}{}  \emph{Sobre C\'odigos y Demimatroides} [\emph{On Codes and Demimatroids}].  Doctoral dissertation, CINVESTAV, M\'exico, D.F., 2019.
\codes{\CSG,~\CMatrd(\CGen):~\CInvar,~\CAppl}%
\itemspace

\author{James Van Buskirk}						%VAN BUSKIRK
\authorspace
\authref  See \hyperlink{Thomas J.\ Lundy}{T.J.~Lundy}.
\itemspace

\author{[Edwin R.\ van Dam]}						%VAN DAM
\authorspace
\authref  See \hyperlink{Edwin R.\ van Dam}{E.R.~van Dam} (under `D').
\itemspace

\author{Pauline van den Driessche}					%VAN DEN DRIESSCHE
\authorspace
\authref   See \hyperlink{Jacques Belair}{J.~B\'elair}, \hyperlink{B.D.\ Bingham}{B.D.~Bingham}, \hyperlink{Thomas Britz}{T.~Britz}, \hyperlink{Steve Butler}{S.~Butler}, \hyperlink{Minerva Catral}{M.~Catral}, \hyperlink{G.J.\ Culos}{G.J.~Culos}, \hyperlink{D.A.\ Grundy, D.D.\ Olesky, \& P.\ van den Driessche}{D.A.~Grundy}, \hyperlink{Clark Jeffries}{C.~Jeffries}, \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}, \hyperlink{Victor Klee}{V.~Klee}, \hyperlink{Keivan Hassani Monfared}{K.~Hassani Monfared}, and \hyperlink{D.D.\ Olesky}{D.D.\ Olesky}.
\itemspace

\author{Hein van der Holst}						%VAN DER HOLST
\authorspace
\authref  See \hyperlink{Marina Arav}{M.~Arav}.
\itemspace

\author{[Jorn van der Pol]}						%VAN DER POL
\authorspace
\authref  See \hyperlink{Jorn van der Pol}{J.~van~der~Pol} (under `P').
\itemspace

\author{Arnout van de Rijt}						%VAN DE RIJT
\authorspace
\bsgitem{2011a}{}  The micro-macro link for the theory of structural balance.  \emph{J.\ Math.\ Sociology} 35 (2011), no.\ 1-3, 94--113.
\MR{2844982} (2012i:91238).  \Zbl{1214.91094}.						
\par
\codes{\CSG:~\CFr}%
\itemspace

\author{Kevin N.\ Vandermeulen}					%VANDERMEULEN
\authorspace
\authref  See \hyperlink{Michael S.\ Cavers}{M.S.~Cavers} and \hyperlink{David A.\ Gregory}{D.A.~Gregory}.
\itemspace

\author{Paul Van Dooren}						%VAN DOOREN
\authorspace
\authref  See \hyperlink{Vincent Antonio Traag}{V.A.~Traag}.
\itemspace

\author{[J.L.\ van Hemmen]}						%VAN HEMMEN
\authorspace
\authref  See \hyperlink{J.L.\ van Hemmen}{J.L.~van~Hemmen} (under `H').
\itemspace

\author{Marc A.A.\ van Leeuwen}					%VAN LEEUWEN
\authorspace
\bsgitem{1996a}{}  The Robinson-Schensted and Sch\"utzenberger algorithms, an elementary approach.  \emph{Electronic J.\ Combin.}\ 3 (1996), no.\ 2, \#R15, 32 pp.
\MR{1392500} (97e:05200).  \Zbl{852.05080}.					
\annot{} Elements of the hyperoctahedral group $\fO_d$ (signed permutations) of even degree $d=2n$ permute $\pm[n]$ and of odd degree $d=2n+1$ permute $[-n,n]$ (pp.\ 22f.  The natural involution is $\pi \mapsto -\bar\pi$, where $\bar\pi$ is the reverse of $\pi$ [reminiscent of signed graph coloring].  [\Cf\ \xref{Bloss}{2003a}{Matthew Bloss} and \xref{Parvathi}{2004a}{M.\ Parvathi}.]
\dated{19 Mar 2011} 
\codes{\Csg:~\CAlgeb}%
\itemspace

\author{A.\ Vannelli}							%VANNELLI
\authorspace
\authref  See \hyperlink{Chuan-Jin Shi}{C.J.~Shi}.
\itemspace

\author{Jean Vannimenus} 						%VANNIMENUS
\authorspace
\authref   See also \hyperlink{B.\ Derrida}{B.~Derrida} and \hyperlink{Gerard Toulouse}{G.~Toulouse}.
\authrefspace

\author{J.\ Vannimenus, S.\ Kirkpatrick, F.D.M.\ Haldane, \& C.\ Jayaprakash}
\authorspace
\bsgitem{1989a}{}  Ground-state morphology of random frustrated $XY$ systems.  \emph{Phys.\ Rev.\ B}\ 39 (1989), no.\ 7, 4634--4643.
\MR{0986455} (89m:82087).  
\annot{} $XY$ means signed graphs with complex-unit vertex spins.
	
\codes{\CPhys:~\Csg}%
\itemspace

\author{J.\ Vannimenus, J.M.\ Maillard, \& L.\ de S\`eze} 
\authorx{J.\ Vannimenus, J.M.\ Maillard, \& L.\ de Seze} 
\authorspace
\bsgitem{1979a}{}  Ground-state correlations in the two-dimensional Ising frustration model.  \emph{J.\ Phys.\ C:  Solid State Phys.}\ 12 (1979), 4523--4532.
\codes{\CPhys:~\CSG}%
\itemspace

\author{J.\ Vannimenus \& G.\ Toulouse} 
\authorspace
\bsgitem{1977a}{}  Theory of the frustration effect:  II.  Ising spins on a square lattice.  \emph{J.\ Phys.\ C:  Solid State Phys.}\ 10 (1977), L537--L541.
\codes{\CSG:~\CPhys}%
\itemspace

\author{[Cyriel van Nuffelen]}						%VAN NUFFELEN
\authorspace
\authref  See \hyperlink{Cyriel van Nuffelen}{C.~v.~Nuffelen} (under `N').
\itemspace

\author{M.E.\ Van Valkenburg} 						%VAN VALKENBURG
\authorspace
\authref   See \hyperlink{W.\ Mayeda}{W.~Mayeda}.
\itemspace

\authorx{Anke van Zuylen}						%VAN ZUYLEN
\author{Anke van Zuylen, Rajneesh Hegde, Kamal Jain, \& David P.\ Williamson}
\authorspace
\bsgitem{2007a}{}  Deterministic pivoting algorithms for constrained ranking and clustering problems.  In:  \emph{Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms} (SODA '07, New Orleans, 2007), pp.\ 405--414.  Assoc.\ for Computing Machinery, New York, and Soc.\ for Industrial and Appl.\ Math., Philadelphia, 2007.
\MR{2482866} (no rev).  \Zbl{1302.68326}.  
\codes{\Csg:~\CClu:~\CAlgor}%
\itemspace

\author{Anke van Zuylen \& David P.\ Williamson}
\authorspace
\bsgitem{2009a}{}  Deterministic pivoting algorithms for constrained ranking and clustering problems.  \emph{Math.\ Oper.\ Res.}\ 34 (2009), no.\ 3, 594--620.
\MR{2555338} (2010j:68136).  \Zbl{1216.68343}.  
\codes{\Csg:~\CClu:~\CAlgor}%
\itemspace

\author{[Stefan H.M.\ van Zwam]}					%VAN ZWAM
\authorspace
\authref  See \hyperlink{Stefan H.M.\ van Zwam}{S.H.M.~van~Zwam} (under ``Z'').
\itemspace

\author{Burak Varan}							%VARAN
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Patricio Vargas}						%VARGAS
\authorspace
\authref  See  \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\authorx{V.\ Vasanthi}						%VASANTHI
\author{V.\ Vasanthi, S.\ Arumugam, Atulya K.\ Nagar, \& Sovan Mitra}
\authorspace
\bsgitem{2015a}{}  Applications of signed graphs to portfolio turnover analysis.  2nd Global Conf.\ Business and Social Sci.\ (Bali, 2015).  \emph{Procedia Social Behavioral Sci.}\ 211 (2015), 1203--1209.
\annot{} Largely an imprecise exposition of \xref{Harary--Lim--Wunsch}{2002a}{Frank Harary, Meng-Hiot Lim, \& Donald C.\ Wunsch} with possible missing edges [hence their reduction of balance testing to triangles is invalid].  The claim about computational speed is wrong.
\dated{12 Feb 2022}
\codes{\CAppl:~\CSG:~\CBal}%
\itemspace

\author{[T.R.\ Vasanth Kumar]}						%VASANTH KUMAR
\authorspace
\authref  See \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Wolmer V.\ Vasconcelos}					%VASCONCELOS
\authorspace
\authref  See \hyperlink{Aron Simis}{A.~Simis}.
\itemspace

\author{B.\ Vasuki}							%VASUKI
\authorspace
\authref  See also \hyperlink{L.\ Shobana}{L.~Shobana}.
\authrefspace

\author{B.\ Vasuki \& L.\ Shobana}
\authorspace
\bsgitem{2018a}{}  Signed product cordial labeling for some families of graphs.  \emph{Int.\ J.\ Simulation Systems Sci.\ Technol.}\ 19 (2018), no.\ 4, art.\ 23, 6 pp. 
\annot{} More examples as in \xref{Baskar, Babujee, and Loganathan}{2011a}{Jayapal Baskar Babujee \& Shobana Loganathan}.
\par
\codes{\CLab:~\CVS:~\CSG,~\CBal}%
\itemspace

\author{Ebrahim Vatandoost}						%VATANDOOST
\authorspace
\authref  See \hyperlink{G.R.\ Omidi}{G.R.~Omidi}.
\itemspace

\authorx{Vijay V.\ Vazirani}						%VAZIRANI
\author{Vijay V.\ Vazirani \& Mihalis Yannakakis}
\authorspace
\bsgitem{1988a}{}  Pfaffian orientations, 0/1 permanents, and even cycles in directed graphs.  
In: Timo Lepist\"o and Arto Salomaa, eds., \emph{Automata, Languages and Programming}  (Proc.\ 15th Int.\ Colloq., Tampere, Finland, 1988),  pp.\ 667--681.  \LNCS 317.  Springer-Verlag, Berlin, 1988.  
\MR{1023669} (90k:68078).  \Zbl{648.68060}.
\annot{} Slightly abridged version of \xrefd{1989a}{Vijay V.\ Vazirani \& Mihalis Yannakakis}.
\codes{\CSD:~\CAdj,~\CBal:~\CAlgor}%
\itemspace

\bsgitem{1989a}{}  Pfaffian orientations, 0--1 permanents, and even cycles in directed graphs.  \emph{Discrete Appl.\ Math.}\ 25 (1989), 179--190.  
\MR{1031270} (91e:05080).  \Zbl{696.68076}.
\annot{} ``Evenness'' of a digraph (i.e., every signing contains a positive cycle) is polynomial-time equivalent to evaluability of a certain 0--1 permanent by a determinant and to parts of the existence and recognition problems for Pfaffian orientations of a graph.  Briefly expounded in \xref{Brundage}{1996a}{Michael Brundage}.]
\codes{\CSD:~\CAdj,~\CBal:~\CAlgor}%
\itemspace

\author{Michalis Vazirgiannis}						%VAZIRGIANNIS
\authorspace
\authref  See \hyperlink{Christos Giatsidis}{C.~Giatsidis} and \hyperlink{Fragkiskos D.\ Malliaros}{F.D.~Malliaros}.
\itemspace

\author{Alina Vdovina}						%VDOVINA
\authorspace
\authref  See \hyperlink{Shiping Liu}{S.-P.~Liu}.
\itemspace

\author{[Renata R.\ Del-Vecchio]}					%VECCHIO
\authorspace
\authref  See \hyperlink{Renata R.\ Del-Vecchio}{R.R.\ Del-Vecchio} (under `D').
\itemspace

\author{J.J.P.\ Veerman}						%VEERMAN
\authorspace
\bsgitem{2018a}{}  Social balance and the Bernoulli equations.  \emph{Amer.\ Math.\ Monthly} 125 (2018), 724--732.
\MR{3859644}.  \Zbl{1403.37099}.  \arXiv{1701.06946}.				
\annot{} Differential equation $\dot X(t) = X(t)^2$ with $X(0)=X_0$, an $n\times n$ invertible matrix, with $\lambda := \max$ real eigenvalue $>0$ having right and left eigenvectors $v$ and $w$.  Thm.\ 2:  $\exists!\ X(t)$ for $t\in[0,\lambda\inv)$; and $(\lambda\inv-1)X(t)\to vw\trans$ as $t\uparrow\lambda\inv$.  Meaning (\S4):  For $t\approx\lambda\inv$, $[n]$ develops up to 4 factions, two ``cohesive'' as in balance and (if $X_0$ is not symmetric) two ``dispersive'' that have antisymmetric relations with the first two.  [The paper's Thm.\ 2 is unnecessarily constrained by a probabilistic statement that requires $2|n$.]  \Cf\ \xref{Marvel, Kleinberg, \etal\!}{2011a}{Seth A.\ Marvel, Jon Kleinberg, Robert D.\ Kleinberg, \& Steven H.\ Strogatz} and \xref{Traag, Dooren, \etal\!}{2013a}{Vincent Antonio Traag, Paul Van Dooren, \& Patrick De Leenheer}.
\dated{12 Jun 2019} 
\codes{\CSD,~\CSG,~\CWD:~\CBal,~\CDyn}%
\itemspace

\author{Fernando Vega-Redondo}					%VEGA-REDONDO
\authorspace
\authref  See \hyperlink{George C.M.A.\ Ehrhardt}{G.C.M.A.~Ehrhardt}.
\itemspace

\author{Guido Veiner}						%VEINER
\authorspace
\bsgitem{1995a}{}  Some problems of approximating symmetric relations by equivalence relation.  \emph{Proc.\ Estonian Acad.\ Sci.\ Eng.}\ 1 (1995), no.\ 2, 105--112.
\MR{1640812} (99d:\-05083).  
URL \url{http://bit.ly/2j7tP4M}
\annot{} Clustering some types of signed $K_n$.
\dated{10 Nov 2017} 
\codes{\Csg:~\CClu}%
\itemspace

\author{Mark Velednitsky}						%VELEDNITSKY
\authorspace
\authref  See \hyperlink{Quico Spaen}{Q.~Spaen}.
\itemspace

\author{Alan Veliz-Cuba}						%VELIZ-CUBA
\authorspace
\authref  See also \hyperlink{Eduardo D.\ Sontag}{E.D.~Sontag}.
\authrefspace

\bsgitem{2011a}{}  Reduction of Boolean network models.  \emph{J.\ Theor.\ Biol.}\ 289 (2011), 167--172.
\MR{2973921} (no rev).  \Zbl{1397.92265}.  
\codes{\CSD:~\CDyn:~\CStr}\codesctd{\CSD:~\CDyn:~\CBiol}%
\itemspace

\author{Alan Veliz-Cuba, Boris Aguilar, Franziska Hinkelmann, \& Reinhard Laubenbacher}
\authorspace
\bsgitem{2014a}{}  Steady state analysis of Boolean molecular network models via model reduction and computational algebra.  \emph{BMC Bioinformatics} 21 (2014), art.\ 221, 8 pp.
\codes{\CSD:~\CDyn}%
\itemspace

\author{Alan Veliz-Cuba \& Reinhard Laubenbacher}
\authorspace
\bsgitem{2012a}{}  On the computation of fixed points in Boolean networks.  \emph{J.\ Appl.\ Math.\ Comput.}\ 39 (2012), 145--153.
\MR{2914469}.  \Zbl{1381.94145}.  
\codes{\CSD:~\CDyn}%
\itemspace

\author{Llu{\'\i}s Vena}						%VENA
\authorx{Lluis Vena}
\authorspace
\authref  See \hyperlink{Andrew Goodall}{A.~Goodall}.
\itemspace

\author{Venkat Venkatasubramanian}					%VENKATASUBRAMANIAN
\authorspace
\authref  See \hyperlink{Mano Ram Maurya}{M.R.\ Maurya}.
\itemspace

\author{V\'eronique Ventos} 						%VENTOS
\authorx{Veronique Ventos}
\authorspace
\authref  See \hyperlink{Pascal Berthome}{P.~Berthom\'e}, \hyperlink{Sylvie Corteel}{S.~Corteel}, and \hyperlink{David Forge}{D.~Forge}.
\itemspace

\author{Maryam Verdian-Rizi}						%VERDIAN-RIZI
\authorspace
\authref  See \hyperlink{Tony Huynh}{T.~Huynh}.
\itemspace

\author{Dirk Vertigan}						%VERTIGAN
\authorspace
\authref  See also \hyperlink{Jim Geelen}{J.~Geelen} and \hyperlink{James G.\ Oxley}{J.~Oxley}.
\authrefspace

\bsgitem{2015a}{}  Dowling geometries representable over rings.  \emph{Ann.\ Combin.}\ 19 (2015), 225--233.
\MR{3319870}.  \Zbl{1310.05054}.  
\annot{} Characterizes representability of the Dowling geometry $Q_n(\fG)$ over a ring, or equivalently a skew partial field, for any finite group $\fG$, thereby solving \xref{Pendavingh and van Zwam}{2013a}{R.A.\ Pendavingh \& S.H.M.\ van Zwam}, Problem 6.5.
\dated{28 Jan 2015} 
\codes{\Cgg:~\CMatrd}%
\itemspace

\author{Adrian Vetta}							%VETTA
\authorspace
\authref  See \hyperlink{Samuel Fiorini}{S.~Fiorini}, \hyperlink{Jim Geelen}{J.~Geelen}, and \hyperlink{Bruce Reed}{B.~Reed}.
\itemspace

\author{K.A.\ Vidya}							%VIDYA
\authorspace
\authref  See \hyperlink{E.\ Sampathkumar}{E.~Sampathkumar}.
\itemspace

\author{Sebastiano Vigna}						%VIGNA
\authorspace
\authref  See \hyperlink{Paolo Boldi}{P.~Boldi}.
\itemspace

\author{Fabien Vignes-Tourneret}					%VIGNES-TOURNERET
\authorspace
\authref  See also \hyperlink{Thomas Krajewski}{T.~Krajewski}.
\authrefspace

\bsgitem{2009a}{}  The multivariate signed Bollob\'as--Riordan polynomial.  \emph{Discrete Math.}\ 309 (2009), no.\ 20, 5968--5981.
\MR{2552629} (2011a:05162).  \Zbl{1228.05183}.  \arXiv{0811.1584}.			
\annot{} Multivariate version of the \xref{Chmutov and Pak}{2007a}{Sergei Chmutov \& Igor Pak} and \xref{Chmutov}{2009a}{Sergei Chmutov} signed ribbon-graph polynomials for orientation-embedded signed graphs.
	
[\Cf\ \xref{Krushkal}{2011a}{Vyacheslav Krushkal}.]
\dated{12 Jan 2012} 
\par
\codes{\CSGc:~\CTop,~\CInvar}%
\itemspace

\bsgitem{2011a}{}  Non-orientable quasi-trees for the Bollob\'as--Riordan polynomial.  \emph{European J.\ Combin.}\ 32 (2011), no.\ 4, 510--532.
\MR{2780852} (2012c:57007).  \Zbl{1226.05104}.  \arXiv{1102.1627}.			
\codes{\Csgc:~\CTop,~\CInvar}
\itemspace

\author{S.\ Vijay}							%VIJAY
\authorspace
\authref  See \hyperlink{V.\ Lokesha}{V.~Lokesha}, \hyperlink{K.V.\ Madhusudhan}{K.V.~Madhusudhan}, and \hyperlink{P.\ Siva Kota Reddy}{P.S.K.~Reddy}.
\itemspace

\author{Ambat Vijayakumar}						%VIJAYAKUMAR A
\authorspace
\authref  See \hyperlink{Zoran Stanic}{Z.~Stani\'c}.
\itemspace

\author{[G.K.\ Vijayakumar]}
\authorspace
\authref   See \hyperlink{G.R.\ Vijayakumar}{G.R.~Vijayakumar}.
\itemspace

\author{G.R.\ Vijayakumar}  						%VIJAYAKUMAR GR
\authorspace
\authref   See also \hyperlink{P.D.\ Chawathe}{P.D.~Chawathe}, \hyperlink{D.K.\ Ray-Chaudhuri, N.M.\ Singhi, \& G.R.\ Vijayakumar}{D.K.~Ray-Chaudhuri}, and \hyperlink{N.M.\ Singhi}{N.M.~Singhi}.
\authrefspace

\bsgitem{1984a}{}  (As ``G.K.~Vijayakumar'')  
A characterization of generalized line graphs and classification of graphs with eigenvalues at least $2$ [misprint for $-2$].  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 9 (1984), 182--192.
\MR{0959067} (89g:05055).  \Zbl{629.05046}.						
\codes{\Csg:~\CEig,~\Clg}%
\itemspace

\bsgitem{1987a}{\dag}  Signed graphs represented by $D_\infty$.  \emph{European J.\ Combin.}\ 8 (1987), 103--112.
\MR{0884068} (88b:05111).  \Zbl{678.05058}.						
\annot{} The finite signed simple graphs represented (see \xrefd{1993a}{G.R.\ Vijayakumar}) by a root system $D_n$ have a characterization by forbidden induced subgraphs, the largest of which has order 6.  [The complete list is given by \xref{Chawathe and Vijayakumar}{1990a}{P.D.\ Chawathe \& G.R.\ Vijayakumar}.  The representable signed graphs are the reduced line graphs of simply signed graphs without loops or half edges; see \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}.]  Also, remarks on signed graphs representable by $\bbR^\infty$.
\codes{\CSG:~\Cadj,~\CGeom,~\Cinc,~\Clg}%
\itemspace

\bsgitem{1992a}{}  Signed graphs represented by root system $E_8$.
Combinatorial Math.\ and Appl.\ (Proc., Calcutta, 1988).  \emph{Sankhya Ser.~A} 54 (1992), 511--517.
\MR{1234728} (94d:05072).  \Zbl{882.05118}.						
\annot{} The finite signed simple graphs represented (see \xrefd{1993a}{G.R.\ Vijayakumar}) by the root system $E_8$ are characterizable by forbidden induced subgraphs.  The largest of these subgraphs has order 10.
\codes{\CSG:~\Cadj,~\CGeom,~\Cinc}%
\itemspace

\bsgitem{1993a}{}  Algebraic equivalence of signed graphs with all eigenvalues $\geq -2$.  \emph{Ars Combin.}\ 35 (1993), 173--191.
\MR{1220518} (93m:05134).  \Zbl{786.05059}.						
\annot{} Finite signed simple graphs only.  $\lambda_{\min}(\Sigma) :=$ least eigenvalue of $A(\Sigma)$.  $\Sigma$ is ``represented'' by $W \subseteq \bbR^\infty$ if the vertices can be mapped into $W$ so that inner products equal signs of edges, where we interpret $\sigma(u,v) \in \{+1, -1, 0\}$.
Thm.\ 1: $\lambda_{\min}(\Sigma) \geq -2 \iff$ $\Sigma$ is represented by $\bbR^\infty$ $\iff$ $\Sigma$ is represented by $D_\infty$ or $E_8$ (the root systems).  
\annotctd{} $\Sigma'$ is ``algebraically equivalent'' to $\Sigma$ if it is obtained from $\Sigma$ by a sequence of switchings and algebraic transforms.  The latter means taking two positively adjacent vertices $a, b$ contained in no negative triangle, switching $b$, removing edges from $b$ to all common neighbors of $a$ and $b$, and adding an edge $xb$, for each $x \in N(a) \setminus N(b)$, with the same sign as $xa$.
Thm.\ 5: If $\Sigma$ is connected and $\lambda_{\min}(\Sigma) > -2$, then $\Sigma$ is algebraically equivalent to the Dynkin diagram of $A_n$, $D_n$, or $E_k$ ($k = 6,7,8$).
\annotctd{} There are other, similar results.
\codes{\CSG:~\Cadj,~\CGeom,~\Cinc}%
\itemspace

\bsgitem{1994a}{}  Representation of signed graphs by root system $E_8$.  \emph{Graphs Combin.}\ 10 (1994), 383--388.
\MR{1307045} (96a:05128).  \Zbl{821.05040}.						
\annot{} An algebraic characterization of the forbidden induced subgraphs for simple signed graphs represented by the root system $E_8$.  \Cf\ \xrefd{1992a}{G.R.\ Vijayakumar} and for definitions \xrefd{1993a}{G.R.\ Vijayakumar}.
\codes{\CSG:~\Cadj,~\CGeom,~\Cinc}%
\itemspace

\bsgitem{2007a}{}  Partitions of the edge set of a graph into internally disjoint paths.  \emph{Australas.\ J.\ Combin.}\ 39 (2007), 241--245.
\MR{2351204} (2008i:05156).  \Zbl{1134.05088}.
\annot{} A connected, contrabalanced biased graph $(\Gamma,\eset)$ has a covering by $\xi+1$ internally disjoint paths, where $\xi =$ cyclomatic number, iff every $(\Gamma\setminus v, \eset)$ has no balanced components.  
[\Question\ 1.  Can this generalize to all connected biased graphs?  Paths should become balanced subgraphs that are ``path-like'' (have at most two vertices of attachment).  $\xi$ should become a measure of the number of independent unbalanced circles.  \Question\ 2.  Is there a recursive decomposition of a 2-connected biased graph into $\xi$ path-like balanced subgraphs, generalizing the standard ear decomposition of a 2-connected, (contrabalanced biased) graph?]  
\dated{8 Mar 2008}
\codes{\Cgg:~\CStr,~\Ccircles}%
\itemspace

\bsgitem{2008a}{}  A graph labeling related to root lattices.  
In:  B.D.\ Acharya, S.\ Arumugam, and Alexander Rosa, eds., \emph{Labelings of Discrete Structures and Applications} (Mananthavady, Kerala, 2006), pp.\ 175--179.  Narosa, New Delhi, 2008.
\MR{2391786} (2009e:05281) (book).  \Zbl{1180.05111}.
\annot{} A ``2-fold labeling'' of a signed simple graph $\Sigma$ is a nonzero $f \in \Nul[2I + A(\Sigma)]$.  Thm.\ 3:  If $\Sigma$ has a 2-fold labeling, it has one such that, for some $v \in V$, $f(v)=1$ and every $f(w)$ is an integer.  Applied to prove via signed graphs the classification of root systems with root length $\surd2$.
\par
\codes{\CSG:~\CEig,~\CGeom}%
\itemspace

\bsgitem{2009a}{}  A method of classifying all simply laced root systems.  \emph{J.\ Algebra Appl.}\ 8 (2009), no.\ 4, 533--537.
\MR{2555519} (2010k:17016).  \Zbl{1172.05337}.
\annot{} Thm.\ 6:  For a connected signed simple graph with eigenvalues $\leq2$, $2I-A(\Sigma)$ is the Gram matrix of a subset of root system $D_n$ or $E_8$.
\par
\codes{\CSG:~\CAdj,~\CGeom}%
\itemspace

\bsgitem{2011a}{}  A property of weighted graphs without induced cycles of nonpositive weights.  \emph{Discrete Math.}\ 311 (2011), no.\ 14, 1385--1387.
\MR{2795549} (2012f:05127).  \Zbl{1238.05128}.  
\annot{} Let $\Gamma$ be 2-connected.  Thm.:  If $\phi: E \to \bbZ_{\leq1}$, extended to $S\subset E$ by $\phi(S) := \sum_{e\in S} f(e)$, satisfies $\phi(C)>0$ for every induced circle, then $\phi(E)>0$.  Cor.:  If $\phi: E \to \bbR$ satisfies $\phi(C)>0$ for every circle (not necessarily induced), then $\phi(E)>0$.  Cor.:  If $\Sigma$ has $\#E^+(C)>\#E^-(C)$ for every induced circle, then $\#E^+(\Sigma) > \#E^-(\Sigma)$, a conjecture from \xref{B.G.~Xu}{2009a}{Baogen Xu}.  [\Cf\ \xref{Balakrishnan and Sudharsanam}{1982a}{R.\ Balakrishnan \& N.\ Sudharsanam} where equality is treated.]
\dated{16 Oct 2011}
\codes{\CSGw:~\CGen}%
\itemspace

\bsgitem{2011b}{}  Equivalence of four descriptions of generalized line graphs.  \emph{J.\ Graph Theory} 67 (2011), no.\ 1, 27--33.
\MR{2809559} (2012e:05318).  \Zbl{1226.05185}.					
\codes{\Csg:~\CLG}%
\itemspace

\bsgitem{2012a}{}  Spectral numbers related to signed graphs.  \emph{Utilitas Math.}\ 87 (2012), 33--40.
\MR{2919984}.  \Zbl{1264.05083}.
\annot{} Let $\Spec(\Sigma) :=$ spectrum of $A(\Sigma)$, $I(\Sigma) := [\min\Spec(\Sigma), \max\Spec(\Sigma)]$.  If $\rho\in\bbR$ is such that $\rho \in I(\Sigma) \implies \rho \in \Spec(\Sigma')$ for some $\Sigma' \subseteq \Sigma$, then $\rho = 0, \pm1, \pm2$.  If $\Sigma'$ must be an induced subgraph, $\rho = 0, \pm1, \pm\surd 2, \pm2$.  (Also, similar results for graphs.)
\codes{\CSG:~\CEig}%
\itemspace

\bsgitem{2013a}{}  Some results on weighted graphs without induced cycles of nonpositive weights.  \emph{Graphs Combin.}\ 29 (2013), no.\ 4, 1101--1111.
\MR{3070078}.  \Zbl{1268.05101}.  
\codes{\CSGw:~\CGen}%
\itemspace

\bsgitem{2014a}{}  From finite line graphs to infinite derived signed graphs.  \emph{Linear Algebra Appl.}\  453 (2014), 84--98.
\MR{3201686}.  \Zbl{1314.05174}.  
\codes{\CSG:~\CLG,~\CEig}%
\itemspace

\author{G.R.\ Vijayakumar (as ``Vijaya Kumar''), S.B.\ Rao, \& N.M.\ Singhi}
\authorspace
\bsgitem{1982a}{} Graphs with eigenvalues at least $-2$.  \emph{Linear Algebra Appl.}\ 46 (1982), 27--42. 
\MR{0664693} (83m:05099).  \Zbl{494.05044}.						
\annot{} A minimal forbidden induced graph has order at most 10, which is best possible.  
\dated{2 Aug 2010} 
\codes{\Csg:~\Cadj,~\CGeom,~\Clg}%
\itemspace

\author{G.R.\ Vijayakumar \& N.M.\ Singhi} 
\authorspace
\bsgitem{1990a}{}  Some recent results on signed graphs with least eigenvalues $\geq -2$.  
In:  Dijen Ray-Chaudhuri, ed., \emph{Coding Theory and Design Theory Part I: Coding Theory} (Proc.\ Workshop IMA Program Appl.\ Combin., Minneapolis, 1987--88), pp.\ 213--218.  IMA Vol.\ Math.\ Appl., Vol.~20.  Springer-Verlag, New York, 1990.
\MR{1047882} (91e:05069).  \Zbl{711.05033}.						
\codes{\CSG:~\CGeom,~\Clg,~\Cadj:~\CExp}%
\itemspace

\author{K.S.\ Vijayan} 						%VIJAYAN
\authorspace
\authref  See \hyperlink{S.B.\ Rao}{S.B.~Rao}.
\itemspace

\author{Viji Paul}							%VIJI P
\authorspace
\authref  See also \hyperlink{Shahul Hameed K}{S.~Hameed~K}.
\authrefspace

\bsgitem{2012a}{}  \emph{Labeling and Set-Indexing Hypergraphs of a Graph and Related Topics}.  Doctoral  thesis, Kannur University, 2012.
\annot{} Ch.\ 5, ``Co-regular signed graphs'':  $\Sigma$ is $(r,k)$-``co-regular'' if $|\Sigma|$ is $r$-regular and $\Sigma^+$ is $\frac12(r+k)$-regular.  
An $r$-regular $\Gamma$ has an $(r,k)$-co-regular signing iff $\Gamma$ has a $\frac12(r+k)$-factor.  Examples are treated.  
Thm.\ 5.4.2:  Every $\Sigma$ is an induced subgraph of an $(r,k)$-co-regular signed graph for all $r,k$ satisfying $r \geq \Delta(\Sigma^+)+\Delta(\Sigma^-)$, $r \equiv k\ \mod2$, and $2\Delta(\Sigma^+)-r \leq k \leq r-2\Delta(\Sigma^-)$ ($\Delta =$ max degree).
\dated{22 Nov 2012} 
\codes{\CSGc:~\CLab}%
\itemspace

\authorx{V.\ Vilfred}							%VILFRED
\author{V.\ Vilfred \& C.\ Jayasekaran}
\authorspace
\bsgitem{2009a}{}  Interchange similar self vertex switchings in graphs.  \emph{J.\ Discrete Math.\ Sci.\ Cryptogr.}\ 12 (2009), no.\ 4, 467--480.
\MR{2589065} (2011f:05206).  \Zbl{1180.05089}.  
\annot{} See \xref{Jayasekaran}{2007a}{C.\ Jayasekaran}.  Examines self-switching vertices that are interchanged by an automorphism of $\Gamma$ (``interchange similar'').
\dated{26 Sept 2012} 
\codes{\Ctg:~\CSw}%
\itemspace

\author{V.\ Vilfred, J.\ Paulraj Joseph, \& C.\ Jayasekaran}
\authorspace
\bsgitem{2008a}{}  Branches and joints in the study of self switching of graphs.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 67 (2008), 111--122.
\MR{2457789} (2010b:05165).  \Zbl{1184.05127}.  
\annot{} See \xref{Jayasekaran}{2007a}{C.\ Jayasekaran}.  Examines when a cut vertex $x$ is self switching, assuming all self-switching vertices are interchanged by automorphisms.
\dated{26 Sept 2012} 
\codes{\Ctg:~\CSw}%
\itemspace

\bsgitem{2010a}{}  Self vertex switchings of trees.  In:  T.\ Tamizh Chelvam \etal, eds., \emph{Algebra, Graph Theory and Their Applications}, pp.\ 118--128.  Narosa Publishing House, New Delhi, 2010.
\Zbl{1220.05019}.  
\annot{} See \xref{Jayasekaran}{2007a}{C.\ Jayasekaran}.  Characterizes forests with a self-switching vertex.
\dated{26 Sept 2012} 
\codes{\Ctg:~\CSw}%
\itemspace

\author{Jacques Villain} 						%VILLAIN
\authorspace
\bsgitem{1959a}{}  La structure des substances magnetiques.  \emph{J.\ Phys.\ Chem.\ Solids} 11 (1959), 303--309.
\annot{} Some physics spin models do, or may, or do not, have negative edges ($J<0$).  Precedes recognition of frustration and switching.
\dated{11 Aug 2018} 
\codes{\CPhys:~\Csg}%
\itemspace

\bsgitem{1977a}{}  Spin glass with non-random interactions.  \emph{J.\ Phys.\ C:  Solid State Phys.}\ 10 (1977), no.~10, 1717--1734.
\annot{} Partition function of ``odd model'' on signed lattice graph, i.e., all ``elementary polygons'' (``plaquettes'' in \xref{Toulouse}{1977a}{Gerard Toulouse}: small chordless circles: squares, triangles, or hexagons in different lattices) are unbalanced.  Spins $S_i:=S(v_i)$ may be Ising (in $\{\pm1\}=S^0$), XY (in $S^1$), or Heisenberg (in $S^2$).  
\S3, ``The Ising version of the odd model on the two-dimensional, square lattice''.  
\S4, ``XY version of the odd model on the two-dimensional, square lattice''.  Unique ground state up to a $\pm1$ variable.  
\S5, ``The odd model on the diamond lattice'':  Ising vs.\ higher-dimensional ground states.  
\S6, ``Magnetic susceptibility of the odd model on the diamond lattice (XY version)''.  
\S7, ``Modified odd models'':  A few balanced plaquettes.  Or, slightly varying spin magnitudes.
\S8, ``The odd model on other types of lattice''.  
App.\ 1:  The odd model switches to periodic form for the lattices treated herein.  
\dated{17 Jun 2012, 10 Aug 2018}
\codes{\CSG:~\CPhys,~\CSw,~\CVS(\CGen)}%
\itemspace

\bsgitem{1977b}{}  Two-level systems in a spin-glass model:  I.  General formalism and two-dimen\-sional model. 
\emph{J.\ Phys.\ C:  Solid State Phys.}\ 10 (1977), 4793--4803.
\codes{\CPhys:~\CSG}%
\itemspace

\bsgitem{1978a}{}  Two-level systems in a spin-glass model:  II.  Three-dimensional model and effect of a magnetic field.  
\emph{J.\ Phys.\ C:  Solid State Phys.}\ 11 (1978), no.~4, 745--752.
\par
\codes{\CPhys:~\CSG}\codesctd{\CGG:~\CPhys,~\CSw,~\CBal}%
\itemspace

\author{Rafael H.\ Villarreal}						%VILLARREAL
\authorspace
\authref  See also \hyperlink{Jose Martinez-Bernal}{J.~Mart\'{i}nez-Bernal}, \hyperlink{Aron Simis}{A.~Simis}, and \hyperlink{Carlos E.\ Valencia}{C.E.~Valencia}.
\authrefspace

\bsgitem{1995a}{}  Rees algebras of edge ideals.  \emph{Commun.\ Algebra} 23 (1995), no.\ 9, 3513--3524.
\MR{1335312} (96e:13005).  \Zbl{836.13014}.  
\annot{} The algebra for $\Gamma$ is closely related to the frame and lift matroids of $-\Gamma$.  
Cor.\ 3.2:  The edge ideal $I(\Gamma)$ is of linear type iff $\bfF(-\Gamma)$ is a free matroid.  
Lem.\ 3.1:  The even cycle space has codimension $\rk \bfF(-\Gamma)$ [as found in \xref{van Nuffelen}{1973a}{Cyriel van Nuffelen}].  
Prop.\ 4.2:  The elementary vectors of $\Nul \Eta(-\Gamma)$, i.e., the integral cycle space, are the indicator vectors of frame circuits [as in \xref{Doob}{1973a}{Michael Doob}]. 
\dated{3 Jun 2015, rev 18 Aug 2018, 22 Oct 2020} 
\codes{\Csg:~\CPar:~\CInc~\CAlgeb,~\Cmatrd}%
\itemspace

\bsgitem{1998a}{}  On the equations of the edge cone of a graph and some applications.  \emph{manu\-scrip\-ta math.}\ 97 (1998), 309--317.
\MR{1654776} (99i:05143).  \Zbl{920.13015}.  
\par
\codes{\Cecyc:~\CGeom,~\CInc}%
\itemspace

\author{J.\ Villegas}							%VILLEGAS
\authorspace
\authref  See \hyperlink{E.E.\ Vogel}{E.E.~Vogel}.
\itemspace

\author{Daniele Vilone}						%VILONE
\authorspace
\authref  See \hyperlink{Filippo Radicchi}{F.~Radicchi}.
\itemspace

\author{Andrew Vince} 						%VINCE
\authorspace
\bsgitem{1983a}{}  Combinatorial maps.  \emph{J.\ Combin.\ Theory Ser.\ B} 34 (1983), 1--21.  
\MR{0701167} (84i:05048).  \Zbl{505.05054}.   
\annot{} See Theorem 6.1. 
\codes{\Csg:~\Cbal:~\CTop}%
\itemspace

\authorx{E.\ Vincent}							%VINCENT
\author{E.\ Vincent, J.\ Hammann, \& M.\ Ocio}
\authorspace
\bsgitem{1992a}{}  Slow dynamics in spin glasses and other complex systems.  In:  D.H.\ Ryan, ed., \emph{Recent Progress in Random Magnets}, pp.\ 207--236.  World Scientific, Singapore, 1992.
\annot{} Surveys experiments with spin glass materials, especially aging behavior.  Observations tend to support a landscape of graph signatures with numerous metastable states, subdividing as temperature decreases.  
[Presumably, the states correspond to clusters of low-frustration states, separated by high-frustration barriers, subdivided into smaller clusters by lower-frustration barriers, and so on.  Mathematical examination is needed.]
\dated{27 Aug 1998, 24 Aug 2012} 
\codes{\CPhys:~\Csg:~\CState:~\CDyn}%
\itemspace

\author{Miguel Angel Virasoro}						%VIRASORO
\authorspace
\authref  See \hyperlink{Marc Mezard}{M.~M\'ezard}.
\itemspace

\author{Lee Inmon Virden}						%VIRDEN
\authorspace
\authref  See \hyperlink{Talmage James Reid}{T.J.~Reid}.
\itemspace

\author{Krishnamurthy Viswanathan}					%VISWANATHAN
\authorspace
\authref  See \hyperlink{Ron M.\ Roth}{R.M.~Roth}.
\itemspace

\author{Fabio Vitale}							%VITALE
\authorspace
\authref  See \hyperlink{Nicolo Cesa-Bianchi}{N.~Cesa-Bianchi} and \hyperlink{Geraud Le Falher}{G.~Le~Falher}.
\itemspace

\author{J.\ Vlach}							%VLACH
\authorspace
\authref  See \hyperlink{Chuan-Jin Shi}{C.J.~Shi}.
\itemspace

\author{Alexander Vogel}						%VOGEL
\authorspace
\authref  See \hyperlink{Eckhard Steffen}{E.~Steffen}.
\itemspace

\author{E.E.\ Vogel [Eugenio E.\ Vogel Matamala]}				%VOGEL
\authorx{E.E.\ Vogel}	
\authorspace
\authref  See also \hyperlink{B.\ Fierro}{B.~Fierro}, \hyperlink{W.\ Lebrecht}{W.~Lebrecht}, \hyperlink{A.J.\ Ramirez-Pastor}{A.J.~Ram{\'\i}rez-Pastor}, \hyperlink{F.\ Roma}{F.~Rom\'a}, \hyperlink{M.C.\ Salas-Solis}{M.C.~Salas-Sol{\'\i}s}, and \hyperlink{J.F.\ Valdes}{J.F.~Vald\'es}.
\authrefspace

\author{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}
\authorspace
\bsgitem{1994a}{} Ground-state properties of finite square and triangular Ising lattices with mixed exchange interactions.   \emph{Phys.\ Rev.\ B} 49 (1994), no.\ 9, 6018--6027.
\annot{} Many small toroidal lattice examples are computed to estimate large-scale behavior of ground states: frustration index, ground state energy $E_g$, etc.  The signs are random, half positive and half negative.  Also near-ground states (energy $E_g+4$), obtained either by switching one vertex of a ground state or as a local minimum in the state space.
\annotctd{} Dictionary (for papers of Vogel \etal):  ``periodic boundary conditions'' = toroidal; ``plaquette'' = face boundary circle; ``curved'' plaquette = frustrated = negative; ``frustration length'' = frustration index; ``ground state'' = minimizing switching function $\zeta$ of $\Sigma$ (i.e.; $\#E^-(\Sigma^\zeta) = \min_\zeta = l$); ``ground state degeneracy'' $W$ = number of ground states; ``magnetization per spin'' (of ground state) = $[\#\zeta\inv(+) - \#\zeta\inv(-)] / nW$; ``remnant entropy'' = $(\ln W)/n$.
	
\dated{4 Jan 2015} 
\codes{\CPhys,~\CSG:~\CState(\Cfr),~\CFr}%
\itemspace

\author{Eugenio E.\ Vogel, Jaime Cartes, Patricio Vargas, \& Dora Altbir}
\authorspace
\bsgitem{2000a}{}  Simulation of hysteresis for $\pm J$ triangular lattices.  \emph{Physica B} 284--288 (2000), 1211--1212.
\codes{\CSG,~\CPhys:~\CState(\Cfr),~\CFr}%
\itemspace

\author{E.E.\ Vogel, J.\ Cartes, P.\ Vargas, D.\ Altbir, S.\ Kobe, T.\ Klotz, \& M.\ Nogala}
\authorspace
\bsgitem{1999a}{}  Hysteresis in $\pm J$ Ising square lattices.  \emph{Phys.\ Rev.\ B} 59 (1999), no.\ 5, 3325--3328.
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\author{E.E.\ Vogel, S.\ Contreras, J.\ Cartes, \& M.A.\ Osorio}
\authorspace
\bsgitem{1996a}{}  Non-frustrated domains in Ising spin glasses with competing interactions.  In:  F.\ Leccabue and V.\ Sagredo, eds., \emph{Magnetism, Magnetic Materials and Their Applications} (Proc., M\'erida, Venezuela, 1995), pp.\ 152--160.  World Scientific, Singapore, 1996.
\annot{} In a signed, toroidal square lattice graph keep the edges that are satisfied in every ground state.  500 $6\times6$ examples were computed.  $h :=$ proportion of retained edges tends to $\approx .5$.  The distribution of component sizes is quite different from that of random subgraphs of similar size.  For large $h$ there is a boundary-linking giant component.  For $h \approx .5$ there tends to be a definite proportion of small components of particular shapes.
\dated{8 Feb 2015} 
\codes{\CSG:~\CFr:~\CState,~\CPhys}%
\itemspace

\author{E.E.\ Vogel, S.\ Contreras, W.\ Lebrecht, \& J.\ Cartes}
\authorspace
\bsgitem{1995a}{}  Order parameters for various Ising lattices with competing $\pm J$ interactions.  \emph{J.\ Magnetism Magnetic Materials} 140--144 (1995), 1793--1794.
\par
\codes{\CSG,~\CPhys:~\CFr:~\CState}%
\itemspace

\author{E.E.\ Vogel, S.\ Contreras, F.\ Nieto, \& A.J.\ Ram{\'\i}rez-Pastor}
\authorx{E.E.\ Vogel, S.\ Contreras, F.\ Nieto, \& A.J.\ Ramirez-Pastor}
\authorspace
\bsgitem{1998a}{}  Percolation in $\pm J$ Ising lattices after removing frustration.  \emph{Physica A} 257 (1998), 256--263.
\codes{\CSG,~\CPhys:~\CState(\Cfr),~\CSw}%
\itemspace

\author{E.E.\ Vogel, S.\ Contreras, M.A.\ Osorio, \& J.\ Cartes}
\authorspace
\bsgitem{1998a}{}  Bond percolation in $\pm J$ Ising square lattices diluted by frustration.  \emph{Phys.\ Rev.\ B} 58 (1998), no.\ 13, 8475--8480.
\annot{} Square lattices on the torus.  ``Diluted'' means removing every edge that is frustrated in one or more ground states.
	
Dictionary:  \cf\ \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}.
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CState(\Cfr),~\CSw}%
\itemspace

\author{E.E.\ Vogel, S.\ Contreras, M.A.\ Osorio, A.J.\ Ram{\'\i}rez-Pastor, \& F.\ Nieto}
\authorx{E.E.\ Vogel, S.\ Contreras, M.A.\ Osorio, A.J.\ Ramirez-Pastor, \& F.\ Nieto}
\authorspace
\bsgitem{1999a}{}  Percolating spin-glass domains in diluted $\pm J$ square lattices.  \emph{Physica A} 266 (1999), 425--429.
\annot{} Square lattices on the torus.  ``Diluted'' means removing every edge that is frustrated in one or more ground states.
	
Dictionary:  \cf\ \xref{Vogel, Cartes, Contreras, Lebrecht, and Villegas}{1994a}{E.E.\ Vogel, J.\ Cartes, S.\ Contreras, W.\ Lebrecht, \& J.\ Villegas}.
\dated{3 Jan 2015} 
\par
\codes{\CSG,~\CPhys:~\CState(\Cfr),~\CSw}%
\itemspace

\author{E.E.\ Vogel \& W.\ Lebrecht}
\authorspace
\bsgitem{1997a}{}  Rapidly converging asymptotic expansions in $\pm J$ Ising lattices.  \emph{Z.\ Physik} 102 (1997), 145--155.
\annot{} The expansions are for functions of signed toroidal triangular, square, and honeycomb lattices such as the ``frustration length'' = minimum length of a $T$-join in the dual graph connecting frustrated plaquettes in pairs, and the proportion of edges that are positive in $\Sigma^\zeta$ for every ground state $\zeta$ .   Applies two theoretical methods and compares to computed examples.
\dated{9 Jan 2015} 
\codes{\CSG,~\CPhys:~\CState,~\CState(\Cfr):~\CInvar}%
\itemspace

\author{E.E.\ Vogel, A.J.\ Ramirez-Pastor, \& F.\ Nieto}
\authorspace
\bsgitem{2002a}{}  Detailed structure of configuration space and its importance on ergodic separation of $\pm J$ Ising lattices.  \emph{Physica A} 310 (2002), 384--396.
\MR{1946320} (2003k:82024).  \Zbl{995.82018}.  
\annot{} ``All'' signed square lattices studied for physical properties like order parameters and grouping of ground states (minimal switchings).  
Related numerical results in \xref{Lebrecht, Vogel, and Vald\'es}{2002a}{W.\ Lebrecht, E.E.\ Vogel, \& J.F.\ Valdes}.
\dated{3 Jan 2015} 
\codes{\CSG,~\CPhys:~\CState(\Cfr)}%
\itemspace

\author{E.E.\ Vogel, J.F.\ Vald\'es, \& W.\ Lebrecht}
\authorx{E.E.\ Vogel, J.F.\ Valdes, \& W.\ Lebrecht}
\authorspace
\bsgitem{2006a}{}  $\pm J$ Ising model on Archimedean $(4, 8^2)$ lattices.  \emph{Physica A} 371 (2006), 150--154.
\codes{\CSG:~\CState(\Cfr),~\CPhys}%
\itemspace

\author{Margit Voigt}							%VOIGT
\authorspace
\authref  See \hyperlink{Arnfried Kemnitz}{A.~Kemnitz}.
\itemspace

\author{G.E.\ Volovik}						%VOLOVIK
\authorspace
\authref  See \hyperlink{I.E.\ Dzyaloshinskii}{I.E.~Dzyaloshinskii}.
\itemspace

\author{Jan Vondr\'ak}						%VONDRAK
\authorx{Jan Vondrak}
\authorspace
\authref  See \hyperlink{Anna Galluccio}{A.~Galluccio}.
\itemspace

\author{Axel von Kamp}						%VON KAMP
\authorspace
\authref  See \hyperlink{Steffen Klamt}{S.~Klamt}.
\itemspace

\author{Heinz-J\"urgen Voss}						%VOSS
\authorx{Heinz-Jurgen Voss}
\authorspace
\authref  See also \hyperlink{Daniel Kral'}{D.~Kr\'al'}.
\authrefspace

\bsgitem{1991a}{}  \emph{Cycles and Bridges in Graphs}.  Math.\ Appl.\ (E.\ Europ.\ Ser.), Vol.\ 49.  Kluwer, Dordrecht, and Deutscher Verlag der Wissenschaften, Berlin, 1991.
\MR{1131525} (92m:05118).  \Zbl{731.05031}.
\annot{} \S3.4, ``The length of bridges of longest odd and even cycles.''  \S3.9, ``The `odd circumference' in bridges of longest odd cycles.''  \S7.6, ``Longest odd and even cycles, \dots.''  \S8.4, ``Odd and even cycles with a given number of diagonals.''  \S10.1, ``Long cycles and long even cycles with many diagonals.''  \S10.3, ``Long odd cycles with many diagonals in non-bipartite graphs.''  
[\Problem.  Generalize results on even and odd circles to signed graphs.  \Cf, e.g., \xref{Conlon}{2004a}{Joseph G.\ Conlon}.]
\codes{\Csg:~\Cpar:~\CCircles}%
\itemspace

\author{Jo\v{z}e Vrabek} 						%VRABEK
\authorspace
\authref   See \hyperlink{Tomaz Pisanski}{T.~Pisanski}.
\itemspace

\author{Eleni-Maria E.\ Vretta}						%VRETTA
\authorspace
\authref  See \hyperlink{Konstantinos Papalamprou}{K.~Papalamprou} and \hyperlink{Leonidas S.\ Pitsoulis}{L.~Pitsoulis}.
\itemspace

\author{Hung Thanh Vu}						%VU HT
\authorspace
\authref  See \hyperlink{Tuyen-Thanh-Thi Ho}{T.T.T.~Ho}.
\itemspace

\author{Damir Vuki\v{c}evic}						%VUKICEVIC
\authorx{Damir Vukicevic}
\authorspace
\authref  See \hyperlink{Tomislav Doslic}{T.~Do\v{s}li\'c}.
\itemspace

\author{Kristina Vu\v{s}kovi\'c}						%VUSKOVIC
\authorx{Kristina Vuskovic}
\authorspace
\authref   See also \hyperlink{Pierre Aboulker}{P.~Aboulker}, \hyperlink{Valerio Boncompagni}{V.~Boncompagni}, \hyperlink{Michele Conforti}{M.~Conforti}, \hyperlink{Ton Kloks}{T.~Kloks}, and \hyperlink{Nicolas Trotignon}{N.~Trotignon}.
\authrefspace

\bsgitem{2010a}{}  Even-hole-free graphs: a survey.  \emph{Appl.\ Anal.\ Discrete Math.}\ 4 (2010), no.\ 2, 219--240.   
\MR{2724633}.  \Zbl{1265.05518}.  
\annot{} [\Question.  How does the concept of an even (or odd) hole generalize to signed graphs?]
\dated{11 Jul 2022} 
\par
\codes{\CSG:~\CBal(\CGen):~\CStr,~\CAlgor:~\CExp}%
\itemspace

\bsgitem{2013a}{}  The world of hereditary graph classes viewed through Truemper configurations.  In:  Simon R.\ Blackburn, Stefanie Gerke, and Mark Wildon, eds., \emph{Surveys in Combinatorics 2013} (24th British Combin.\ Conf., London, 2013), Ch.\ 7, pp.\ 265--325.  London Math.\ Soc.\ Lect.\ Note Ser., Vol.\ 409.  Cambridge Univ.\ Press, Cambrige, Eng., 2013.
\MR{3184115}.  \Zbl{1301.05279}.  
\codes{\Csg:~\CEig:~\CExp}%
\itemspace


\label{LETTER W} \hypertarget{LETTER W}{}	% LETTER  W


\author{Michelle L.\ Wachs}						%WACHS
\authorspace
\authref  See also \hyperlink{Anders Bjorner}{A.~Bj\"orner}, \hyperlink{Eric Gottlieb}{E.~Gottlieb}, and \hyperlink{Jeffrey B.\ Remmel}{J.B.~Remmel}.
\authrefspace

\bsgitem{2007a}{}  Poset topology: tools and applications.  In:  E.\ Miller, V.\ Reiner, and B.\ Sturmfels, eds., \emph{Geometric Combinatorics}, pp.\ 497--615.  American Math.\ Soc.\ and Institute for Advanced Study (IAS),
Providence, R.I., and Princeton, N.J., 2007.
\MR{2383132} (no rev).  \Zbl{1135.06001}.  \arXiv{math/0602226}.			
\par
\codes{\Csg:~\Csg:~\CTop,~\CAlgeb:~\CExp}%
\itemspace

\author{Donald K.\ Wagner} 						%WAGNER DK
\authorspace
\authref   See also \hyperlink{Vijaya Chandru}{V.~Chandru} and \hyperlink{Collette R.\ Coullard}{C.R.~Coullard}.
\authrefspace

\bsgitem{1985a}{\dag\dag}  Connectivity in bicircular matroids.  \emph{J.\ Combin.\ Theory Ser.~B} 39 (1985), 308--324.  
\MR{0815399} (87c:05041).  \Zbl{584.05019}.						
\annot{} Prop.\ 1 and Thm.\ 2 show that $n$-connectivity of the bicircular matroid $BG(\Gamma)$ is equivalent to ``$n$-biconnectivity'' of $\Gamma$.  
\annotctd{} When do two $3$-biconnected graphs have isomorphic bicircular matroids?  \S5 proves that $3$-biconnected graphs with $> 4$ vertices have isomorphic bicircular matroids iff one is obtained from the other by a sequence of operations called ``edge rolling'' and ``$3$-star rotation''.  This is the bicircular analog of Whitney's circle-matroid isomorphism theorem, but it is complicated.  [An important theorem, generalized to all bicircular matroids in \xref{Coullard, del Greco, and Wagner}{1991a}{Collette R.\ Coullard, John G.\ del Greco, \& Donald K.\ Wagner}.  \emph{Major Research Problems}.  Generalize to frame matroids of biased graphs.  Find the analog for lift matroids.]
\codes{\CBic:~\CStr}%
\itemspace

\bsgitem{1988a}{}  Equivalent factor matroids of graphs.  \emph{Combinatorica} 8 (1988), 373--377.  
\MR{0981894} (90d:05071).  \Zbl{717.05022}.						
\annot{} ``Factor matroid'' = even-cycle matroid $\bfF(-\Gamma)$.  Decides when $\bfF(-\Gamma) \cong \bfM(\Beta)$ where $\Beta$ is a given bipartite, 4-connected graph.
\codes{\CECyc:~\CStr}%
\itemspace

\author{H.\ Wagner}							%WAGNER H
\authorspace
\authref  See \hyperlink{K.\ Druhl}{K.~Dr\"uhl}.
\itemspace

\author{Magnus Wahlstr\"om}						%WAHLSTROM
\authorx{Magnus Wahlstrom}
\authorspace
\authref  See \hyperlink{Sebastian Bocker}{S.~B\"ocker} and \hyperlink{Stefan Kratsch}{S.~Kratsch}.
\itemspace

\author{Bronislaw Wajnryb} 						%WAJNRYB
\authorspace
\authref   See \hyperlink{Ron Aharoni}{R.~Aharoni}.
\itemspace

\author{Oliver Waldmann}						%WALDMANN
\authorspace
\authref  See \hyperlink{Kartik Chandra Mondal}{K.C.~Mondal}.
\itemspace

\authorx{M.H.\ Waldor}						%WALDOR
\author{M.H.\ Waldor, W.F.\ Wolff, \& J.\ Zittartz}
\authorspace
\bsgitem{1985a}{}  Ising models on the pentagon lattice.  \emph{Z.\ Phys.\ B} 59 (1985), no.\ 1, 43--51.
\MR{0788876} (86e:82054).  
\annot{} Physics of all-positive and all-negative (``fully frustrated'': all face boundary circles are negative) signs as examples.  \S~II, ``Thermodynamics'', b) ``Homogeneous case'':  The all-negative signature has multiple ground states that have energy $-J$ ($J =$ bond strength) per vertex because the frustrated edges form a perfect matching in a ground state.  
[I.e., frustration index $l(-\Gamma) = \#V$ for these pentagonal lattice graphs, assuming no boundary as, e.g., when the lattice is toroidal.]
\dated{18 Jun 2012} 
\codes{\CPhys:~\CSG:~\CPar:~\CState(\Cfr)}%
\itemspace

\authorx{H.B.\ Walikar}						%WALIKAR
\author{H.B.\ Walikar, Satish V.\ Motammanavar, \& B.D.\ Acharya}
\authorspace
\bsgitem{2015a}{}  Signed domination in signed graphs.  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 40 (2015), 107--128.
\Zbl{1358.05135}.  
\annot{} Like \xref{B.D.\ Acharya}{2012b}{B.\ Devadas Acharya}, but a signed domination function is $\mathbf{f} \in \{\pm1\}^V$.  NASC for a signed path, star, circle, or caterpillar to admit such a function.
Every signed graph is an induced subgraph of a signed graph that admits one.
\dated{18 May 2018, 29 Dec 2020} 
\codes{\CSG:~\CLab,~\CDom}%
\itemspace

\author{Derek A.\ Waller} 						%WALLER*
\authorspace
\authref   See also \hyperlink{F.W.\ Clarke}{F.W.~Clarke}.
\authrefspace

\bsgitem{1976a}{\dag}  Double covers of graphs.  \emph{Bull.\ Austral.\ Math.\ Soc.}\ 14 (1976), 233--248.  
\MR{0406876} (53 \#10662).  \Zbl{318.05113}.					
\annot{} The prehistory of signed covering graphs (\cf\ \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, \S6); the language has changed.  
\annotctd{} Signed covers $\tilde\Sigma$ in terms of covering projections $\downarrow$ viewed as graph homomorphisms $\to$.  \S2: Examples ``Kronecker double cover'' = signed cover of $-\Gamma$; disjoint union = signed cover of $+\Gamma$.  General definition.  
\S3, ``Antipodal double covers'', i.e., the pair covering a vertex are the unique pair at max distance in $\tilde\Sigma$.  [\Problem.  If $\Gamma$ has an antipodal cover, what are the signs?]  Thm.\ 3.1:  If $\Delta_i$ antipodally covers $\Gamma_i$ ($i=1,2$), then $\Delta_1 \,\square\, \Delta_2$ antipodally covers a graph.  
\annotctd{} \S4, ``Pullbacks of double covers'':  Prop.\ 4.1:  If $\Gamma_2 \to \Gamma_1$ and $\Delta_1 \downarrow \Gamma_1$, then $\exists\ \Delta_2 \to \Delta_1$ that $\downarrow \Gamma_2$.  Choice of spanning tree $T$ means fixing $\sigma|_T \equiv +$.  [``Nontrivial'' edge = negative.]  Structural results for connected base graphs.
\S5, ``Classification of double covers'':  Hyperoctahedron $K_{n(2)} \downarrow \pm K_{n}$ (with ``type 1'' [positive] and ``type 2'' [negative] edges) and is ``universal'' for $n$-coloring.  Prop.\ 5.3:  $\Delta \downarrow \Gamma \implies \chi(\Delta) \leq \chi(\Gamma)$.  Thm.\ 5.5:  All double covers are pullbacks of $K_{n(2)} \downarrow \pm K_{n}$.  Monomorphisms $\alpha', \beta': C_n \to |\pm K_n|$ are equivalent iff they give the same sign in $\pm K_n$.  Thm.\ 5.6:  Double covers are isomorphic iff they induce  switching equivalent signatures [which seems obvious in signed-graph language].
\dated{30 Jan 2022} 
\codes{\Csg:~\CCov:~\Csw,~\Ccol}%
\itemspace

\author{Zach Walsh}							%WALSH
\authorspace
\bsgitem{2022a}{}  A new matroid lift construction and an application to group-labeled graphs.  \emph{Electronic J.\ Combin.}\ 29 (2022), no.\ 1, art.\ P1.6, 18 pp.
\MR{4395251}.  
\arXiv{2104.08257}.  
\codes{\CGG:~\CMatrd}%
\itemspace

\authorx{Georg R.\ Walther}						%WALTHER G
\author{Georg R.\ Walther, Matthew Hartley, \& Maya Mincheva}
\authorspace
\bsgitem{2014a}{}  GraTeLPy: graph-theoretic linear stability analysis.  \emph{BMC Systems Biol.}\ 8 (2014), art.\ 22, 11 pp.
\codes{\CSD:~\CQM:~\CDyn,~\CChem,~\CBiol}%
\itemspace

\author{O.\ Walther}							%WALTHER O
\authorspace
\authref  See \hyperlink{Q.\ Zheng}{Q.~Zheng}.
\itemspace

\author{Changping Wang}						%WANG CP
\authorspace
\bsgitem{2008a}{}  The signed matchings in graphs.  \emph{ Discuss.\ Math.\ Graph Theory}\ 28 (2008), no.\ 3, 477--486.
\MR{2514204} (2010f:05147).  \Zbl{1175.05114}.  
\annot{} Fix $\Gamma$.  Signed matching: $\sigma_m$ such that all $d^\pm(v) \leq 1$.  Signed matching number $\beta_1'(\Gamma) := \max_{\sigma_m} (\#E^+ - \# E^-)$.  Finding a max $\sigma_m$ is strongly polynomial.  Bounds on $\beta_1'$; exact values for $P_n$, $C_n$, $K_n$, $K_{p,q}$.  Signed edge cover: $\sigma^c$ such that all $d^\pm(v) \geq 1$.  Signed edge cover number $\rho_1'(\Gamma) := \min (\#E^+ - \# E^-)$.  $\beta_1'(\Gamma) + \rho_1'(\Gamma)$ need not $=n$.
\dated{17 Dec 2020} 
\codes{\Csg,~\CAlgor}%
\itemspace

\bsgitem{2010a}{}  Signed $b$-matchings and $b$-edge covers of strong product graphs.  \emph{Contrib.\ Discrete Math.}\ 5 (2010), no.\ 2, 1--10.
\MR{2791285} (2012h:05255).  \Zbl{1317.05156}.  
\annot{} Generalizing \xrefd{2008a}{Changping Wang}, $b$- means $\geq b$ or $\leq b$.  Bounds for strong product.  Gallai theorem for $K_n$ and $K_{p,q}$.
\dated{17 Dec 2020} 
\codes{\Csg}%
\itemspace

\bsgitem{2013a}{}  The signed $k$-submatchings in graphs.  \emph{Graphs Combin.}\ 29 (2013), 1961--1971.
\MR{3119952}.  \Zbl{1282.05191}.  
\annot{} Generalizing \xrefd{2008a}{Changping Wang}, $\sigma_S$ such that $d^\pm(v) \leq 1$ for $\geq k$ vertices.  Sharp bounds on $\beta_S^k(\Gamma) := \max_{\sigma_S} (\#E^+ - \# E^-)$ and some exact values.  
[More in \xref{S.\ Akbari, M.\ Dalirrooyfard, K.\ Ehsani, \& R.\ Sherkati}{2016a}{S.\ Akbari, M.\ Dalirrooyfard, K.\ Ehsani, \& R.\ Sherkati}.]
\dated{17 Dec 2020} 
\codes{\Csg}%
\itemspace

\author{Chong-Jun Wang}						%WANG CJ
\authorspace
\authref  See \hyperlink{Yi Wang}{Y.~Wang}.
\itemspace

\author{Chun-Chieh Wang}						%WANG CC
\authorspace
\authref  See \hyperlink{Li-Hsuan Chen}{L.-H.~Chen}.
\itemspace

\author{Cuihua Wang}						%WANG CH
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Dan Wang}							%WANG D
\authorspace
\authref  See \hyperlink{Wei Chen}{W.~Chen}.
\itemspace

\author{Dengyin Wang}						%WANG DY
\authorspace
\authref  See \hyperlink{Fenglei Tian}{F.L.~Tian} and \hyperlink{Sai Wang}{S.~Wang}.
\itemspace

\author{Dijian Wang}							%WANG DJ
\authorspace
\authref  See also \hyperlink{Yaoping Hou}{Y.-P.~Hou}.
\authrefspace

\author{Dijian Wang \& Yaoping Hou}
\bsgitem{2018a}{}  On the sum of Laplacian eigenvalues of a signed graph.  \emph{Linear Algebra Appl.}\ 555 (2018), 39--52.
\MR{3834190}.  \Zbl{1393.05184}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{2019a}{}  Bicyclic signed graphs with at most one odd cycle and maximal energy.  \emph{Discrete Appl.\ Math.}\ 260 (2019), 244--255.
\MR{3944625}.  \Zbl{1409.05133}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2020a}{}  Integral signed subcubic graphs.  \emph{Linear Algebra Appl.}\ 593 (2020), 29--44.
\MR{4066149}.  \Zbl{1436.05066}.  
\annot{} There are one infinite family and 14 sporadic signed graphs with integral $\Spec A(\Sigma)$.
\dated{9 Dec 2020} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\bsgitem{2021a}{}  Laplacian integral subcubic signed graphs.  \emph{Electronic J.\ Linear Algebra} 37 (2021), 163--176.
\Zbl{1464.05171}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\bsgitem{20xxa}{}  Unicyclic signed graphs with maximal energy.  Submitted.  \emph{}\ 
\arXiv{1809.06206}.							
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Dijian Wang, Wenkuan Dong, Yaoping Hou, \& Deqiong Li}
\authorspace
\bsgitem{20xxa}{}  On signed graphs whose spectral radius does not exceed $\sqrt{2+\sqrt5}$.  Submitted.  \emph{}\ 
\arXiv{2203.01530}.  
\annot{} All $\Sigma$ with $\max|\lambda_i(A(\Sigma))| \leq \sqrt{2+\sqrt5} = \sqrt\tau+\sqrt\tau\inv$ ($\tau$ = golden ratio), a Hoffman spectral limit number.
\dated{4 Mar 2022} 
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Dijian Wang, Yaoping Hou, \& Deqiong Li}
\authorspace
\bsgitem{20xxa}{}  Extremed [\sic] signed graphs for triangle.  Submitted.  
\arXiv{2212.11460}.  
\annot{} A Tur\'an-type theorem for excluded negative triangles.  Also, upper bound on spectral radius.
\codes{\CSG:~\CStr,~\CAdj:~\CEig}%
\itemspace

\authorx{Guangfu Wang}						%WANG GF
\author{Guangfu Wang, Shuchao Li, Wei Wei, \& Siqi Zhang}
\authorspace
\bsgitem{2019a}{}  On the characteristic polynomials and $H$-ranks of the weighted mixed graphs.  \emph{Linear Algebra Appl.}\ 581 (2019), 383--404.
\MR{3987452}.  \Zbl{1419.05107}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.  
Formulas for coefficients of weighted characteristic polynomial.  $\rk A(\Phi,w)$ determined in some cases.
\dated{15 Dec 2020} 
\par
\codes{\Cgg,~\CWG:~\CAdj}%
\itemspace

\author{Hai-Feng Wang}						%WANG HF
\authorspace
\authref  See \hyperlink{Miao-Lin Ye}{M.L.~Ye}.
\itemspace

\authorx{Haotian Wang}						%WANG HT
\author{Haotian Wang, Feng Luo, \& Jie Gao}
\authorspace
\bsgitem{20xxa}{}  Co-evolution of opinion and social tie dynamics towards structural balance.  Submitted.  \emph{} (SODA 2022, virtual).
\codes{\CSD:~\CKG:~\CBal,~\CDyn}%
\itemspace

\author{Hua Wang}							%WANG H
\authorspace
\authref  See \hyperlink{Jing Huang}{J.~Huang}.
\itemspace

\author{Jianfeng Wang}						%WANG JF
\authref  See also \hyperlink{Francesco Belardo}{F.~Belardo} and \hyperlink{Lu Lu}{L.~Lu}.
\authrefspace

\author{Jianfeng Wang \& Francesco Belardo}
\authorspace
\bsgitem{2011a}{}  A note on the signless Laplacian eigenvalues of graphs.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 10, 2585--2590.
\MR{2811140} (2012d:05242).  \Zbl{1225.05176}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{JianFeng Wang, Francesco Belardo, QiongXiang Huang, \& Bojana Borovi\-\'canin}
\authorx{JianFeng Wang, Francesco Belardo, QiongXiang Huang, \& Bojana Borovicanin}
\authorspace
\bsgitem{2010a}{}  On the two largest $Q$-eigenvalues of graphs.  \emph{Discrete Math.}\ 310 (2010), no.\ 21, 2858--2866.
\MR{2677645} (2011j:05218).  \Zbl{1208.05079}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jianfeng Wang, Francesco Belardo, QiongXiang Huang, \& Enzo M.\ Li Marzi}
\authorspace
\bsgitem{2013a}{}  On graphs whose Laplacian index does not exceed $4.5$.  \emph{Linear Algebra Appl.}\ 438 (2013), no.\ 4, 1541--1550.
\MR{3005240}.  \Zbl{1259.05119}.  
\annot{} See \xref{Wang, Huang, Belardo, and Li Marzi}{2009a}{JianFeng Wang, QiongXiang Huang, Francesco Belardo, \& Enzo M.\ Li Marzi}.
\dated{20 Dec 2011} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{JianFeng Wang, Francesco Belardo, Wei Wang, \& QiongXiang Huang}
\authorspace
\bsgitem{2013a}{}  On graphs with exactly three $Q$-eigenvalues at least two.  \emph{Linear Algebra Appl.}\ 438 (2013), 2861--2879.
\MR{3018045}.  \Zbl{1259.05118}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jianfeng Wang \& Qiongxiang Huang}
\authorspace
\bsgitem{2011a}{}  Maximizing the signless Laplacian spectral radius of graphs with given diameter or cut vertices.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 7, 733--744.
\MR{2871248}.  \Zbl{1222.05182}.						
\annot{} Thm.\ 3.1:  Fixing diameter $d$, the largest eigenvalue $q_1(\Lap(-\Gamma))$ is maximized uniquely by a path of length $d$ with a clique joined to its three middle vertices ($K_n$ if $d\leq2$).  Thm.\ 3.2:  Fixing $\nu := n-d$, $q_1 \nearrow 4\nu^2/(2\nu-1)$.  
Thm.\ 4.1:  Fixing the number $k$ of cutpoints, $q_1$ is maximized uniquely by $K_{n-k}$ with paths of nearly equal length attached to every vertex.  Fix $\mu:=n-k$.  Thm.\ 4.2:  If $k\leq n/2$, $q_1 < 2(\mu-1) + k\mu/[2(\mu-1)^2-n].$  Thm.\ 4.3:  If $n/2 < k \leq n-3$, $q_1 \nearrow 2\mu-1 + 1/(2\mu-3).$
[\Question.  Can these results generalize to signed graphs?]
\dated{16 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jianfeng Wang, Qiongxiang Huang, Xinhui An, \& Francesco Belardo}
\authorspace
\bsgitem{2010a}{}  Some results on the signless Laplacians of graphs.  \emph{Appl.\ Math.\ Lett.}\ 23 (2010), no.\ 9, 1045--1049.
\MR{2659136} (2011e:05161).  \Zbl{1209.05148}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{JianFeng Wang, QiongXiang Huang, Francesco Belardo, \& Enzo M.\ Li Marzi}%
\authorspace
\bsgitem{2009a}{}  On graphs whose signless Laplacian index does not exceed 4.5.  \emph{Linear Algebra Appl.}\ 431 (2009), no.\ 1-2, 162--178.
\MR{2522565} (2010g:05238).  \Zbl{1171.05035}.					
\annot{} The signless Laplacian $Q = \Lap(-\Gamma)$ is employed to derive results on the Laplacian $\Lap(\Gamma)$.  Continued in \xref{Wang, Belardo, Huang, and Li Marzi}{2013a}{Jianfeng Wang, Francesco Belardo, QiongXiang Huang, \& Enzo M.\ Li Marzi}.
\dated{9 Feb 2013} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  On the spectral characterizations of $\infty$-graphs.  \emph{Discrete Math.}\ 310 (2010), 1845--1855.
\MR{2629903} (2011m:05188).  \Zbl{1231.05174}.  
\annot{} The spectra of $\Lap(-\Gamma)$ and $\Lap(+\Gamma)$ are compared, where $\Gamma$ is a tight handcuff (a figure-eight graph).  
[\Conjecture.  The results certainly hold for $\Lap(-\Gamma)$ vs.\ all $\Lap(\Gamma,\sigma)$ since only the circle signs diffentiate $+\Gamma$ from $(\Gamma,\sigma)$.]
\dated{20 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Jian-Sheng Wang}						%WANG JS
\author{Jian-Sheng Wang and Robert H.\ Swendsen}
\authorspace
\bsgitem{1988a}{}  Low-temperature properties of the $\pm J$ Ising spin glass in two dimensions.  \emph{Phys.\ Rev.\ B} 38 (1988), no.\ 7, 4840--4844.
\codes{\CPhys:~\Csg:~\CFr}%
\itemspace

\author{Jianfeng Wang, Jing Wang, \& Maurizio Brunetti}
\authorspace
\bsgitem{20xxa}{}  The Hoffman program of graphs: old and new.  Submitted.  \emph{}\ 
\arXiv{2012.13079}.  
\annot{} Survey and new results on graphs with spectral radius bounded by a spectral limit point.  \S6, ``Signed-adjacency matrix'' [i.e., signed-graph adjacency matrix]:  $A(\Sigma)$.  \S4, ``Signless Laplacian matrix'':  $\Lap(-\Gamma)$.  \S5, ``Hermitian adjacency matrix'':  $A(\Phi)$ where $\Phi$ has gains in $\{\pm1,\pm\mathrm{i}\}$.  \S7, ``Skew-adjacency matrix'':  Essentially, gains $\pm\mathrm{i}$.
\dated{4 Mar 2022} 
\codes{\CSG,~\Cgg:~\CAdj,~\CLap:~\CEig}\codesctd{\CSG,~\Cgg:~\CAdj,~\CLap:~\CEig:~\CExp}%
\itemspace

\author{Jing Wang}							%WANG J
\authorspace
\authref  See \hyperlink{Jianfeng Wang}{J.-F.~Wang}.
\itemspace

\author{Jue Wang}							%WANG J
\authorspace
\authref  See also \hyperlink{Beifang Chen}{B.~Chen}.
\authrefspace

\bsgitem{2007a}{}  \emph{Algebraic Structures of Signed Graphs}.  Doctoral dissertation, Hong Kong University of Science and Technology, 2007.
\annot{} Cycle and cocycle spaces, interpretations and relationships.  New formulation of matrix-tree generalization.
\codes{\CSG:~\CInc,~\CStr}%
\itemspace

\author{Kyle Wang}							%WANG K
\authorspace
\authref  See \hyperlink{Gina Chen}{G.~Chen}.
\itemspace

\author{Larry X.W.\ Wang}						%WANG LXW
\authorspace
\authref  See \hyperlink{William Y.C.\ Chen}{W.Y.C.~Chen}.
\itemspace

\author{Li Wang}							%WANG L
\authorspace
\authref  See \hyperlink{Shasha Feng}{S.-S.~Feng}.
\itemspace

\author{Ligong Wang}						%WANG LG
\authorspace
\authref  See also \hyperlink{Yanqing Chen}{Y.Q.~Chen}, \hyperlink{Ke Li}{K.~Li}, \hyperlink{Yong Lu}{Y.~Lu}, and \hyperlink{Xiuwen Yang}{X.-W.~Yang}.
\authrefspace

\author{Ligong Wang, Guopeng Zhao, \& Ke Li}
\authorspace
\bsgitem{2014a}{}  Seidel integral complete $r$-partite graphs.  \emph{Graphs Combin.}\ 30 (2014), 479--493.
\MR{3167023}.  \Zbl{1298.05213}.  
\annot{} I.e., $\Sigma = K_n(-K_{n_1,\ldots,n_r})$.
\codes{\CSG:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Long Wang}							%WANG L
\authorspace
\authref  See \hyperlink{Xue Lin}{X.~Lin} and \hyperlink{Xiaobin Ma}{X.B.~Ma}.
\itemspace

\authorx{Longqin Wang}						%WANG LQ
\author{Longqin Wang, Zhengke Miao, \& Chao Yan}
\authorspace
\bsgitem{2009a}{}  Local bases of primitive non-powerful signed digraphs.  \emph{Discrete Math.}\ 309 (2009), no.\ 4,  748--754.
\MR{2502184} (2010f:05092).  \Zbl{1168.05029}.					
\codes{\CSD:~\Cqm}%
\itemspace

\author{Longqin Wang, Lihua You, \& Hongping Ma}
\authorspace
\bsgitem{2011a}{}  Primitive non-powerful sign pattern matrices with base 2.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 6, 693--700.
\MR{2801362} (2012j:05272).  \Zbl{1223.05104}.					
\par
\codes{\CSD:~\CQM}%
\itemspace

\author{Lusheng Wang}						%WANG LS
\authorspace
\authref  See \hyperlink{Xueliang Li}{X.L.~Li}.
\itemspace

\author{Qingwen Wang}						%WANG QW
\authorspace
\authref  See \hyperlink{Guihai Yu}{G.H.~Yu}.
\itemspace

\authorx{Sai Wang}							%WANG S
\author{Sai Wang, Dengyin Wang, \& Fenglei Tian}
\authorspace
\bsgitem{2021a}{}  Relations between the positive inertia index of a $\bbT$-gain graph and that of its underlying graph.  \emph{J.\ Math.\ Res.\ Appl.}\ 41 (2021), no.\ 3, 221--237.
\MR{4267167}.  
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\author{Sai Wang, Dein Wong, \& Fenglei Tian}
\authorspace
\bsgitem{2021a}{}  Graphs determined by their $T$-gain spectra.  \emph{Bull.\ Austral.\ Math.\ Soc.}\ 103 (2021), no.\ 2, 195--203.
\MR{4229487}.  \Zbl{1460.05119}.  
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\author{Shanfeng Wang}						%WANG SF
\authorspace
\authref  See \hyperlink{Qing Cai}{Q.~Cai}.
\itemspace

\author{Shaohui Wang}						%WANG SH
\authorspace
\authref  See \hyperlink{Jia-Bao Liu}{J.-B.~Liu}.
\itemspace

\authorx{Shilin Wang}						%WANG SL
\author{Shilin Wang \& Bo Zhou}
\authorspace
\bsgitem{2013a}{}  The signless Laplacian spectra of the corona and edge corona of two graphs.  \emph{Linear Multilinear Algebra} 61 (2013), no.\ 2, 197--204.
\MR{3003050}.  \Zbl{1259.05120}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Shiying Wang}						%WANG SY
\author{Shiying Wang, Jing Li, Wei Han, \& Shangwei Lin}
\authorspace
\bsgitem{2010a}{}  The base sets of quasi-primitive zero-symmetric sign pattern matrices with zero trace.  \emph{Linear Algebra Appl.}\ 433 (2010), 595--605.
\MR{2653824} (2011g:15056).  \Zbl{1195.15031}.					
\codes{\CQM:~\CSD}%
\itemspace

\author{Shujing Wang}						%WANG SJ
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.C.~Li}.
\itemspace

\author{Suijie Wang}							%WANG SJ
\authorspace
\authref  See \hyperlink{Xiaogang Liu}{X.G.~Liu}.
\itemspace

\author{Tianfei Wang}						%WANG TF
\authorspace
\bsgitem{2007a}{}  Several sharp upper bounds for the largest Laplacian eigenvalue of a graph.  \emph{Sci.\ in China Ser.\ A Math.}\ 50 (2007), no.\ 12, 1755--1764.
\MR{2390486} (2009a:05131) (\qv).  \Zbl{1134.05064}.  
\annot{} See MR for the formulas, which apply to $q_1(\Lap(-\Gamma))$ [hence to signed simple graphs].  The proofs use a normalized Laplacian.
\dated{16 Jan 2012} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace


\author{Wei Wang}							%WANG W
\authorspace
\authref  See \hyperlink{Xiao-Jun Tian}{X.-J.~Tian}.
\itemspace

\authorx{Wei Wang}							%WANG W [MR 36]
\author{Wei Wang [mr36]}
\authorspace
\authref  See also \xref{Wang, Qiu, Qian, and Wang}{2020a}{Wei Wang, Lihong Qiu, Jianguo Qian, \& Wei Wang}.
\authrefspace

\author{Wei Wang \& Jianguo Qian}
\authorspace
\bsgitem{2020a}{}  A generalization of Noel-Reed-Wu Theorem to signed graphs. \emph{Discrete Math.}\ 343 (2020), no.\ 6, art.\ 111833, 12 pp.
\MR{4062291}.  \Zbl{1437.05092}.  \arXiv{1810.09741}.				
\codes{\CSG:~\CCol:~\CGen}%
\itemspace

\author{Wei Wang, Jianguo Qian, \& Toshiki Abe}
\authorspace
\bsgitem{2019a}{}  Alon-Tarsi number and modulo Alon-Tarsi number of signed graphs.  \emph{Graphs Combin.}\ 35 (2019), no.\ 5, 1051--1064.
\MR{4003656}.  \Zbl{1426.05059}.  \arXiv{1809.02907}.				
\annot{} 
Previous version \arXiv{1809.02907v1}.
\codes{\CSG:~\CCol:~\CGen}%
\itemspace

\author{Wei Wang, Zhidan Yan, \& Jianguo Qian}
\authorspace
\bsgitem{2021a}{}  Eigenvalues and chromatic number of a signed graph.  \emph{Linear Algebra Appl.}\ 619 (2021), 137--145.
\MR{4223364}.  \Zbl{1466.05125}.  
\annot{} Thm.:  $\chi(\Sigma) \geq 1+\lambda_1/|\lambda_n|,\ n/(n-\lambda_1)$, where $\lambda_1$, $\lambda_n$ = max, min adjacency eigenvalues.
\dated{19 Mar 2021} 
\codes{\CSG:~\CCol,~\CAdj:~\CEig}%
\itemspace

\author{Wei Wang [mr21]}\\						%WANG W [MR 21]
\author{Wei Wang, Lihong Qiu, Jianguo Qian, \& Wei Wang}
\authorspace
\bsgitem{2020a}{}  Generalized spectral characterization of mixed graphs.  \emph{Electronic J.\ Combin.}\ 27 (2020), no.\ 4, art.\ 4.55, 31 pp.
\MR{4245230}.  \Zbl{1472.05102}.  \arXiv{1911.13004}.  
\annot{} $\Phi$ with gains in $\{1,\pm\mathrm{i}\}$ as in \xref{Liu and Li}{2015a}{Jianxi Liu \& Xueliang Li} and \xref{Mohar}{2016a}{Bojan Mohar}.  ``Generalized spectrum'' = $(\Spec A(\Phi), \Spec(J-I-A(\Phi)))$.  Characterization up to isomorphism, not switching isomorphism or conjugation (thus must be self-conjugate = digraph self-converse).  [Conjugation preserves spectra.  Switching preserves $\Spec A(\Phi)$ but not $\Spec(J-I-A(\Phi))$.]  
[For gain graphs, this is unnatural; spectral characterization up to switching and conjugation is natural.]
\dated{23 Dec 2020} 
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{X.L.\ Wang}							%WANG XL
\authorspace
\authref  See \hyperlink{Y.\ Chen}{Y.~Chen}.
\itemspace

\authorx{Xiao Wang}							%WANG X
\author{Xiao Wang, You Lu, Cun-Quan Zhang, \& Shenggui Zhang}
\authorspace
\bsgitem{2019a}{}  Six-flows on almost balanced signed graphs.  \emph{J.\ Graph Theory} 92 (2019), 394--404.
\MR{4030965}.  \Zbl{1443.05087}.  
\codes{\CSG:~\CFlows}%
\itemspace

\author{Xiao Wang, You Lu, \& Shenggui Zhang}
\authorspace
\bsgitem{2021a}{}  Flow number of signed Halin graphs.  \emph{Appl.\ Math.\ Comput.}\ 393 (2021), art.\ 125751, 12 pp.
\MR{4172776}.  
\annot{} If it has a nowhere-zero flow, it has a nowhere-zero 5-flow, thus satisfying \xref{Bouchet's}{1983a}{Andre Bouchet} conjecture.
\dated{26 Nov 2020} 
\codes{\CSG:~\CFlows}%
\itemspace

\author{Xin Wang}							%WANG X
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Xing Ke Wang}						%WANG XK
\authorspace
\authref  See \hyperlink{Shang Wang Tan}{S.W.~Tan}.
\itemspace

\author{Xioafeng Wang}						%WANG XF
\authorspace
\authref  See \hyperlink{Er Ling Wei}{E.L.~Wei}.
\itemspace

\author{Yajun Wang}							%WANG YJ
\authorspace
\authref  See \hyperlink{Yanhua Li}{Y.-H.~Li}.
\itemspace

\author{Yi Wang}							%WANG Y
\authorspace
\authref  See also \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}, \hyperlink{Lily L.\ Liu}{L.L.~Liu}, \hyperlink{Gui-Dong Yu}{G.-D.~Yu}, \hyperlink{Bo-Jun Yuan}{B.-J.~Yuan}, and \hyperlink{Jun Zhou}{J.~Zhou}.
\authrefspace

\author{Yi Wang \& Yi-Zheng Fan}
\authorspace
\bsgitem{2012a}{}  The least eigenvalue of signless Laplacian of graphs under perturbation.  \emph{Linear Algebra Appl.}\ 436 (2012), no.\ 7, 2084--2092.
\MR{2889977}.  \Zbl{1238.05168}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Yi Wang, Shi-Cai Gong, \& Yi-Zheng Fan}
\authorspace
\bsgitem{2018a}{}  On the determinant of the Laplacian matrix of a complex unit gain graph.  \emph{Discrete Math.}\ 341 (2018), no.\ 1, 81--86.
\MR{3713385}.  \Zbl{1372.05097}.  
\codes{\CGG:~\CLap}%
\itemspace

\author{Yi Wang \& Yeong-Nan Yeh}
\authorspace
\bsgitem{2005a}{}  Polynomials with real zeros and P\'olya frequency sequences.  \emph{J.\ Combin.\ Theory Ser. A} 109 (2005), 63--74.
\MR{2110198} (2005i:05013).  \Zbl{1057.05007}.  
\annot{} The Whitney numbers $W_m(n,k):=W_k(\Lat\fQ K_n\full)$ ($m=\#\fG$) are Example 5.
\dated{9 Apr 2016} 
\codes{\Cgg:~\CMatrd:~\CInvar}%
\itemspace

\author{Yi Wang, Bo-Jun Yuan, Shuang-Dong Li, \& Chong-Jun Wang}
\authorspace
\bsgitem{2017a}{}  Mixed graphs with $H$-rank 3.  \emph{Linear Algebra Appl.}\ 524 (2017), 22--34.
\MR{363017}.  \Zbl{1361.05076}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.  ``Rank'' = $\rk A(\Phi)$.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CAdj}%
\itemspace

\author{Yishui Wang}							%WANG YS
\authorspace
\authref  See \hyperlink{Sai Ji}{S.~Ji}.
\itemspace

\author{Yitian Wang}							%WANG YT
\authorspace
\authref  See \hyperlink{Zhiyun Cheng}{Z.-Y.~Cheng}.
\itemspace

\author{Yuanpei Wang}						%WANG YP
\authorspace
\authref  See \hyperlink{Hongyan Cai}{H.-Y.~Cai}.
\itemspace

\author{Yue Wang}							%WANG Y
\authorspace
\authref  See \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}.
\itemspace

\author{Yunzhuo Wang}						%WANG YZ
\authorspace
\authref  See \hyperlink{Wenzhong Liu}{W.-Z.~Liu}.
\itemspace

\author{Zhouningxin Wang}						%WANG ZNX
\authorspace
\authref  See \hyperlink{Chiara Cappello}{C.~Cappello}, \hyperlink{Jiaao Li}{J.-A.~Li}, and \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\author{Egon Wanke}						%WANKE
\authorspace
\authref  See also \hyperlink{Franz Hofting}{F.~H\"ofting}.
\authrefspace

\bsgitem{1993a}{}  Paths and cycles in finite periodic graphs.  
In:  Andrzej M.~Borzyszkowski and Stefan Soko{\l}owski, eds., \emph{Mathematical Foundations of Computer Science 1993} (Proc., 18th Int.\ Sympos., MFCS '93, Gda\'nsk, 1993), pp.\ 751--760.  
\MR{1265106} (95c:05077).  \Zbl{925.05038}.
\annot{} Broadly resembles \xref{H\"ofting and Wanke}{1994a}{Franz Hofting \& Egon Wanke} but omits those edges of $\tilde\Phi$ that are affected by the modulus $\alpha$.  
\codes{\CGD(\CCov):~\CAlgor}%
\itemspace

\author{Ian M.\ Wanless}						%WANLESS
\authorspace
\bsgitem{2005a}{}  Permanents of matrices of signed ones.  \emph{Linear Multilinear Algebra} 52 (2005), no.\ 6, 427--433.
\MR{2162064} (2006g:15013).  \Zbl{1085.15006}.  
\annot{} $\per(B) := \sum_M \sigma(M)$, summed over all perfect matchings $M \subseteq \Sigma = (K_{n,n},\sigma)$, where $A(\Sigma) = (\begin{smallmatrix} B & O \\ O & B \end{smallmatrix})$.
\dated{22 Aug 2012} 
\codes{\Csg:~\CAdj}%
\itemspace

\author{G.H.\ Wannier} 						%WANNIER*
\authorspace
\bsgitem{1950a}{}  Antiferromagnetism.  The triangular Ising net.  \emph{Phys.\ Rev.}\ (2) 79 (1950), 357--364.  
\MR{0039627} (12, 576).  \Zbl{038.41904} (38, p.\ 419d).				
Errata.  \emph{Phys.\ Rev.\ B} 7 (1973), 5017.
\annot{} Physical consequences of frustration with Ising spins, i.e.\ $\zeta: V \to \signnos$, in the all-negative triangular lattice graph.  
[Also see \xref{Houtappel}{1950a}{R.M.F.\ Houtappel}, \xrefd{1950b}{R.M.F.\ Houtappel}, \xref{Newell}{1950a}{G.F.\ Newell}, \xref{I.\ Sy\^ozi}{1950a}{Itiro Syozi}.]
\dated{16 Jun 2012}
\codes{\CPhys:~\CSG:~\Cpar:~\CState(\Cfr)}%
\itemspace

\authorx{T.\ Wanschura}						%WANSCHURA
\author{T.\ Wanschura, D.A.\ Coley, \& S.\ Migowsky}
\authorspace
\bsgitem{1996a}{}  Ground-state energy of the $\pm J$ spin glass with dimension greater than three.  \emph{Solid State Commum.}\ 99 (1996), no.\ 4, 247--248.
\annot{} Calculation by genetic algorithm for certain lattice graphs.
\dated{3 Jan 2015} 
\codes{\CPhys,~\CSG:~\CState(\Cfr):~\CAlgor}%
\itemspace

\author{Dan Warner}							%WARNER
\authorspace
\authref  See \hyperlink{Charles R.\ Johnson}{C.R.~Johnson}.
\itemspace

\author{Jacqueline M.\ Warren}						%WARREN
\authorspace
\authref  See \hyperlink{Etienne Leclerc}{E.~Leclerc} and \hyperlink{Gary MacGillivray}{G.~MacGillivray}.
\itemspace

\authorx{Stanley Wasserman}						%WASSERMAN S
\author{Stanley Wasserman \& Katherine Faust, eds.}
\authorspace
\bsgitem{1994a}{}  \emph{Social Network Analysis:  Methods and Applications}.  Structural Anal.\ Soc.\ Sci., 8.  Cambridge Univ.\ Press, Cambridge, Eng., 1994.
\Zbl{980.24676}.  
\annot{} \S1.2:  ``Historical and theoretical foundations.'' A brief summary of various network methods in sociometry, signed graphs and digraphs among them.  
Ch.\ 4 by \xref{Dawn Iacobucci}{1994a}{Dawn Iacobucci}.  Elementary.  
\S4.4:  ``Signed graphs and signed directed graphs.''  Mathematical basics.  
\S4.5: ``Valued graphs and valued directed graphs.''  Mentions unweighted and positively weighted signed (di)graphs.  
Ch.\ 6:  ``Structural balance and transitivity.''  Application of balance of signed (di)graphs and of ensuing notions like clusterability, historically evolving into transitivity of unsigned digraphs.  History and evaluation.  
\S6.1: ``Structural balance.''  Balance, indices of imbalance.  
\S6.2: ``Clusterability.''  All graphs, and complete graphs, as in \xref{Davis}{1967a}{James A.\ Davis}.  
\S6.3: ``Generalizations of clusterability.''  \S6.3.2: ``Ranked clusterability.''  As in \xref{Davis and Leinhardt}{1972a}{James A.\ Davis \& Samuel Leinhardt}.
\dated{28 Apr 2009}
\codes{\CPsS,~\CSG,~\CSD:~\CBal,~\CFr,~\CClu,~\CGen:~\CExp,~\CRef}%
\itemspace

\author{Yusuke Watanabe}						%WATANABE
\authorspace
\bsgitem{2011a}{}  Uniqueness of belief propagation on signed graphs.  In:  J.\ Shawe-Taylor \etal, eds., \emph{Advances in Neural Information Processing Systems 24} (Proc., NIPS 2011, Granada, Spain), art.\ 4422, 9 pp.  Neural Information Processing Systems Foundation, 2011.
\codes{\CSG}%
\itemspace

\author{William C.\ Waterhouse} 					%WATERHOUSE
\authorspace
\bsgitem{1977a}{}  Some errors in applied mathematics.  \emph{Amer.\ Math.\ Monthly} 84 (January, 1977), no.~1, 25--27.  
\Zbl{376.9001} (\qv).  
\annot{} Criticizes \xref{Roberts and Brown}{1975a}{Fred S.\ Roberts \& Thomas A.\ Brown}, \xrefd{1977a}{Fred S.\ Roberts \& Thomas A.\ Brown}.  See rebuttal in the Zbl review. 
\itemspace

\author{John J.\ Watkins}						%WATKINS JJ
\authorspace
\authref  See \hyperlink{Robin J.\ Wilson}{R.J.~Wilson}.
\itemspace

\author{William Watkins}						%WATKINS W
\authorspace
\authref  See \hyperlink{Magnhild Lien}{M.~Lien}.
\itemspace

\author{Kevin D.\ Wayne}						%WAYNE
\authorspace
\authref  See \hyperlink{Eva Tardos}{\`E.~Tardos}.
\itemspace

\author{Nikolai Weaver}						%WEAVER
\authorspace
\authref  See \hyperlink{Erica Flapan}{E.~Flapan}.
\itemspace

\authorx{Jeffrey R.\ Weeks}						%WEEKS
\author{Jeffrey R.\ Weeks \& Kenneth P.\ Bogart}
\authorspace
\bsgitem{1979a}{}  Consensus signed digraphs.  \emph{SIAM J.\ Appl.\ Math.}\ 36 (1979), 1--14.  
\MR{0519178} (81i:92026).  \Zbl{411.05042}.					
\codes{\CSD}%
\itemspace

\author{Chunyan Wei}						%WEI CY
\authorspace
\authref  See \hyperlink{Jiaao Li}{J.-A.~Li}.
\itemspace

\authorx{Er Ling Wei}						%WEI EL
\author{Er Ling Wei, Wen Tang, \& Dong Ye}
\authorspace
\bsgitem{2014a}{}  Nowhere-zero 15-flow in 3-edge-connected bidirected graphs.  \emph{ Acta Math.\ Sinica (Engl.\ Ser.)}\ 30 (2014), no.\ 4, 649--660.
\MR{3176918}.  \Zbl{1294.05088}.  
\annot{} A nowhere-zero 15-flow exists.  (\Cf\ \xref{Wei, Tang, and Wang}{2011a}{Erling Wei, Wenliang Tang, \& Xioafeng Wang})
\dated{16 Apr 2014} 
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\author{Erling Wei, Wenliang Tang, \& Xioafeng Wang}
\authorspace
\bsgitem{2011a}{}  Flows in 3-edge-connected bidirected graphs.  \emph{Frontiers Math.\ China} 6 (2011), no.\ 2, 339--348.
\MR{2780896} (2012b:05137).  \Zbl{1226.05130}.  
\annot{} A nowhere-zero 25-flow exists.  (Progress on Bouchet's conjecture, \xref{Bouchet}{1983a}{Andre Bouchet}.)
\dated{6 Jun 2011} 
\codes{\CSG:~\COri,~\CFlows}%
\itemspace

\author{Fuyi Wei}							%WEI FY
\authorspace
\authref  See \hyperlink{Fi-yi Wei}{F.-Y.~Wei}.
\itemspace

\author{Fi-yi Wei}							%WEI FY
\authorspace
\authref  See also \hyperlink{Mu Huo Liu}{M.H.~Liu}.
\authrefspace

\author{Fi-yi Wei \& Muhuo Liu}
\authorspace
\bsgitem{2011a}{}  Ordering of the signless Laplacian spectral radii of unicyclic graphs.  \emph{Australasian J.\ Combin.}\ 49 (2011), 255--264.
\MR{2790977} (2011m:05189).  \Zbl{1228.05208}.					
\annot{} Unicyclic $\Gamma$ with $\lambda^1(\Lap(-\Gamma)) \geq n-2$.
[\Problem.  Generalize to signed graphs, or complex unit gain graphs.  \Cf\ \xref{Reff}{2012a}{Nathan Reff}.]
\dated{19 May 2019} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace


\author{Li Juan Wei}							%WEI LJ
\authorspace
\authref  See \hyperlink{Yaoping Hou}{Y.P.~Hou}.
\itemspace

\authorx{Po-Sun Wei}						%WEI PS
\author{Po-Sun Wei and Bang Ye Wu}
\authorspace
\bsgitem{2012a}{}  Balancing a complete signed graph by changing minimum number of edge signs.  
In:  \emph{Proceedings of the 29th Workshop on Combinatorial Mathematics and Computation Theory} (Taipei, 2012), pp.\ 1--8.  
National Taipei College of Business, Taipei, Taiwan, 2012.  
URL \url{http://par.cse.nsysu.edu.tw/~algo/paper/paper_list12.htm}
\annot{} Algorithm for frustration index $l(K_n,\sigma)$ by changing edge signs.
\dated{5 Jun 2017} 
\codes{\CSG:~\CKG:~\CFr:~\CAlgor}%
\itemspace

\author{Wei Wei}							%WEI W
\authorspace
\authref  See also \hyperlink{Shuchao Li}{S.-L.~Li} and \hyperlink{Guangfu Wang}{G.-F.~Wang}.
\authrefspace

\author{Wei Wei \& Shuchao Li}
\authorspace
\bsgitem{2020a}{}  Relation between the Hermitian energy of a mixed graph and the matching number of its underlying graph.  \emph{Linear Multilinear Algebra}\ 68 (2020), no.\ 7, 1395--1410.
\MR{4137053}.  \Zbl{1459.05194}.  
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\author{Wei Wei, Shuchao Li, \& Hongping Ma}
\authorspace
\bsgitem{2021a}{}  Bounds on the nullity, the $H$-rank and the Hermitian energy of a mixed graph.  \emph{Linear Multilinear Algebra}\ 69 (2021), no.\ 13, 2469--2490.
\MR{4301424}.  \Zbl{1472.05104}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.
\codes{\Cgg:~\CAdj,~\CEig}%
\itemspace

\author{Martin Weigt}						%WEIGT
\authorspace
\authref  See \hyperlink{Alexander K.\ Hartmann}{A.K.~Hartmann}.
\itemspace

\author{Gerry M.\ Weiner}						%WEINER
\authorspace
\authref  See \hyperlink{John S.\ Maybee}{J.S.~Maybee}.
\itemspace

\author{Volkmar Welker}						%WELKER
\authorspace
\bsgitem{1997a}{}  Colored partitions and a generalization of the braid arrangement.  \emph{Electronic J.\ Combin.}\ 4 (1997), no.\ 1, art.\ R4, 12 pp.
\MR{1435130} (98b:57026).  \Zbl{883.52010}.
\annot{} The arrangement is the affine part (that is, where $x_0 = 1$) of the projective representation of $\bfF(\Phi)$, where $\Phi$ is the complex multiplicative gain graph $\Phi = \{1\}K_{n+1} \cup \{re_{0i} : 1 \leq i \leq n \text{ and } 2 \leq r \leq s \}$.  Here the vertex set is $\{0, 1, \ldots, n\}$, $s$ is any positive integer, and $re_{0i}$ (in the paper, $e_{0i}(r)$) denotes an edge $v_0v_i$ with gain $r$.  The topics of interest are those related to the complex complement.  The study is based on the combinatorics of the intersection semilattice [that is, the geometric semilattice $\Latb \Phi$ of balanced flats], including the Poincar\'e polynomial of the arrangement [equivalent to the balanced chromatic polynomial of $\Phi$].
\codes{\Cgg:~\CMatrd,~\CGeom,~\CInvar}%
\itemspace

\author{Albert L.\ Wells, Jnr.}						%WELLS
\authorspace
\authref  See also \hyperlink{Peter J.\ Cameron}{P.J.~Cameron} and \hyperlink{Ying Cheng}{Y.~Cheng}.
\authrefspace

\bsgitem{1982a}{}  \emph{Regular Generalized Switching Classes and Related Topics}.  D.Phil.\ thesis, Oxford University, 1982.
\codes{\CSG:~\CSw,~\CAdj,~\CEig,~\CEnum,~\CTG,~\CGeom,~\CCov,~\CAut}%
\itemspace

\bsgitem{1984a}{\dag}  Even signings, signed switching classes, and $(-1,1)$-matrices.  \emph{J.\ Combin.\ Theory Ser.~B} 36 (1984), 194--212.  
\MR{0746549} (85i:05206).  \Zbl{527.05007}.   					
\par
\codes{\CSG:~\CSw,~\CEnum,~\CAut}%
\itemspace

\author{D.J.A.\ Welsh [Dominic Welsh]}					%WELSH
\authorx{D.J.A.\ Welsh}
\authorspace
\authref   See also \hyperlink{Laszlo Lovasz}{L.~Lov\'asz} and \hyperlink{W.\ Schwarzler}{W.~Schw\"arzler}.
\authrefspace

\bsgitem{1976a}{}  \emph{Matroid Theory}.  L.M.S.\ Monographs, Vol.~8.  Academic Press, London, 1976.
\MR{0427112} (55 \#148).  \Zbl{343.05002}.					
Repr.:  Dover Publications, Mineola, N.Y., 2010.
\annot{} \S11.4: ``Partition matroids determined by finite groups'', sketches the most basic parts of \xref{Dowling}{1973b}{Thomas A.\ Dowling}.
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\bsgitem{1992a}{}  On the number of knots and links.  
In: G.~Hal\'asz, L.~Lov\'asz, D.~Mikl\'os, and T.~Sz\"onyi, eds., \emph{Sets, Graphs and Numbers} (Proc., Budapest, 1991), pp.\ 713--718.  Colloq.\ Math.\ Soc.\ J\'anos Bolyai, Vol.~60.  J\'anos Bolyai Math.\ Soc., Budapest, and North-Holland, Amsterdam, 1992.
\MR{1218230} (94f:57010).  \Zbl{799.57001}.
\annot{} The signed graph of a link diagram is employed to get an upper bound.
\par
\codes{\CSGc:~\CEnum}%
\itemspace

\bsgitem{1993a}{}  \emph{Complexity: Knots, Colourings and Counting}.  London Math.\ Soc.\ Lect.\ Note Ser., 186.  Cambridge Univ.\ Press, Cambridge, Eng., 1993.
\MR{1245272} (94m:\-57027).  \Zbl{799.68008}.
\annot{} Includes very brief treatments of some appearances of signed graphs.
\annotctd{} \S2.2, ``Tait colourings'', defines the signed graph of a link diagram, mentioned again in observation (2.3.1) on alternating links and Prop\, (5.2.16) on ``states models'' (from \xref{Schw\"arzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh}).  \S5.6, ``Thistlethwaite's nontriviality criterion'': the criterion depends on the signed graph.
\annotctd{} \S2.5, ``The braid index and the Seifert index of a link'', defines the Seifert graph, a signed graph based on splitting the link diagram.
\par
\codes{\CSGc,~\CKnot}%
\annotctd{} \S5.7, ``Link invariants and statistical mechanics'', defines a relatively simple spin model for signed graphs, with an arbitrary finite number of possible spin values.  The partition function is related to link diagrams.
\annotctd{} \S4.2, ``The Ising model'', introduces the basic concepts in mathematical terms.  \S6.4, ``The complexity of the Ising model'', ``Computing ground states of spin systems'', pp.\ 105--107, discusses finding a ground state of the Ising model.  This is described as the min-weight cut problem with weights the negatives [this is an error] of the Ising bond interaction values: that is, the weighted frustration index problem in the negative [erroneous] of the Ising graph.  It is the max-cut problem when the Ising graph is balanced (ferromagnetic) [should be antibalanced (antiferromagnetic)].  For external magnetic field, follows \xref{Barahona}{1982a}{Francisco Barahona}.
\par
\codes{\Csg:~\CState(\Cfr),~\CFr,~\CPhys}%
\annotctd{} \S3.6, ``Ice models'', counts ``ice configurations'' (certain graph orientations) via poise gains modulo $3$, although the counting function is not gain-graphic.
\codes{\Cgg,~\CInvar,~\CPhys}%
\annotctd{} \S4.4: ``The Ashkin--Teller--Potts model''.  This treatment of the Potts model has a different Hamiltonian from that of \xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz}.  [It does not seem that Welsh intends to admit edge signs.  If they are allowed then the Hamiltonian (without edge weights) is $- \sum \sigma(e_{ij})(\delta(s_i, s_j)$ $- 1)$.  Up to halving and a constant term, this is \xref{Doreian and Mrvar's}{1996a}{Patrick Doreian \& Andrej Mrvar} clusterability measure $P(\pi)$, with $\alpha = .5$, of the vertex partition induced by the state.]  [Also \cf\ \xref{Fischer and Hertz}{1991a}{K.H.\ Fischer \& J.A.\ Hertz}.]
\codes{\Cclu,~\CPhys}%
\itemspace

\bsgitem{1993b}{}  The complexity of knots.  
In: John Gimbel, John W.~Kennedy and Louis V.~Quintas, eds., \emph{Quo Vadis, Graph Theory?}, pp.\ 159--171.  Ann.\ Discrete Math., Vol.~55.  North-Holland, Amsterdam, 1993.
\MR{1217989} (94c:57021).  \Zbl{801.68086}.					
\annot{} Link diagrams $\leftrightarrow$ dual pairs of sign-colored plane graphs: based on \xref{Yajima and Kinoshita}{1957a}{Takeshi Yajima \& Shin'ichi Kinoshita}.  Unsolved algorithmic problems about knots based on link diagrams; in particular, triviality of diagrams is equivalent to \Problem\ 4.2:  A polynomial-time algorithm to decide whether the graphical Reidemeister moves can convert a given signed plane graph to one with edges all of one sign.
\codes{\CSGc:~\CDual,~\CKnot:~\CAlgor,~\CExp}%
\itemspace

\bsgitem{1993c}{}  Knots and braids: some algorithmic questions.
In: Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 109--123.  Contemp.\ Math., Vol.~147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.
\MR{1224698} (94g:57014).  \Zbl{792.05058}.					
\annot{} \S1 presents the sign-colored graph of a link diagram and \S5, ``Reidemeister graphs'', describes \xref{Schw\"artzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh}.  \S3 defines the sign-colored Seifert graph.  
\codes{\CSGc.~\CSc(\CMatrd):~\CInvar,~\CAlgor,~\CKnot:~\CExp}%
\itemspace

\bsgitem{1994a}{}  The computational complexity of knot and matroid polynomials.  \emph{Discrete Math.}\ 124 (1994), 251--269.
\MR{1258858} (95d:57008).  \Zbl{816.57008}.  
\annot{} Pp.\ 258--259:  Sign-colored graph of an alternating link diagram.
\dated{27 Feb 2017} 
\codes{\CKnot:~\CSGc:~\CExp}%
\itemspace

\bsgitem{1997a}{}  Knots.  In: Lowell W.~Beineke and Robin J.~Wilson, eds., \emph{Graph Connections: Relationships between Graph Theory and other Areas of Mathematics}, Ch.~12, pp.\ 176--193.  The Clarendon Press, Oxford, 1997.
\MR{1634542} (99a:05001) (book).  \Zbl{878.57001}.
\annot{} Mostly describes the signed graph of a link diagram and its relation to knot theory, including knot properties deducible directly from the signed graph, the Kauffman bracket and two-variable polynomials, etc.  Similar to relevant parts of \xrefd{1993a}{D.J.A.\ Welsh}.
\codes{\CSGc:~\CKnot:~\CInvar:~\CExp}%
\itemspace

\author{Michael Welsh}						%WELSH
\authorspace
\bsgitem{2011a}{}  \emph{Golden-mean and Secret Sharing Matroids.}  Master's thesis, Victoria Univ.\ of Wellington, 2011.
\annot{} Examples include:  Def.\ 2.0.9:  Dowling geometry $Q_n(\GF(3)^times)$.  \S3.1.2, ``Spikes'', i.e., $\bfL_0(2C_n,\cB)$.
\dated{27 Feb 2017} 
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\author{Stefan Weltge}							%WELTGE
\authorspace
\authref  See \hyperlink{Michele Conforti}{M.~Conforti}.
\itemspace

\author{Emo Welzl}							%WELZL
\authorspace
\authref  See \hyperlink{Herbert Edelsbrunner}{H.~Edelsbrunner} and \hyperlink{Jurriaan Hage}{J.~Hage}.
\itemspace

\authorx{Chao Wen}							%WEN C
\author{Chao Wen, Qiang Sun, Shunzhe Zhang, Huanhuan Guan, \& Chao Zhang}
\authorspace
\bsgitem{20xxa}{}  The homomorphism and colouring of the direct product of signed graphs.  Submitted.  \emph{}\ 
\annot{} The weak Hedetniemi-type conjecture $\chi(\Sigma_1 \times \Sigma_2) \leq \min\{\chi(\Sigma_1),\chi(\Sigma_2)\}$ is disproved.  It is valid if $\Sigma_2$ is balanced, with = if also $|\Sigma_1|$ maps into $|\Sigma_2|$.
\dated{5 Feb 2023} 
\codes{\CSG:~\CCol,~\CHom}%
\itemspace

\authorx{Qin Wen}							%WEN Q
\author{Qin Wen, Qin Zhao, \& Huiqing Liu}
\authorspace
\bsgitem{2015a}{}  The least signless Laplacian eigenvalue of non-bipartite graphs with given stability number.  \emph{Linear Algebra Appl.}\ 476 (2015), 148--158.
\MR{3327137}.  \Zbl{1314.05120}.  
\codes{\CPar:~\CLap:~\CEig}%
\itemspace

\author{D.\ de Werra} 						%WERRA
\authorspace
\authref   See \hyperlink{C.\ Benzaken}{C.~Benzaken}.
\itemspace

\author{Hans V.\ Westerhoff}						%WESTERHOFF
\authorspace
\authref  See \hyperlink{Boris N.\ Kholodenko}{B.N.~Kholodenko}.
\itemspace

\author{Peter Whalen}						%WHALEN
\authorspace
\authref  See \hyperlink{Robin Thomas}{R.~Thomas}.
\itemspace

\author{Joyce Jiyoung Whang}						%WHANG J JY
\authorspace
\authref  See \hyperlink{Kai-Yang Chiang}{K.-Y.~Chiang}.
\itemspace

\author{Arthur T.\ White} 						%WHITE AT
\authorspace
\bsgitem{1984a}{}  \emph{Graphs, Groups and Surfaces}.  Completely revised and enlarged edn.  
North Holland Math.\ Stud., Vol.~8.  North-Holland, Amsterdam, 1984.  
\MR{0780555} (86d:05047).  \Zbl{551.05037}.					
\annot{} Ch.~10:  ``Voltage graphs''.
\codes{\CGG:~\CTop,~\CCov}%
\itemspace

\bsgitem{1994a}{}  An introduction to random topological graph theory.  \emph{Combinatorics, Probability and Computing} 3 (1994), 545--555.  
\MR{1314074} (95j:05083).  \Zbl{815.05027}.
\annot{} Take a graph $\Gamma$ with cyclomatic number $k$ and randomly sign it so that each edge is negative with probability $p$.  The probability that $(\Gamma, \sigma)$ is balanced $= 2^{-k}$ if $p = \frac{1}{2}$ [obvious] and $\leq [\max(p, 1-p)]^k$ in general [not obvious] (this has an interesting asymptotic consequence due to \hyperlink{John Gimbel}{Gimbel}, given in this paper).  [Related: \xref{Frank and Harary}{1979a}{Ove Frank \& Frank Harary}.]
\par
\codes{\CSG:~\CRand,~\CBal}%
\itemspace

\bsgitem{2001a}{}  \emph{Graphs of Groups on Surfaces:  Interactions and Models}.  North-Holland Math.\ Stud., 188.  North-Holland (Elsevier), Amsterdam, 2001.
\MR{1852593} (2002k:\-05001).  \Zbl{1054.05001}.
\annot{} \S10-2, ``Voltage graphs'':  Voltage graphs and the covering graph.  Thm.\ 10-8 is similar to \xref{Biggs}{1974a}{Norman Biggs}, Thm.\ 19.5.  Construction of surface embeddings.
\S11-3, ``Nonorientable voltage graph imbeddings'':  Rotation schemes supplemented by edge signatures as in \xref{Ringel}{1977a}{Gerhard Ringel}, \xref{Stahl}{1978a}{Saul Stahl}, and \xref{Zaslavsky}{1992a}{Thomas Zaslavsky}.
\codes{\CGG,~\CSG:~\CTop,~\CCov}%
\itemspace

\bsgitem{2009a}{}  Embeddings and geometries.  
In:  Lowell W.\ Beineke and Robin J.\ Wilson, eds., \emph{Topics in Topological Graph Theory}, Ch.\ 12, pp.\ 245--267.  Encycl.\ Math.\ Appl., Vol.\ 128.  Cambridge Univ.\ Press, Cambridge, Eng., 2009.
\MR{2581549} (no rev).  \Zbl{1225.05086}.  
\annot{} The voltage graph (i.e., gain graph) construction is used to generate embeddings of finite geometries.
\dated{12 Jun 2013} 
\par
\codes{\CTop:~\CGG,~\CCov:~\CExp}%
\itemspace

\author{Jacob A.\ White}						%WHITE JA
\authorspace
\bsgitem{20xxa}{}  Burnside chromatic polynomials of group-invariant graphs.  \emph{Discuss.\ Math.\ Graph Theory} (to appear).
\annot{} Unlabelled coloring of a gain graph, in the form of its group covering graph, in the more general form of a graph with a group action.  
Incidentally, develops a renark in \xref{Zaslavsky}(2009a){Thomas Zaslavsky}.
\dated{24 Nov 2020} 
\codes{\CGG:~\CCov,~\CGen:~\CCol,~\CInvar}%
\itemspace

\author{Neil L.\ White} 						%WHITE NL
\authorspace
\authref  See also \hyperlink{Anders Bjorner}{A.~Bj\"orner}.
\authrefspace

\bsgitem{1986a}{}  A pruning theorem for linear count matroids.  \emph{Congressus Numerantium} 54 (1986), 259--264.  
\MR{0885285} (88c:05047).  \Zbl{621.05009}.
\codes{\CMatrd:~\CBic:~\CGen}%
\itemspace

\author{Neil White \& Walter Whiteley}
\authorspace
\bsgitem{1983a}{\dag}  A class of matroids defined on graphs and hypergraphs by counting properties.  Unpublished manuscript, 1983.
\annot{} See \xref{Whiteley}{1996a}{Walter Whiteley} for an exposition and extension.  [Generalization in \xref{Pikhurko}{2001a}{Oleg Pikhurko}.]
\codes{\CMatrd:~\CBic:~\CGen}%
\itemspace

\author{Walter Whiteley} 						%WHITELEY
\authorspace
\authref   See also \hyperlink{Neil L.\ White}{N.L.~White}.
\authrefspace

\bsgitem{1988a}{}  The union of matroids and the rigidity of frameworks.  \emph{SIAM J.\ Discrete Math.} 1 (1988), no.\ 2, 237--255.
\MR{0941354} (89d:05055).  \Zbl{671.05026}.  
\par
\codes{\Cgg:~\CGeom,~\CMatrd}%
\itemspace

\bsgitem{1991a}{}  The combinatorics of bivariate splines.  
In:  Peter Gritzman and Bernd Sturmfels, eds., \emph{Applied Geometry and Discrete Mathematics:  The Victor Klee Festschrift}, pp.\ 587--608.  DIMACS Ser.\ Discrete Math.\ Theor.\ Comput.\ Sci., Vol.\ 4.  American Math.\ Soc., Providence, R.I., 1991.
\MR{1116378} (92m:41038).  \Zbl{741.41014}.  
\annot{} ``Balance'' used for circles with identity gain (in a gain graph with additive matrix gains), independently of \xref{Harary}{1953a}{Frank Harary}.  \S3, ``Splines and matrices on graphs'':  The matrix gains are $L_{hi}^{r+1}$ (p.\ 592) and the balance equation is $(*)$ (p.\ 593).
\dated{12 Jun 2012} 
\codes{\Cgg:~\Cbal}%
\itemspace

\bsgitem{1996a}{}  Some matroids from discrete applied geometry.  
In:  Joseph E.~Bonin, James G.~Oxley, and Brigitte Servatius, eds., \emph{Matroid Theory} (Proc., Seattle, 1995), pp.\ 171--311.  Contemp.\ Math., Vol.\ 197.  Amer.\ Math.\ Soc., Providence, R.I., 1996.
\MR{1411692} (97h:05040).  \Zbl{860.05018}.					
\annot{} Appendix: ``Matroids from counts on graphs and hypergraphs'', which expounds and extends \xref{Lor\'ea}{1979a}{M.\ Lorea}, \xref{Schmidt}{1979a}{Rudiger Schmidt}, and especially \xref{White and Whiteley}{1983a}{Neil White \& Walter Whiteley}, describes matroids on the edge sets of graphs (and hypergraphs) that generalize the bicircular matroid.  The definition: given $m \geq 0$ and $k \in \bbZ$, $S$ is independent iff $\eset \subset S' \subseteq S$ implies $\#S' \leq m\#V(S') + k$.  [Suggested name: ``Linearly bounded matroids,'' since they are defined by a linear bound on the rank.]
\par
\codes{\CMatrd:~\CBic:~\CGen}\codesctd{\CRef}%
\itemspace

\author{Geoff Whittle} 						%WHITTLE
\authorspace
\authref   See also \hyperlink{Rong Chen}{R.~Chen}, \hyperlink{Jim Geelen}{J.~Geelen}, \hyperlink{Dillon Mayhew}{D.~Mayhew}, \hyperlink{James G.\ Oxley}{J.~Oxley}, and \hyperlink{Charles Semple}{C.~Semple}.
\authrefspace

\bsgitem{1989a}{}  Dowling group geometries and the critical problem.  \emph{J.\ Combin.\ Theory Ser.\ B} 47 (1989), 80--92.  
\MR{1007716} (90g:51008).  \Zbl{628.05018}.					
\annot{} A Dowling-lattice version of Crapo and Rota's critical problem, with new structure of \xref{Dowling's}{1973b}{Thomas A.\ Dowling} rank-$n$ matroid $Q_n(\fG) = \bfF(\fG K_n\full)$ of a finite group $\fG$.  \S3, ``Tangential $k$-blocks over finite groups'':  Some minimal submatroids of $Q_n(\fG)$ whose critical exponent is $k$ (``tangential $k$-blocks over $\fG$''), such as $Q_n(\{+,-\})$ and, for $\fH \leq \fG$ and some values of $n$, $\bfF(\fH K_n)$ (Thm.\ 3.8, Cor.\ 3.9).  Thm.\ 3.10 on structure of $\bfF(\fG K_n)$.
\dated{25 May 2009, 13 Oct 2022}
\codes{\CGG:~\CMatrd:~\CInvar}%
\itemspace

\bsgitem{1989b}{}  A generalisation of the matroid lift construction.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 316 (1989), 141--159.
\MR{0957084} (90b:05038).  \Zbl{684.05014}.					
\annot{} Examples include bicircular and frame matroids.  
	
\codes{\CGG:~\CMatrd,~\CBic}%
\itemspace

\bsgitem{2005a}{}  Recent work in matroid representation theory.  \emph{Discrete Math.}\ 302 (2005), 285--296. 
\MR{2179649} (2006m:05053).  \Zbl{1076.05022}.					
annot{} P.\ 288:  The ``free spike $\Phi_r$'' is $\bfL(2C_r,\eset)$.  
Pp.\ 290--291:  Biased graphs and the bias [i.e., frame] matroid.  \Conjecture\ 5.2:  With few exceptions, a highly connected matroid that is representable over more than one characteristic is a frame or dual frame matroid.  
P.\ 294:  The ``free swirl $\Psi_k$'' is $\bfF(2C_k,\eset)$.  $U_{3,6} = \bfL(2C_3,\eset) = \bfF(2C_3,\eset)$ [the latter because there are no vertex-disjoint unbalanced circles].  
\dated{25 May 2009} 
\codes{\Cgg:~\CMatrd:~\CExp}%
\itemspace

\author{G\'abor Wiener}						%WIENER
\authorspace
\authref  See \hyperlink{T.\ Fleiner}{T.~Fleiner}.
\itemspace

\author{Avi Wigderson}						%WIGDERSON
\authorspace
\authref  See \hyperlink{Shlomo Hoory}{S.~Hoory}.
\itemspace

\author{Chris Wiggins}						%WIGGINS
\authorspace
\authref  See \hyperlink{Etay Ziv}{E.~Ziv}.
\itemspace

\author{J.K.\ Williams}						%WILLIAMS JK
\authorspace
\authref  See also \hyperlink{B.G.S.\ Doman}{B.G.S.~Doman}.
\authrefspace

\bsgitem{1981a}{}  Ground state properties of frustrated Ising chains.  \emph{J.\ Phys.\ C} 14 (1981), 4095--4107.
\annot{} \S2, ``The random-bond Ising chain in a uniform field: ($T=0$)'':  A path with random edge signs, weighted $J$, magnetic field $B$ [interpretable as an extra all-positive vertex, as in ??].  Continued in \xref{Doman and Williams}{1982a}{B.G.S.\ Doman \& J.K.\ Williams}, \S2.
\dated{28 Aug 2012} 
\codes{\CPhys,~\CSG,~\CWG:~\CState(\Cfr)}%
\itemspace

\author{David P.\ Williamson}						%WILLIAMSON
\authorspace
\authref  See \hyperlink{Anke van Zuylen}{A.~van~Zuylen}.
\itemspace

\author{Andrew Timothy Wilson}					%WILSON AT
\authorspace
\authref  See \hyperlink{Emily Leven}{E.~Leven}.
\itemspace

\author{Mark C.\ Wilson}						%WILSON MC
\authorspace
\authref  See \hyperlink{Samin Aref}{S.~Aref}.
\itemspace

\author{Richard C.\ Wilson}						%WILSON RC
\authorspace
\authref  See \hyperlink{Pierre-Louis Giscard}{P.-L.~Giscard}.
\itemspace


\author{Robin J.\ Wilson}						%WILSON RJ
\authorspace
\authref  See also \hyperlink{Lowell W.\ Beineke \& Robin J.\ Wilson, eds.}{L.W.~Beineke}.
\authrefspace

\author{Robin J.\ Wilson \& John J.\ Watkins}
\authorspace
\bsgitem{1990a}{}  \emph{Graphs: An Introductory Approach.  A First Course in Discrete Mathematics}.  Wiley, New York, 1990.
\MR{1038804} (91b:05001).  \Zbl{712.05001}.
\annot{} \S3.2:  ``Social Sciences'' (pp.\ 51--53) applies signed graphs.  \S5.1:  ``Signed digraphs'' (pp.\ 96--98) discusses positive and negative feedback (i.e., positive and negative cycles) in applications.  Based on \xref{Open University}{1981a}{The Open University}.
\codes{\CSG,~\CPsS,~\CSD:~\CExp}%
\itemspace

\author{Steve Wilson}						%WILSON S
\authorspace
\bsgitem{1989a}{}  Cantankerous maps and rotary embeddings of $K_n$.  \emph{J.\ Combin.\ Theory Ser.\ B} 47 (1989), no.\ 3, 262--273.
\MR{1026064} (90j:05115).  \Zbl{687.05018}.					
\annot{} Cantankerous map: signed expansion graph $\pm\Gamma$, orientation embedded in a surface, whose map automorphisms act transitively on flags.  Rotary map: map with automorphisms that are cyclic permutations around a face and around a vertex on the face.  Thm.: A rotary map is either cantankerous or a kind of branched covering.  
[See \xref{C.H.~Li and \v{S}ir\'a\v{n}}{2007a}{Cai Heng Li \& Jozef Siran} for more on cantankerous maps.]
\dated{rev.\ 31 Jul 2014}
\par
\codes{\Csg:~\CTop:~\CAut}%
\itemspace

\author{Marco Winkler}						%WINKLER
\authorspace
\authref  See \hyperlink{Evimaria Terzi}{E.~Terzi}.
\itemspace

\author{Shmuel Winograd}						%WINOGRAD
\authorspace
\authref   See \hyperlink{Richard M.\ Karp}{R.M.~Karp}.
\itemspace

\author{Wayland H.\ Winstead}						%WINSTEAD
\authorspace
\authref   See \hyperlink{James R.\ Burns}{J.R.~Burns}.
\itemspace

\author{Anthony Wirth}						%WIRTH
\authorspace
\authref  See \hyperlink{Moses Charikar}{M.~Charikar} and \hyperlink{Tom Coleman}{T.~Coleman}.
\itemspace

\authorx{Pepijn Wissing}						%WISSING
\author{Pepijn Wissing \& Edwin R.\ van Dam}
\authorspace
\bsgitem{2022a}{}  Unit gain graphs with two distinct eigenvalues and systems of lines in complex space.  \emph{Discrete Math.}\ 345 (2022), art.\ 112827, 28 pp.
\MR{4379244}.  
\arXiv{2105.09149}.  
\codes{\Cgg:~\CGeom,~\CAdj:~\CEig}%
\itemspace

\bsgitem{2022b}{}  Spectral fundamentals and characterizations of signed directed graphs.  \emph{J.\ Combin.\ Theory Ser.\ A} 187 (2022), art.\ 105573, 40 pp.
\MR{4352598}.  \Zbl{1480.05087}.  \arXiv{2009.12181}.  
\codes{\CSG:~\CGG:~\CAdj:~\CEig}%
\itemspace

\author{H.S.\ Witsenhausen} 						%WITSENHAUSEN
\authorspace
\authref   See \hyperlink{Y.\ Gordon}{Y.~Gordon}.
\itemspace

\authorx{C.\ Witzgall}							%WITZGALL
\author{C.\ Witzgall \& C.T.\ Zahn, Jr.}
\authorspace
\bsgitem{1965a}{}  Modification of Edmonds' maximum matching algorithm.  \emph{J.\ Res.\ Nat.\ Bur.\ Standards (U.S.A.) Sect.~B} 69B (1965), 91--98.  
\MR{0188107} (32 \#5548).  \Zbl{141.21901}.
\codes{\Cpar:~\Cori}%
\itemspace

\author{Jakub Onufry Wojtaszczyk}					%WOJTASZCZYK
\authorspace
\authref  See \hyperlink{Marek Cygan}{M.~Cygan}.
\itemspace

\author{W.F.\ Wolff}							%WOLFF
\authorspace
\authref  See also \hyperlink{P.\ Hoever}{P.~Hoever} and \hyperlink{M.H.\ Waldor}{M.H.~Waldor}.
\authrefspace

\author{W.F.\ Wolff \& J.\ Zittartz}
\authorspace
\bsgitem{1982a}{}  Correlations in inhomogeneous Ising models.  I.  General methods, the ``fully-frustrated square lattice'' and the ``chessboard'' model.  \emph{Z.\ Phys.\ B} 47 (1982), no.\ 4, 341--352.
\MR{0675258} (85d:82104).  
\annot{} \S~III, ``The fully-frustrated square lattice model (FFS)'':  Square lattice graph signed so every square (``plaquette'') is negative.  
\S~IV, ``The chessboard model'':  Square lattice graph with alternate squares negative and positive.
\dated{28 Aug 2012} 
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\bsgitem{1983a}{}  Spin glasses and frustration models:  Analytical results.  
In:  J.L.\ van Hemmen and I.\ Morgenstern, eds., \emph{Heidelberg Colloquium on Spin Glasses} (Proc., Heidelberg, 1983), pp.\ 252--271.  Lect.\ Notes in Physics, Vol.\ 192.  Springer-Verlag, Berlin, 1983.
\annot{} Early theoretical physics study of frustrated graphs based on \xref{Toulouse}{1977a}{Gerard Toulouse}.  Signed square lattice with translational sign symmetry and limited variation of signs and edge weights.  \S~II, ``Layered Ising models''.
Dictionary: ``plaquette'' = square, ``frustration index'' = sign of a plaquette.
\dated{24 May 2012} 
\codes{\CPhys,~\CSG:~\CState(\Cfr)}%
\itemspace

\author{Paul Wollan}							%WOLLAN
\authorspace
\authref  See \hyperlink{Bertrand Guenin}{B.~Guenin}, \hyperlink{Tony Huynh}{T.~Huynh}, and \hyperlink{Ken-ichi Kawarabayashi}{K.~Kawarabayashi}.
\itemspace

\author{Dein Wong}							%WONG DI
\authorspace
\authref  See \hyperlink{Xiaobin Ma}{X.B.~Ma}, \hyperlink{Fenglei Tian}{F.L.~Tian}, \hyperlink{Sai Wang}{S.~Wang}, and \hyperlink{Feng Xu}{F.~Xu}.
\itemspace

\author{Tsai-Lien Wong}						%WONG TL
\authorspace
\authref  See \hyperlink{Ligang Jin}{L.-G.~Jin} and \hyperlink{Hao Qi}{H.~Qi}.
\itemspace

\author{R.\ Kevin Wood}						%WOOD RK
\authorspace
\authref  See \hyperlink{Gerald G.\ Brown}{G.G.~Brown}.
\itemspace

\author{Bart{\l}omiej Wr\'oblewski}					%WROBLEWSKI
\authorx{Bartlomiej Wroblewski}
\authorspace
\authref  See \hyperlink{Robert Janczewski}{R.~Janczewski}.
\itemspace

\author{Bang Ye Wu}							%WU BY
\authorspace
\authref  See also \hyperlink{Chun-Hsiang Bai}{C.-H.~Bai}, \hyperlink{Jia-Fen Chen}{J.-F.~Chen},  \hyperlink{Li-Hsuan Chen}{L.-H.~Chen}, and \hyperlink{Po-Sun Wei}{P.-S.~Wei}.
\authrefspace

\bsgitem{2013b}{}  Balancing a complete signed graph by editing edges and deleting nodes.  In:  R.-S.\ Chang \etal, eds., \emph{Advances in Intelligent Systems and Applications} (Proc.\ ICS 2012, Hualien, Taiwan, 2012), Vol.\ 1, pp.\ 79--88.  Smart Innovation, Systems and Technologies, Vol.\ 20.  Springer, Berlin, 2013.
\codes{\CSG:~\CKG:~\CFr:~\CAlgor}%
\itemspace

\author{Bang Ye Wu \& Li-Hsuan Chen}
\authorspace
\bsgitem{2015a}{}  Parameterized algorithms for the 2-clustering problem with minimum sum and minimum sum of squares objective functions.  \emph{Algorithmica} 72 (2015), 818--835.
\MR{3355837}.  \Zbl{1328.68098}.  
\annot{} Equivalent:  Find minimizing bipartition for $(K_n,\sigma)$.  
Objective: minimize $\sum_v c(v)$ (thus finding $l(K_n,\sigma)$) or $\sum_v c(v)^2$.
	
\annotctd{} Dictionary:  ``graph'' = positive subgraph of $(K_n,\sigma)$; ``editing'' = edge sign changes; ``$2$-clustering'' = bipartition $V = X \cup Y$, ``conflict number'' $c(x)$, for $x\in X$ (say), = \#(edges $+xx'$) + \#(edges $-xy$).  
\dated{13 Jun 2017}
\codes{\Csg:~\Ckg:~\CFr(\CGen):~\CAlgor}%
\itemspace

\author{Baofeng Wu}							%WU BF
\authorspace
\authref  See \hyperlink{Peikang Zhang}{P.-K.~Zhang}.
\itemspace

\author{Chai Wah Wu}						%WU CW
\authorspace
\bsgitem{2005a}{}  On Rayleigh--Ritz ratios of a generalized Laplacian matrix of directed graphs.  \emph{Linear Algebra Appl.}\ 402 (2005), 207--227.
\MR{2141085} (2005m:05108).  \Zbl{1063.05065}.
\annot{} The graphs are weighted mixed graphs, i.e., bidirected graphs without introverted edges, and the matrices are digraph matrices, i.e., (weighted) outdegree matrices.  The ``Laplacian'' is $D-A$ where $A$ is the adjacency matrix and $D$ is the diagonal outdegree matrix.  
	
\dated{23 Mar 2009}
\par
\codes{\Csg,~\Csd:~\Cori:~\Cinc,~\CEig}%
\itemspace

\author{Chong Wu}							%WU C
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Guangmei Wu}						%WU GM
\authorspace
\authref  See \hyperlink{Chuanchao Huang}{C.-C.~Huang}.
\itemspace

\author{Jianliang Wu}						%WU JL
\authorspace
\authref  See \hyperlink{Xingqin Qi}{X.Q.~Qi}.
\itemspace

\authorx{Jianshe Wu}						%WU JS
\author{Jianshe Wu, Licheng Jiao, Chao Jin, Fang Liu, Maoguo Gong, Ronghua Shang, \& Weisheng Chen}
\authorspace
\bsgitem{2012a}{}  Overlapping community detection via network dynamics.  \emph{Phys.\ Rev.\ E} 85 (2012), art.\ 016115, 7 pp.
\annot{} Detects possibly overlapping clusters in $\Gamma$ via $(K_n,\sigma)$ with $E^+=E(\Gamma)$.  Each vertex gets a randomly phased oscillator.  Edge weights $w_+>0$, $w_-\geq0$.  Oscillators $\to$ in-phase on $+$ edges, out-of-phase on $-$ edges, revealing clusters.
\dated{16 Jun 2018} 
\codes{\Csg:~\Ckg,~\CWG:~\CClu:~\CDyn}%
\itemspace

\author{Jianshe Wu, Long Zhang, Yong Li, \& Yang Jiao}
\authorspace
\bsgitem{2016a}{}  Partition signed social networks via clustering dynamics.  \emph{Physica A} 443 (2016), 568--582.
\MR{3416828}.  \Zbl{1400.91496}.  
\codes{\CSG:~\CClu:~\CAlgor}%
\itemspace

\author{Jingwen Wu}							%WU JW
\authorspace
\authref  See \hyperlink{Yong Lu}{Y.~Lu}.
\itemspace

\authorx{Leting Wu}							%WU LT
\author{Leting Wu, Xintao Wu, Aidong Lu, \& Yuemeng Li}
\authorspace
\bsgitem{2014a}{}  On spectral analysis of signed and dispute graphs.  In:  \emph{2014 IEEE International Conference on Data Mining}, pp.\ 1049--1054.  IEEE, 2014.
\annot{} Extended abstract of \xrefd{2017a}{Leting Wu, Xintao Wu, Aidong Lu, \& Yuemeng Li}.
\codes{\CSG:~\CClu:~\CAdj:~\CEig,~\CGeom}%
\itemspace

\bsgitem{2017a}{}  On spectral analysis of signed and dispute graphs: Application to community structure.  \emph{IEEE Trans.\ Knowledge Data Engineering} 29 (2017), no.\ 7, 1480--1493.
\codes{\CSG:~\CClu:~\CAdj:~\CEig,~\CGeom}%
\itemspace

\author{Leting Wu, Xiaowei Ying, Xintao Wu, Aidong Lu, \& Zhi-Hua Zhou}
\authorspace
\bsgitem{2011a}{}  Spectral analysis of $k$-balanced signed graphs.  In: Joshua Zhexue Huang, Longbing Cao and Jaideep Srivastava, eds., \emph{Advances in Knowledge Discovery and Data Mining} (Proc.\ 15th Pacific-Asia Conf., PAKDD 2011, Shenzen, Part II), pp.\ 1--12.  \LNCS 6635.  Springer, Berlin, 2011.
\annot{} Spectral analysis of clusterable signed graphs.  
Dictionary:  ``$k$-balance'' = $k$-clusterability.
\dated{26 Apr 2012} 
\codes{\CSG:~\CClu~\CEig}%
\itemspace

\bsgitem{20xxa}{}  Examining spectral space of complex networks with positive and negative links.  \emph{Int.\ J.\ Social Network Mining} (to appear).
\codes{\CSG:~\CEig:~\CClu~\CBal}%
\itemspace

\author{Lifang Wu}							%WU LF
\authorspace
\authref  See \hyperlink{Weichan Liu}{W.-C.~Liu}.
\itemspace

\author{Qi Wu}							%WU Q
\authorspace
\authref  See also \hyperlink{Yong Lu}{Y.~Lu}.
\authrefspace

\author{Qi Wu, Yong Lu, \& Bit-Shun Tam}
\authorspace
\bsgitem{2022a}{}  On connected signed graphs with rank equal to girth.  \emph{Linear Algebra Appl.}\ 651 (2022), 90--115.
\MR{4445167}.  \Zbl{1493.05148}.  
\codes{\CSG:~\CAdj}%
\itemspace

\author{Qiang Wu}							%WU Q
\authorspace
\authref  See \hyperlink{Gui Zhen Liu}{G.Z.~Liu}.
\itemspace

\author{Shu-Hui Wu}							%WU SH
\authorspace
\authref  See \hyperlink{Bit-Shun Tam}{B.S.~Tam}.
\itemspace

\author{Sun Wu}							%WU S
\authorspace
\authref  See \hyperlink{Wei-Kuan Shih}{W.-S.~Shih}.
\itemspace

\authorx{Xiao Li Wu}							%WU XL
\author{Xiao Li Wu, Jing Jing Jiang, Ji Ming Guo, \& Shang Wang Tan}
\authorspace
\bsgitem{2011a}{}  The minimal signless Laplacian spectral radius of graphs with diameter $n-4$.  (In Chinese.)  \emph{Acta Math.\ Sinica (Chin.\ Ser.)}\ 54 (2011), no.\ 4, 601--608.
\MR{2868198} (2012i:05176).  \Zbl{1265.05423}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xintao Wu}							%WU XT
\authorspace
\authref  See \hyperlink{Yuemeng Li}{Y.-M.~Li} and \hyperlink{Leting Wu}{L.T.~Wu}.
\itemspace

\author{Yarong Wu}							%WU YR
\authorspace
\authref  See \hyperlink{Guanglong Yu}{G.L.~Yu}.
\itemspace

\authorx{Yanzhi Wu}							%WU YZ
\author{Yanzhi Wu, Lu Liu, Jiangping Hu, \& Gang Feng}
\authorspace
\bsgitem{2018a}{}  Adaptive antisynchronization of multilayer reaction--diffusion neural networks.  \emph{IEEE Trans.\ Neural Networks Learning Sys.}\ 29 (2018), no.\ 4, 807--818.
\MR{3789355}.  
\annot{} Multilayer signed digraph.
\codes{\CSD,~\Csg:~\CBal,~\CDyn}%
\itemspace

\authorx{Yezhou Wu}							%WU YZ
\author{Yezhou Wu \& Dong Ye}
\authorspace
\bsgitem{2018a}{}  Circuit covers of cubic signed graphs.  \emph{J.\ Graph Theory} 89 (2018), 40--54.
\MR{828127}.  \Zbl{1398.05101}.  \arXiv{1609.03620}.				
\codes{\CSG:~\CStr}%
\itemspace

\bsgitem{2020a}{}  Minimum $T$-Joins and signed-circuit covering.  \emph{SIAM J.\ Discrete Math.}\ 34 (2020), no.\ 2, 1192--1204.
\MR{4101365}.  \Zbl{1441.05098}.  \arXiv{1803.03696}.  
\codes{\CSG:~\Cmatrd}%
\itemspace

\author{Yezhou Wu, Dong Ye, Wenan Zang, \& Cun-Quan Zhang}
\authorspace
\bsgitem{2014a}{}  Nowhere-zero 3-flows in signed graphs.  \emph{SIAM J.\ Discrete Math.}\ 28 (2014), no.\ 3, 1628--1637.
\MR{3264564}.  \Zbl{1408.05069}.  
\codes{\CSG:~\CFlows}%
\itemspace

\author{Yuhan Wu}							%WU YH
\authorspace
\authref  See \hyperlink{Shu Yong Yi}{S.Y.~Yi} and \hyperlink{Lihua You}{L.H.~You}.
\itemspace

\author{Zhaoyang Wu}						%WU ZY
\authorspace
\bsgitem{2003a}{}  On the number of spikes over finite fields.  \emph{Discrete Math.}\ 265 (2003), 261--296.
\MR{1969378} (2004b:05057).  \Zbl{1014.05015}.
\annot{} A spike is $\bfL_0(\Omega)$ where $\|\Omega\| = 2C_n$.  
\codes{\Cgg:~\CMatrd:~\CEnum}%
\itemspace

\author{Donald C.\ Wunsch}						%WUNSCH
\authorspace
\authref  See \hyperlink{Frank Harary, Meng-Hiot Lim, Amit Agarwal, \& Donald C.\ Wunsch}{Harary, Lim, \etal}
\itemspace


\label{LETTER X} \hypertarget{LETTER X}{}	% LETTER  X


\author{Chengyi Xia}							%XIA CY
\authorspace
\authref  See \hyperlink{Shasha Feng}{S.-S.~Feng}.
\itemspace

\author{Wen Xia}							%XIA W
\authorspace
\authref  See \hyperlink{Xueliang Li}{X.-L.~Li}.
\itemspace

\author{Kai-nan Xiang}						%XIANG KN
\authorspace
\authref  See \hyperlink{Rong Chen}{R.~Chen}.
\itemspace

\authorx{Bai Xiao}							%XIAO B
\author{Bai Xiao, Song Yi-Zhe, \& Peter Hall}
\authorspace
\bsgitem{2011a}{}  Learning invariant structure for object identification by using graph methods.  \emph{Computer Vision Image Understanding}\ 115 (2011), 1023--1031.
\annot{} Empirical tests of usefulness of the eigenvalues (the ``feature vector'') of $\Lap(-\Gamma)$.
\dated{24 Jan 2012} 
\codes{\CPar:~\CEig:~\CAppl}%
\itemspace

\author{Min Xiao}							%XIAO M
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Peng Xiao}							%XIAO P
\authorspace
\authref  See \hyperlink{Yong Lu}{Y.~Lu}.
\itemspace

\authorx{Mengmeng Xie}						%XIE MM
\author{Mengmeng Xie \& Chuixiang Zhou}
\authorspace
\bsgitem{2020a}{}  Signed circuit cover of bridgeless signed graphs.  \emph{Graphs Combin.}\ 36 (2020), 1423--1443.
\MR{4148421}.  \Zbl{1458.05101}.  
\codes{\CSG:~\CMatrd}%
\itemspace

\authorx{Rundan Xing}						%XING RD
\author{Rundan Xing \& Bo Zhou}
\authorspace
\bsgitem{2015a}{}  Laplacian and signless Laplacian spectral radii of graphs with fixed domination number.  \emph{Math.\ Nachrichten} 288 (2015), no.\ 4, 476--480.
\MR{3320461}.  \Zbl{1396.05080}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Zhuang Xiong}						%XIONG ZA
\authorspace
\authref  See also \hyperlink{Li Ou}{L.~Ou}.
\authrefspace

\author{Zhuang Xiong \& Yaoping Hou}
\authorspace
\bsgitem{2022a}{}  Eigenvalue-free interval for Seidel matrices of threshold graphs.  \emph{Applied Math.\ Comput.}\ 427 (2022), art.\ 127177.
\MR{4411535}.  
\annot{} Esp., $(-\sqrt2,\sqrt2)$ contains no eigenvalues of $A(\Sigma)$ other than $\pm1$.  
[\Cf\ \xref{Milica An{\dj}eli\'c, Tamara Koledin, \& Zoran Stani\'c}{2019a}{Milica Andelic, Tamara Koledin, \& Zoran Stanic}.]
\dated{20 Jun 2022} 
\codes{\Csg:~\CKG:~\CAdj:~\CEig}%
\itemspace

\author{Baogen Xu}							%XU BG
\authorspace
\authref  See also \hyperlink{Shuchao Li}{S.-C.~Li}.
\authrefspace

\bsgitem{2001a}{}  On signed edge domination numbers of graphs.  \emph{Discrete Math.} 239 (2001), 179--189. 
MR 1850997 (2002e:05104).  Zbl 979.05081.				
\annot{} Let $s:=\sum_e \sigma(e) = \#E^+ - \#E^-$ and $s_e := \sum\{\sigma(f): f \in N[e]\}$.  The signed edge domination number of $\Gamma$ is $\min \{s :  \sigma \text{ s.t.\ }s_e>0,\ \forall e \big\}.$  Question:  What is $g(n):=\min\{s: \#V=n\}$?  \Cf\ \xref{Akbari, Bolouki, \etal\!}{2009a}{S.\ Akbari, S.\ Bolouki, P.\ Hatami, \& M.\ Siami}.
\dated{17 Feb 2011} 
\codes{\CSGw}
\itemspace

\bsgitem{2009a}{}  On signed cycle domination in graphs.  \emph{Discrete Math.}\ 309 (2009), no.\ 4, 1007--1012.
\MR{2502161} (2010e:05236).  \Zbl{1180.05082}.					
\annot{} See \xref{Vijayakumar}{2011a}{G.R.\ Vijayakumar}.
\dated{10 Feb 2013} 
\codes{\CSGw}%
\itemspace

\author{Brian Xu}							%XU B
\authorspace
\authref  See \hyperlink{Lucas J.\ Rusnak}{L.J.~Rusnak}.
\itemspace

\author{Dachuan Xu}							%XU DC
\authorspace
\authref  See \hyperlink{Sai Ji}{S.~Ji}.
\itemspace

\authorx{Feng Xu}							%XU F
\author{Feng Xu, Qi Zhou, Dein Wong, \& Fenglei Tian}
\authorspace
\bsgitem{2020a}{}  Complex unit gain graphs of rank 2.  \emph{Linear Algebra Appl.}\ 597 (2020), 155--169.
\MR{4080071}.  \Zbl{1437.05157}.  
\annot{} Rank of $A(\Phi)$.
\codes{\CGG:~\CAdj}%
\itemspace

\author{Guang-Hui Xu}						%XU GH
\authorspace
\authref  See \hyperlink{Shi-Cai Gong}{S.C.~Gong}.
\itemspace

\author{Guangjun Xu}						%XU GJ
\authorspace
\authref  See \hyperlink{Hongyan Cai}{H.-Y.~Cai}.
\itemspace

\author{Jing Xu}							%XU J
\authorspace
\authref  See \hyperlink{Bo-Jun Yuan}{B.-J.~Yuan}.
\itemspace

\author{Meiling Xu}							%XU ML
\authorspace
\authref  See \hyperlink{Guanglong Yu}{G.L.~Yu}.
\itemspace

\author{Rongxing Xu}						%XU RX
\authorspace
\authref  See \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\authorx{Rui Xu}							%XU R
\author{Rui Xu \& Cun-Quan Zhang}
\authorspace
\bsgitem{2005a}{}  On flows in bidirected graphs.  \emph{Discrete Math.}\ 299 (2005), 335--343.
\MR{2168714} (2006e:05081).  \Zbl{1073.05033}.					
\annot{} $\Sigma$ has a nowhere-zero 6-flow if it is coloop-free and edge 6-connected, thus solving \xref{Bouchet's}{1983a}{Andre Bouchet} conjecture in this case.
\dated{5 Feb 2010} 
\codes{\CSG:~\CFlows}%
\itemspace

\author{Shaoji Xu}							%XU SJ
\authorspace
\authref  See also \hyperlink{Fred S.\ Roberts}{F.S.~Roberts}.
\authrefspace

\bsgitem{1998a}{}  \emph{Cycle Space: Cycle Bases, Signed Graphs and Marked Graphs}.  Doctoral dissertation, Rutgers Center for Operations Research, Rutgers University, 1998.
\par
\codes{\CSG,~\CVS:~\CBal,~\Cfr,~\CAlgor,~\CPsS}
\itemspace

\bsgitem{1998b}{}  The line index and minimum cut of weighted graphs.  \emph{European J.\ Operational Res.}\ 109 (1998), no.\ 3, 672--685. 
\Zbl{972.05026}.  
\codes{\CWG,~\CSG:~\Cfr:~\CAlgor}%
\itemspace

\author{Wei-Ru Xu}							%XU WR
\authorspace
\authref  See \hyperlink{Yong Lu}{Y.~Lu}.
\itemspace

\author{Yuan Xu}							%XU Y
\authorspace
\authref  See \hyperlink{Di Peng}{D.~Peng}.
\itemspace

\author{Zhi-Ming Xu}							%XU ZM
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Dong Xue}							%XUE D
\author{Dong Xue, Sandra Hirche, \& Ming Cao}
\authorspace
\bsgitem{2020a}{}  Evolution of social power over influence networks containing antagonistic interactions.  \emph{Inform.\ Sci.}\ 540 (2020), 449--468.
\MR{4124416}.  \Zbl{1479.91287}.  
\codes{\CSG:~\CDyn}%
\itemspace

\author{Jie Xue}							%XUE J
\authorspace
\authref  See \hyperlink{Ruifang Liu}{R.-F.~Liu}.
\itemspace


\label{LETTER Y} \hypertarget{LETTER Y}{}	% LETTER  Y


\author{Sandeep Kumar Yadav}					%YADAV
\authorspace
\authref  See \hyperlink{Bibhas Adhikari}{B.~Adhikari}.
\itemspace

\authorx{Takeshi Yajima}				 		%YAJIMA
\author{Takeshi Yajima \& Shin'ichi Kinoshita}
\authorspace
\bsgitem{1957a}{}  On the graphs of knots.  \emph{Osaka Math.\ J.}\ 9 (1957), 155--163.  
\MR{0098385} (20 \#4845).  \Zbl{080.17002} (80, p.\ 170b).				
\annot{} Examines the relationship between the two dual sign-colored graphs, $\Sigma$ and $\Sigma'$, of a link diagram (\xref{Bankwitz}{1930a}{C.\ Bankwitz}), translating the Reidemeister moves into graph operations and showing that they will convert $\Sigma$ into $\Sigma'$.
\codes{\CSGc:~\CKnot}%
\itemspace

\author{Takeo Yamada}						%YAMADA
\authorspace
\bsgitem{1988a}{}  A note on sign-solvability of linear system of equations.  \emph{Linear Multilinear Algebra} 22 (1988), no.\ 4, 313--323.
\MR{0943759} (89g:15005).  \Zbl{723.05111}.  
\codes{\CQM:~\CSol:~\CSD}%
\itemspace

\author{Takeo Yamada \& Harunobu Kinoshita}
\authorspace
\bsgitem{2002a}{}  Finding all the negative cycles in a directed graph.  \emph{Discrete Appl.\ Math.}\ 118 (2002), 279--291.
\MR{1892974} (2002m:05187).  \Zbl{999.05057}.
\annot{} In a real-weighted digraph, ``negative'' means the sum of weights is negative.
\codes{\CWG}%
\itemspace

\author{Yuuzi Yamada}						%YAMADA Y
\authorspace
\authref  See \hyperlink{Ojiro Nagai}{O.~Nagai}.
\itemspace

\author{Chiaki Yamaguchi}						%YAMAGUCHI C
\authorspace
\bsgitem{2013a}{}  Conjectured exact percolation thresholds of the Fortuin--Kasteleyn cluster for the $\pm J$ Ising spin glass model.  \emph{Physica A} 392 (2013), no.\ 6, 1263--1268.
\MR{3019857} (no rev).  \arXiv{1004.0654}.  
\codes{\CPhys:~\CSG}%
\itemspace

\bsgitem{2014a}{}  Analytical estimates of the locations of phase transition points in the ground state for the bimodal Ising spin glass model in two dimensions.  \emph{Progress Theor.\ Exper.\ Phys.}\ 2014 (2014), art.\ 053A02, 9 pp.
\arXiv{1309.6554}.  
\par
\codes{\CPhys:~\CSG,~\CState}%
\itemspace

\author{Yutaro Yamaguchi}						%YAMAGUCHI Y
\authorspace
\authref  See \hyperlink{Yasushi Kawase}{Y.~Kawase}.
\itemspace

\author{Takeo Yamamoto}						%YAMAMOTO T
\authorspace
\authref  See \hyperlink{Tota Nakamura}{T.~Nakamura}.
\itemspace

\author{Chao Yan}							%YAN C
\authorspace
\authref  See \hyperlink{Guanglong Yu}{G.L.~Yu} and \hyperlink{Longqin Wang}{L.Q.~Wang}.
\itemspace

\author{Eric Yan}							%YAN E
\authorspace
\authref  See \hyperlink{Lucas J.\ Rusnak}{L.J.~Rusnak}.
\itemspace

\authorx{Jing-Ho Yan}						%YAN JH
\author{Jing-Ho Yan, Ko-Wei Lih, David Kuo, \& Gerard J.\ Chang}
\authorspace
\bsgitem{1997a}{}  Signed degree sequences of signed graphs.  \emph{J.\ Graph Theory} 26 (1997), 111--117.  
\MR{1469358} (98i:05160).  \Zbl{980.04848}.					
\annot{} Net degree sequences of signed simple graphs.  Thm.\ 2 improves the Havel--Hakimi-type theorem from \xref{Chartrand, Gavlas, Harary, and Schultz}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz} by determining the length parameter.  Thm.\ 7 characterizes the net degree sequences of signed trees.  [There seems to be room to strengthen the characterization and generalize to weighted degree sequences: see notes on \xref{Chartrand, Gavlas, \etal{}}{1994a}{Gary Chartrand, Heather Gavlas, Frank Harary, \& Michelle Schultz}.]
\par
\codes{\CSGw:~\Cori:~\CInvar}%
\itemspace

\author{Zhidan Yan}							%YAN ZD
\authorspace
\authref  See \hyperlink{Wei Wang [mr36]}{W.~Wang}.
\itemspace

\authorx{Aimei Yang}							%YANG AM
\author{Aimei Yang \& Josh Bentley}
\authorspace
\bsgitem{2017a}{}  A balance theory approach to stakeholder network and apology strategy.  \emph{Public Relations Rev.}\ 43 (2017), no.\ 2, 267--277.
\codes{\CPsS:~\CSG:~\CBal}%
\itemspace

\author{Alex Yang}							%YANG A
\authorspace
\authref  See \hyperlink{Vinciane Chen}{V.~Chen}.
\itemspace

\author{Arthur L.B.\ Yang}						%YANG ALB
\authorspace
\authref  See \hyperlink{William Y.C.\ Chen}{W.Y.C.~Chen}.
\itemspace

\authorx{Bo Yang}							%YANG B
\author{Bo Yang, William K.\ Cheung, \& Jiming Liu}
\authorspace
\bsgitem{2007a}{}  Community mining from signed social networks.  \emph{IEEE Trans.\ Knowledge Data Engineering} 19 (2007), no.\ 10, 1333--1348.
\annot{} An algorithm for an approximate clustering of a (weighted) signed (di)graph.  Input: The graph and a length parameter $l$.  Step 1:  Construct transition probabilities $p{ij} := [\sigma{ij}w_{ij}]^+/d(v_i)$.  Step 2:  Apply the probabilities in a random walk of length $\leq l$ on positive edges; the matrix of $l$-step probabilities is $(p_{ij})^l$.  Combine in a cluster the vertices that have high probabilities from a given starting point.  ``High'' and $l$ are based on the network structure.
\annotctd{} Also, a cut algorithm for approximate clustering.  A cluster is $X \subset V$ such that the total net degree $d^\pm(\Sigma\ind X) \geq d^\pm(X,X^c)$ and $d^\pm(X^c,X) \leq d^\pm(\Sigma\ind X^c)$.
\dated{11 Feb 2009}
\codes{\CSG:~\CWG:~\CClu:~\CAlgor}%
\itemspace

\author{Bo Yang, Xueyan Liu, Yang Li, \& Xuehua Zhao}
\authorspace
\bsgitem{2017a}{}  Stochastic blockmodeling and variational Bayes learning for signed network analysis.  \emph{IEEE Trans.\ Knowledge Data Eng.}\ 29 (2017), no.\ 9, 2026--2039.
\codes{\CSG:~\CClu}%
\itemspace

\author{Bo Yang, Xuehua Zhao, \& Xueyan Liu}
\authorspace
\bsgitem{2015a}{}  Bayesian approach to modeling and detecting communities in signed network.  In:  \emph{Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence} (AAAI-15, Austin, 2015), pp.\ 1952--1958.  AAAI Press, Palo Alto, Calif., 2015.  URL\\ \url{https://www.aaai.org/ocs/index.php/AAAI/AAAI15/paper/view/9376}
\par
\codes{\CSG:~\CClu}%
\itemspace

\author{Dan Yang}							%YANG D
\authorspace
\authref  See \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}.
\itemspace

\authorx{Fan Yang}							%YANG F
\author{Fan Yang \& Sanming Zhou}
\authorspace
\bsgitem{2016a}{}  Nowhere-zero 9-flows in 3-edge-connected signed graphs.  Manuscript, 2016.
\arXiv{1508.04620}.  
\annot{} Withdrawn due to incomplete proof.  [Can someone complete it?]
\dated{20 Jul 2020} 
\codes{\CSG:~\CFlows}%
\itemspace


\author{Laurence T.\ Yang}						%YANG LT
\authorspace
\authref  See \hyperlink{Fei Hao}{F.~Hao}.
\itemspace

\author{Ruixian Yang}						%WANG RX
\authorspace
\authref  See \hyperlink{Chuanchao Huang}{C.-C.~Huang}.
\itemspace

\authorx{Shiju Yang}							%YANG SJ
\author{Shiju Yang, Chuandong Li, Xiping He, \& Wanli Zhang}
\authorspace
\bsgitem{2022a}{}  Variable-time impulsive control for bipartite synchronization of coupled complex networks with signed graphs.  \emph{Appl.\ Math.\ Computation} 420 (2022), art.\ 126899, 16 pp.
\MR{4361759}.  
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\author{Shiju Yang, Chuandong Li, Yu Li, Ting Yang, \& Bo Li}
\authorspace
\bsgitem{20xxa}{}  The fast fixed-time bipartite synchronization of coupled delayed neural networks with signed graphs.  Submitted.  \emph{}\ 
\annot{} $\{V_1,V_2\} =$ Harary bipartition of balanced signed digraph $\vec\Sigma$.  Nodes $v_i$ have states $z_i(t) \in \bbR^m$ for time $t\geq0$.  Arc weights and a control function govern the evolution of $z$.  ``Bipartite synchronization'' means $\exists\ y(t),\ t_0$ for which $z_i(t)=y(t), -y(t)$ when $v_i \in V_1, V_2$ respectively, $\forall\ t>t_0$.  
[One of many papers on the topic.]
\dated{30 Jan 2022} 
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\authorx{Shuang Hong Yang}						%YANG SH
\author{Shuang Hong Yang, Alex Smola, Bo Long, Hongyuan Zha, \& Yi Chang}
\authorspace
\bsgitem{2012a}{}  Friend or frenemy?  Predicting signed ties in social networks.  In:  \emph{Proceedings of the 35th International ACM SIGIR Conference on Research and Development in Information Retrieval} (SIGIR '12), pp.\ 555--564.  ACM, New York, 2012.
\codes{\CSG:~\CPsS}%
\itemspace

\author{Ting Yang}							%YANG T
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.-C.~Li} and \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\authorx{Weiling Yang}						%YANG WL
\author{Weiling Yang \& Fuji Zhang}
\authorspace
\bsgitem{2007a}{} The Kauffman bracket polynomial of links and universal signed plane graph. 
In: Jin Akiyama \etal, eds., \emph{Discrete Geometry, Combinatorics and Graph Theory} (7th China-Japan Conf., CJCDGCGT 2005, Tianjin and Xi'an, China, 2005), pp.\ 228--244. 
\LNCS 4381.  Springer, Berlin, 2007. 
\MR{2364767} (2009b:57031).  \Zbl{1149.05308}.					
\annot{} The ``chain polynomials'' of sign-colored plane graphs with cyclomatic number $\leq 5$ are obtained systematically.
[\Cf\ \xref{Jin and Zhang}{2005a}{Xian'an Jin \& Fuji Zhang}, \xrefd{2007a}{Xian'an Jin \& Fuji Zhang}.]
\dated{5 July 2009} 
\codes{\CSGc:~\CInvar}%
\itemspace

\authorx{Xiuwen Yang}						%YANG XW
\author{Xiuwen Yang \& Ligong Wang}
\authorspace
\bsgitem{2021a}{}  Iota energy orderings of bicyclic signed digraphs.  \emph{Trans.\ Combin.}\ 10 (2021), no.\ 3, 187--200.
\MR{4257184}.  
\arXiv{2004.01412}.  
\annot{} \Cf\ \xref{Hafeez and Khan}{2018a}{Sumaira Hafeez \& Mehtab Khan}.
\codes{\CSD:~\CAdj:~\CEig}%
\itemspace

\authorx{Yujun Yang}							%YANG YJ
\author{Yujun Yang \& Dong Ye}
\authorspace
\bsgitem{2018a}{}  Inverses of bipartite graphs.  \emph{Combinatorica} 38 (2018), no.\ 5, 1251--1263.
\MR{3884787}.  \Zbl{1424.05127}.  \arXiv{1611.06535}.				
\annot{} Elegant solution to problem of \xref{Godsil}{1985a}{C.D.\ Godsil}:  Iff $\Gamma$ excludes an ``odd flower''.  
Dictionary:  ``diagonally similar to a non-negative matrix'' = balanced.  
\Question.  Is the answer the same if $\Gamma$ is allowed to be a signed graph?
\dated{11 Dec 2018} 
\codes{\Csg:~\CAdj,~\CBal,~\Csw}%
\itemspace

\author{Mihalis Yannakakis}  						%YANNAKAKIS
\authorspace
\authref   See also \hyperlink{Vijay V.\ Vazirani}{V.V.~Vazirani}.
\authrefspace

\authorspace
\bsgitem{1985a}{}  On a class of totally unimodular matrices.  \emph{Math.\ Operations Res.}\ 10 (1985), 280--304.
\MR{0793885} (87h:90245).  \Zbl{565.90042}.					
\annot{} ``Restricted totally unimodular matrices'':  Bipartite.  Structural characterization by decomposition into incidence matrices (and transposes) of balanced bidirected graphs.  
	
\codes{\Csg:~\CStr}%
\annotctd{} Structure of ``restricted balanceable graphs'', defined as bipartite and signable so a circle is positive iff it is evenly even.
	
\dated{19 Jan 2015} 
\par
\codes{\CSGw,~\Csg:~\CStr}%
\itemspace

\author{Milhalis Yannakakis [Mihalis Yannakakis]}
\authorspace
\authref  See \hyperlink{Mihalis Yannakakis}{Mihalis Yannakakis}.
\itemspace

\author{Yan Hong Yao}						%YAO YH
\authorspace
\authref  See \hyperlink{Lin Feng}{L.~Feng}.
\itemspace

\author{Yuan Yao}							%YAO Y
\authorspace
\authref  See \hyperlink{Zilin Jiang}{Z.-L.~Jiang}.
\itemspace

\author{Zahra Yarahmadi}						%YARAHMADI
\authorspace
\bsgitem{2010a}{}  The bipartite edge frustration of extension of splice and link graphs.  \emph{Appl.\ Math.\ Lett.}\  23 (2010), no.\ 9, 1077--1081.
\MR{2659141} (2011e:05219).  \Zbl{1210.05115}.  
\annot{} 
Dictionary:  ``bipartite edge frustration'' of $\Gamma$ = frustration index $l(-\Gamma)$.
\par
\codes{\Cpar:~\CFr}%
\itemspace

\author{Zahra Yarahmadi \& Ali Reza Ashrafi}
\authorspace
\bsgitem{2011a}{}  The bipartite edge frustration of graphs under subdivided edges and their related sums.  \emph{Computers Math.\ Appl.}\  62 (2011), no.\ 1, 319--325.
\MR{2821848} (no rev).  \Zbl{1228.05132}.  
\codes{\Cpar:~\CFr}%
\itemspace

\bsgitem{2011b}{}  Extremal properties of the bipartite vertex frustration of graphs.  \emph{Appl.\ Math.\ Lett.}\  24 (2011), 1774--1777.
\MR{2812210} (2012h:05163).  \Zbl{1234.05134}.  
\annot{} 
Dictionary:  ``bipartite vertex frustration'' of $\Gamma$ = frustration number $l_0(-\Gamma)$.
\codes{\Cpar:~\CFr}%
\itemspace

\author{Z.\ Yarahmadi, T.\ Do\v{s}li\'c, \& A.R.\ Ashrafi}
\authorx{Z.\ Yarahmadi, T.\ Doslic, \& A.R.\ Ashrafi}
\authorspace
\bsgitem{2010a}{}  The bipartite edge frustration of composite graphs.  \emph{Discrete Appl.\ Math.}\ 158 (2010), no.\ 14, 1551--1558.
\MR{2659170} (2011g:05301).  \Zbl{1215.05094}.  
\par
\codes{\Cpar:~\CFr}%
\itemspace

\author{Geva Yashfe}						%YASHFE
\authorspace
\authref  See \hyperlink{Lukas Kuhne}{L.~K\"uhne}.
\itemspace

\author{T.\ Yasuda}							%YASUDA
\authorspace
\bsgitem{2015a}{}  \emph{Inferring Chromosome Structures With Bidirected Graphs Constructed From Genomic Structural Variations.}  Ph.D.\ Thesis, University of Tokyo, 2015.
\par
\codes{\Csg:~\COri:~\CBiol}%
\itemspace

\author{Stephen S.\ Yau}						%YAU SS
\authorspace
\authref  See \hyperlink{Fei Hao}{F.~Hao}.
\itemspace

\author{Dong Ye}							%YE D
\authorspace
\authref  See \hyperlink{Er Ling Wei}{E.L.~Wei}, \hyperlink{Yujun Yang}{Y.-J.~Yang}, \hyperlink{Yezhou Wu}{Y.-Z.~Wu}, and \hyperlink{Li Zhang}{L.~Zhang}.
\itemspace

\authorx{Miao-Lin Ye}						%YE ML
\author{Miao-Lin Ye, Yi-Zheng Fan, \& Hai-Feng Wang}
\authref  See also \hyperlink{Jia Sheng}{J.~Sheng}.
\authrefspace

\bsgitem{2010a}{}  Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 6, 1180--1186.
\MR{2661684} (2011i:05142).  \Zbl{1207.05125}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Peter Ye}							%YE P
\authorspace
\authref  See \hyperlink{Lucas Rusnak}{L.J.~Rusnak}.
\itemspace

\author{Yeong-Nan Yeh}						%YEH YN
\authorspace
\authref   See \hyperlink{Ivan Gutman}{I.~Gutman}, \hyperlink{Yiting Jiang}{Y.-T.~Jiang}, \hyperlink{Shyi-Long Lee}{S.L.~Lee}, and \hyperlink{Yi Wang}{Y.~Wang}.
\itemspace

\author{Aylin Yener}							%YENER
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Anders Yeo}							%YEO
\authorspace
\authref  See \hyperlink{Noga Alon}{N.~Alon} and \hyperlink{Gregory Gutin}{G.~Gutin}.
\itemspace

\authorx{Shu Yong Yi}						%YI SY
\author{Shu Yong Yi \& Li Hua You}
\authorspace
\authref  See also \hyperlink{Lihua You \& Shuyong Yi}{L.H.~You}.
\authrefspace

\bsgitem{2011a}{}  The local bases and bases of primitive anti-symmetric signed digraphs with no loops.  \emph{J.\ South China Normal Univ.\ Natur.\ Sci.\ Ed.}\ 2011 (2011), no.\ 1, 39--42.
\MR{2839257} (no rev).  \Zbl{1249.05262}.  
\codes{\CSD:~\CAdj,~\Cqm}%
\itemspace

\author{Shuyong Yi, Lihua You, \& Yuhan Wu}
\authorspace
\bsgitem{2012a}{}  The bases and base set of primitive symmetric loop-free signed digraphs.  \emph{J.\ Math.\ Res.\ Appl.}\ 32 (2012), no.\ 3, 313--326.
\MR{2985369}.  \Zbl{1265.05287}.  
\codes{\CSD,~\CDyn,~\Cqm}%
\itemspace

\author{Xiaowei Ying}						%YING XW
\authorspace
\authref  See \hyperlink{Leting Wu}{L.T.~Wu}.
\itemspace

\author{Xuerong Yong}						%YONG XR
\authorspace
\authref  See \hyperlink{Xiaogang Liu}{X.G.~Liu} and \hyperlink{Yuanping Zhang}{Y.P.~Zhang}.
\itemspace

\author{En Sup Yoon}						%YOON ES
\authorspace
\authref  See \hyperlink{Gibaek Lee}{G.~Lee}.
\itemspace

\author{Yeoin Yoon}							%YOON YI
\authorspace
\authref  See \hyperlink{Jeong-Rae Kim}{J.-R.~Kim}.
\itemspace

\author{Young-Jin Yoon}						%YOON YJ
\authorspace
\bsgitem{1997a}{}  A characterization of supersolvable signed graphs.  \emph{Commun.\ Korean Math.\ Soc.}\ 12 (1997), 1069--1073.
\MR{1643925} (99j:05165).  \Zbl{945.05051}.					
\annot{} An attempt to characterize supersolvability of $\bfF(\Sigma)$ in terms of \linebreak{} [bias-]simplicial vertices.  
[Fundamental conceptual and technical errors vitiate the entire paper; see \xref{Koban}{2004a}{Lori Koban}.  For correct results see \xref{Zaslavsky}{2001a}{Thomas Zaslavsky} and \xref{Koban}{2004a}{Lori Koban}.]
\codes{\CSG:~\CMatrd:~\CStr}%
\itemspace

\author{Yusuke Yoshie}						%YOSHIE
\authorspace
\authref  See \hyperlink{Etsuo Segawa}{E.~Segawa}.
\itemspace

\authorx{Takeo Yoshikawa}						%YOSHIKAWA
\author{Takeo Yoshikawa, Takashi Iino, \& Hiroshi Iyetomi}
\authorspace
\bsgitem{2011a}{}  Market structure as a network with positively and negatively weighted links.  
In:  Junzo Watada, Gloria Phillips-Wren, Lakhmi C.\ Jain, and Robert J.\ Howlett, eds., \emph{Intelligent Decision Technologies} (Proc.\ 3rd Int.\ Conf., IDT'2011), pp.\ 511--518.  Smart Innovation, Systems and Technologies, Vol.\ 10.  Springer-Verlag, Berlin, 2011.
\annot{} Preliminary report of \xrefd{2012a}{Takeo Yoshikawa, Takashi Iino, \& Hiroshi Iyetomi}.
\dated{26 Jun 2012} 
\par
\codes{\CSG,~\CWG:~\CClu:~\CAppl}%
\itemspace

\bsgitem{2012a}{}  Observation of frustrated correlation structure in a well-developed financial market.  \emph{Progress Theor.\ Phys.}\ Suppl.\ No.\ 194 (2012), 55--63.
\annot{} Application of correlation clustering to the Tokyo stock market.  The ``frustration'' of a clustering $\pi = \{B_1,\ldots,B_k\} \in \Pi_V$ in a weighted signed graph $(\Sigma,w)$ is $F(\pi) := -\sum_i \sum_{e\in E{:}B_i} w_e$ (\cf\ \xref{Traag and Bruggeman}{2009a}{V.A.\ Traag \& Jeroen Bruggeman}).
\dated{26 Jun 2012} 
\codes{\CSG,~\CWG:~\CClu:~\CAppl}%
\itemspace

\author{Huazheng You}						%YOU HZ
\authorspace
\authref  See \hyperlink{Wenzhong Liu}{W.-Z.~Liu}.
\itemspace

\author{Lihua You}							%YOU LH
\authorspace
\authref  See also \hyperlink{Feng Cheng}{F.~Cheng}, \hyperlink{Yu Liu}{Y.~Liu}, \hyperlink{Longqin Wang}{L.Q.~Wang},and \hyperlink{Shu Yong Yi}{S.Y.~Yi}.
\authrefspace

\author{Lihua You, Jiayu Shao, \& Haiying Shan}
\authorspace
\bsgitem{2007a}{}  Bounds on the bases of irreducible generalized sign pattern matrices.  \emph{Linear Algebra Appl.}\ 427 (2007), 285--300.
\MR{2351360} (2008g:15009).  \Zbl{1179.15034}.					
\par
\codes{\CQM:~\CSD}%
\itemspace

\author{Lihua You \& Yuhan Wu}
\authorspace
\bsgitem{2011a}{}  Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs.  \emph{Linear Algebra Appl.}\ 434 (2011), no.\ 5, 1215--1227.
\MR{2763581} (2012c:05201).  \Zbl{1204.05051}.					
\codes{\CSD:~\Cqm}%
\itemspace

\author{Lihua You \& Shuyong Yi}
\authorspace
\bsgitem{2012a}{}  The characterization of primitive symmetric loop-free signed digraphs with the maximum base.  \emph{Electronic J.\ Linear Algebra}\ 23 (2012), 122--136.
\MR{2889576}.  \Zbl{1252.05078}.  
\codes{\CSD:~\CAdj,~\CQM}%
\itemspace

\author{Zhifu You}							%YOU ZF
\authorspace
\authref  See also \hyperlink{Bolian Liu}{B.L.~Liu}.
\authrefspace

\author{Zhifu You \& Bolian Liu}
\authorspace
\bsgitem{2011a}{}  The signless Laplacian separator of graphs.  \emph{Electronic J.\ Linear Algebra} 22 (2011), 151--160.
\MR{2781043} (2012a:05209).  \Zbl{1226.05174}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{A.P.\ Young}							%YOUNG AP
\authorspace
\authref  See \hyperlink{R.N.\ Bhatt}{R.N.~Bhatt}, \hyperlink{K.\ Binder}{K.~Binder} and \hyperlink{Matteo Palassini}{M.~Palassini}.
\itemspace

\author{Michael Young}						%YOUNG M
\authorspace
\authref  See \hyperlink{Matthias Beck}{M.~Beck}.
\itemspace

\author{J.W.T.\ Youngs} 						%YOUNGS*
\authorspace
\bsgitem{1968a}{}  Remarks on the Heawood conjecture (nonorientable case).  \emph{Bull.\ Amer.\ Math.\ Soc.}\ 74 (1968), 347--353.  
\MR{0220623} (36 \#3675).  \Zbl{161.43303}.
\annot{} Introducing ``cascades'': current graphs with bidirected edges.  A ``cascade'' is a bidirected graph, not all positive, that is provided with both a rotation system (hence it is orientation embedded in a surface) and a current (which is a special kind of bidirected flow).  Dictionary: ``broken'' = negative edge.  
[Also see \xref{Ringel}{1974a}{Gerhard Ringel}.]
\codes{\Csg:~\COri:~\CAppl,~\CFlows}%
\itemspace

\bsgitem{1968b}{}  The nonorientable genus of $K_n$.  \emph{Bull.\ Amer.\ Math.\ Soc.}\ 74 (1968), 354--358.  
\MR{0220624} (36 \#3676).  \Zbl{161.43304}.  
\annot{} ``Cascades'': see \xrefd{1968a}{J.W.T.\ Youngs}.
\codes{\Csg:~\COri:~\CAppl}%
\itemspace

\author{Aimei Yu}							%YU AM
\authorspace
\authref  See \hyperlink{Shengjie He}{S.J.~He} \hyperlink{Guojun Li}{G.J.~Li}, and \hyperlink{W.J.\ Zhang}{W.J.~Zhang}.
\itemspace

\author{Cheng-Ching Yu}						%YU CC
\authorspace
\authref  See \hyperlink{Chung-Chien Chang}{C.C.~Chang}.
\itemspace

\authorx{Guanglong Yu}						%YU GL
\author{Guanglong Yu, Shuguang Guo, \& Meiling Xu}
\authorspace
\bsgitem{2013a}{}  On the least signless Laplacian eigenvalues of some graphs.  \emph{Electronic J.\ Linear Algebra} 26 (2013), no.\ 1, art.\ 37, 560--573.
\MR{3104546}.  \Zbl{1282.05162}.  
\annot{} The graphs minimizing $\lambda_n(\Lap(-\Gamma))$ given $n$ and the matching number $\mu$ or edge cover number.  Thm.\ 3.5 finds the (few) connected, unbalanced, antibalanced signed graphs $-\Gamma$ with given $n,\mu$ that minimize $\lambda_n(\Lap(-\Gamma))$.  [\Problem.  Generalize to connected, unbalanced signed graphs.]
\dated{20 Jan 2015} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Guanglong Yu, Zhengke Miao, Chao Yan, \& Jinlong Shu}
\authorspace
\bsgitem{2013a}{}  Gaps in the base set of primitive nonpowerful sign patterns.  \emph{Linear Multilinear Algebra} 61 (2013), no.\ 6, 801--810.
\MR{3005658}.  \Zbl{1273.05136}.  
\codes{\CSD,~\CQM}%
\itemspace

\author{Guanglong Yu, Yarong Wu, \& Jinlong Shu}
\authorspace
\bsgitem{2011a}{}  Signless Laplacian spectral radii of graphs with given chromatic number.  \emph{Linear Algebra Appl.}\ 435 (2011), no.\ 8, 1813--2096.
\MR{2810629} (2012e:05247).  \Zbl{1221.05244}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011b}{}  Sharp bounds on the signless Laplacian spectral radii of graphs.  \emph{Linear Algebra Appl.}\ 434 (2011), no.\ 3, 683--687.
\MR{2746075} (2012e:05246).  \Zbl{1225.05178}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\authorx{Gui-Dong Yu}						%YU GD
\author{Gui-Dong Yu, Yi-Zheng Fan, \& Yi Wang}
\authorspace
\bsgitem{2014a}{}  Quadratic forms on graphs with application to minimizing the least eigenvalue of signless Laplacian over bicyclic graphs.  \emph{Electronic J.\ Linear Algebra} 27 (2014), no.\ 1, art.\ 13, 213--236.
\MR{3194952}.  \Zbl{1288.05168}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Guihai Yu}							%YU GH
\authorspace
\authref  See also \hyperlink{Lihua Feng}{L.H.~Feng}.
\authrefspace

\bsgitem{2008a}{}  On the maximal signless Laplacian spectral radius of graphs with given matching number.  \emph{Proc.\ Japan Acad.\ Ser.\ A Math.\ Sci.}\ 84 (2008), no.\ 9, 163--166.
\MR{2483600} (2009m:15012).  \Zbl{1175.05090}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Guihai Yu, Matthias Dehmer, Frank Emmert-Streib, \& Herbert Jodlbauer}
\authorspace
\bsgitem{2019a}{}  Hermitian normalized Laplacian matrix for directed networks.  \emph{Inform.\ Sci.}\ 495 (2019), 175--184.
\MR{3948704}.  \Zbl{1451.05150}.  
\annot{} It is a normalized Laplacian for digraph gains given by $1$ if $e_{ij},e_{ji} \in E$, gain $\mathrm{i}$ if only $e_{ij} \in E$.  [\Cf\ \xref{Liu and Li}{2015a}{Jianxi Liu \& Xueliang Li}, \xref{Guo and Mohar}{2017a}{Krystal Guo \& Bojan Mohar}.]  Results about eigenvalues, e.g., symmetry, special values 0 and 2, edge interlacing, and expressions for coefficients of characteristic polynomial.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CLap:~\CEig}%
\itemspace

\author{Guihai Yu, Lihua Feng, Aleksandar Ili\'c, \& Dragan Stevanovi\'c}
\authorx{Guihai Yu, Lihua Feng, Aleksandar Ilic, \& Dragan Stevanovic}
\authorspace
\bsgitem{2015a}{}  The signless laplacian spectral radius of bounded degree graphs on surfaces.  \emph{Appl.\ Anal.\ Discrete Math.}\ 9 (2015), 332--346.
\MR{3444911}.  \Zbl{1464.05258}.  
\codes{\Csg:~\Cpar:~\CLap:~\CEig,~\CTop}%
\itemspace

\author{Guihai Yu, Lihua Feng, \& Hui Qu}
\authorspace
\bsgitem{2016a}{}  Signed graphs with small positive index of inertia.  \emph{Electronic J.\ Linear Algebra} 31 (2016), art.\ 18, 232--243.
\MR{3504407}.  \Zbl{1339.05173}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Guihai Yu, Lihua Feng, Qingwen Wang, \& Aleksandar Ili\'c}
\authorspace
\bsgitem{2014a}{}  The minimal positive index of inertia of signed unicyclic graphs.  \emph{Ars Combin.}\ 117 (2014), 245--255. 
\MR{3243844}.  \Zbl{1349.05153}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Guihai Yu, Xin Liu, \& Hui Qu}
\authorspace
\bsgitem{2017a}{}  Singularity of Hermitian (quasi-)Laplacian matrix of mixed graphs.  \emph{Appl.\ Math.\ Comput.}\  293 (2017), 287--292.
\MR{3549669} (no rev).  \Zbl{1411.05176}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$ ($\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges).  $\Lap(\Phi)$ and $\Lap(-\Phi)$ are positive semidefinite and singularity is characterized [\cf\ \xref{Guihai Yu \& Hui Qu}{2015a}{Guihai Yu \& Hui Qu}].  Formula for determinant.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CLap}%
\itemspace

\author{Guihai Yu \& Hui Qu}
\authorspace
\bsgitem{2015a}{}  Hermitian Laplacian matrix and positive of mixed graphs.  \emph{Appl.\ Math.\ Comput.}\ 269 (2015), 70--76.
\MR{3396760} (no rev).  \Zbl{1410.05145}.  
\annot{} Based on \xref{Liu and Li}{2015a}{Jianxi Liu \& Xueliang Li}, \xref{Guo and Mohar}{2017a}{Krystal Guo \& Bojan Mohar}, and \xref{Harary}{1953a}{Frank Harary}.  
\S2, ``The positive of mixed graphs''.  Thms.\ 2--3:  For complete $\|\Phi\|$ [an unnecessary restriction], $\Phi$ is balanced iff $V = V_1 \cupdot V_2$ so all edges in $V_i$ are positive and all $V_1V_2$ edges have gain $\mathrm{i}$  [note: $\not\exists$ negative edges], iff all triangles are balanced.  Thm.\ 4:  $\Phi$ is balanced iff all $uv$-paths have the same gain.
\S3, ``Hermitian Laplacian matrix of mixed graphs''.  Switching equivalence.  Incidence matrix [special case of \xref{Zaslavsky}{2003b}{Thomas Zaslavsky}, \S2].  Laplacian matrix from incidence matrix, hence positive semidefinite (Thm.\ 9).  Elementary eigenvalue properties and nice bounds [compare \xref{Reff}{2012a}{Nathan Reff}] for $A$ and $\Lap$.
Dictionary:  ``positive'' = balanced.
\dated{15 Dec 2020} 
\codes{\Cgg:~\CBal,~\CAdj,~\CInc,~\CLap:~\CEig}%
\itemspace

\bsgitem{2018a}{}  More on spectral analysis of signed networks.  \emph{Complexity} 2018 (2018), art.\ 3467158, 6 pp.
\Zbl{1407.05154}.  
\annot{} Spectral symmetry around 1 and meaning of eigenvalue 2 of normalized Laplacian matrix.
\annotctd{} Major error:  Their incidence matrix $S$ is that of $-\Sigma$, i.e., $\Eta(-\Sigma)$ (\xref{Zaslavsky}{1982a}{Thomas Zaslavsky}); thus their Laplacian $\Lap(\Sigma)$ is really $\Lap(-\Sigma)$.  Thm.\ 2: For connected $\Sigma$, $\Lap(\Sigma)$ is singular iff all $uv$-walks have the same sign [I.e., $\Sigma$ is balanced; known from \xref{Zaslavsky}{1982a}{Thomas Zaslavsky} in matroid form].  Wrong proof; the calculation reverses signs.  Their Laplacian should be $\Lap(-\Sigma)$ throughout.  Further results are known or wrong; e.g., Thm.\ 9 to the extent correct is \xref{Zaslavsky}{1982a}{Thomas Zaslavsky}, Thm.\ 8A.4.  [The new results can and should be corrected.]
\dated{10 Nov 2018} 
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Guihai Yu, Hui Qu, \& Jianhua Tu}
\authorspace
\bsgitem{2015a}{}  Inertia of complex unit gain graphs.  \emph{Appl.\ Math.\ Comput.}\ 265 (2015), 619--629.
\MR{3373510} (no rev).  \Zbl{1410.05174}.  
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\author{Guihai Yu, Xiying Yuan, \& Hui Qu}
\authorspace
\bsgitem{2019a}{}  Signed $k$-uniform hypergraphs and tensors.  \emph{ Linear Algebra Appl.}\ 580 (2019), 1--13.
\MR{3975082}.  \Zbl{1420.05108}.  
\codes{\CSH:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Jianming Yu}							%YU JM
\authorspace
\authref  See \hyperlink{Guangfeng Jiang}{G.~Jiang}.
\itemspace

\author{Wei-Hsuan Yu}						%YU WH
\authorspace
\authref  See \hyperlink{Meng-Yue Cao}{M.-Y.~Cao}.
\itemspace

\author{Weiqiang Yu}						%YU WQ
\authorspace
\authref  See \hyperlink{Reza Naserasr}{R.~Naserasr}.
\itemspace

\author{Yuantian Yu}							%YU YT
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.-C.~Li}.
\itemspace

\author{B.\ Yuan}							%YUAN B
\authorspace
\authref  See \hyperlink{Y.\ Chen}{Y.~Chen}.
\itemspace

\author{Bo-Jun Yuan}						%YUAN BJ
\authorspace
\authref  See also \hyperlink{Yi Wang}{Y.~Wang}.
\authrefspace

\author{Bo-Jun Yuan, Yi Wang, \& Jing Xu}
\authorspace
\bsgitem{2020a}{}  Characterizing the mixed graphs with exactly one positive eigenvalue and its application to mixed graphs determined by their $H$-spectra.  \emph{Appl.\ Math.\ Comput.}\ 380 (2020), art.\ 125279, 7 pp.
\MR{4085149}.  \Zbl{1460.05120}.  
\annot{} $\Phi$ with gain group $\{\pm1,\pm\mathrm{i}\}$: $\phi(e)=1$ for undirected, $\mathrm{i}$ for directed edges.  Examples where spectrum determines $\Phi$ [\cf\ \xref{Mohar}{2016a}{Bojan Mohar}].
\dated{15 Dec 2020} 
\codes{\Cgg:~\CAdj:~\CEig}%
\itemspace

\authorx{Lili Yuan}							%YUAN LL
\author{Lili Yuan, Jixiang Meng, \& Eminjan Sabir}
\authorspace
\bsgitem{2021a}{}  The antistrong property for special digraph families.  \emph{Graphs Combin.}\ 37 (2021), 2511--2519.
\MR{4338743}.  \Zbl{1479.05133}.  
\annot{} ``Antistrong'' means there is an antidirected trail from any vertex to any other vertex.  Antistrong Cartesian and lexicographic products and tournaments.  [\Problem.  Generalize to bidirected graphs.]
\dated{22 Nov 2021} 
\codes{\Cgg:~\CPaths(\CGen)}%
\itemspace

\author{Shuhan Yuan}						%YUAN SH
\authorspace
\authref  See \hyperlink{Yuemeng Li}{Y.-M.~Li}.
\itemspace

\author{Xiying Yuan}							%YUAN XY
\authorspace
\authref  See also \hyperlink{Vladimir Nikiforov}{V.~Nikiforov}, \hyperlink{Zhenan Shao}{Z.-N.~Shao}, and \hyperlink{Guihai Yu}{G.H.~Yu}.
\authrefspace

\bsgitem{2014a}{}  Maxima of the $Q$-index: forbidden odd cycles.  \emph{Linear Algebra Appl.}\ 458 (2014), 207--216.
\MR{3231816}.  \Zbl{1295.05147}.  \arXiv{1401.4363}.				
Corrigendum.  \emph{Ibid.}\ 65 (2015), 426--429.
\MR{3274687}.  \Zbl{1304.05097}.  \arXiv{1401.4363}.				
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xi-Ying Yuan, Yue Liu, \& Miaomiao Han}
\authorspace
\bsgitem{2011a}{}  The Laplacian spectral radius of trees and maximum vertex degree.  \emph{Discrete Math.}\ 311 (2011), no.\ 8-9, 761--768.
\MR{2774232} (2011m:05191).  \Zbl{1216.05013}.  
\annot{} \S3:  $Q := \Lap(-\Gamma)$ is used to prove results about trees.
\dated{21 Jan 2012} 
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\author{Xiying Yuan, Yanqi Mao, \& Lele Liu}
\authorspace
\bsgitem{2021a}{}  Maximal signed graphs with odd signed cycles as star complements.  \emph{Appl.\ Math.\ Comput.}\ 408 (2021), art.\ 126367, 11 pp. 
\MR{4264737}.  
\annot{} 
[See \xref{Rowlinson and Stani\'c}{2022a}{Peter Rowlinson \& Zoran Stanic}.]
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Zihan Yuan}							%YUAN ZH
\authorspace
\authref  See \hyperlink{Shengxiang Lv}{S.-X.~Lv}.
\itemspace

\authorx{Raphael Yuster}						%YUSTER
\author{Raphael Yuster \& Uri Zwick}
\authorspace
\bsgitem{1994a}{}  Finding even cycles even faster.
In:  Serge Abiteboul and Eli Shamir, eds., \emph{Automata, Languages and Programming} (Proc.\ 21st Int.\ Colloq., ICALP 94, Jerusalem, 1994), pp.\ 532--543.  Lect.\ Notes Computer Sci., Vol.~820.  Springer-Verlag, Berlin, 1994.
\MR{1334128}.  \Zbl{844.00024} (book).  						
\annot{} Abbreviated version of \xrefd{1997a}{Raphael Yuster \& Uri Zwick}.
\codes{\Cpar:~\CCircles:~\CAlgor}%
\itemspace

\bsgitem{1997a}{}  Finding even cycles even faster.  \emph{SIAM J.\ Discrete Math.}\ 10 (1997), 209--222.  
\MR{1445033} 98d:05137.  \Zbl{867.05065}.					
\annot{} For fixed \emph{even} $k$, a very fast algorithm for finding a $k$-gon.  Also, one for finding a shortest even circle.  [\Question.  Are these the all-negative cases of similarly fast algorithms to find positive $k$-gons, or shortest positive circles, in signed graphs?]
\codes{\Cpar:~\CCircles:~\CAlgor}%
\itemspace

\author{Sergey Yuzvinsky}						%YUZVINSKY
\authorspace
\bsgitem{2004a}{}  Realization of finite abelian groups by nets in $\bbP^2$.  \emph{Compos.\ Math.}\ 140 (2004), no.\ 6, 1614--1624.
\MR{2098405} (2005g:52057).  \Zbl{1066.52027}.
\annot{} Prop.\ 3.3:  A $k$-net in $\bbC\bbP^2$ whose classes are pencils is the canonical representation of the jointless Dowling geometry $Q^\dag(\bbZ_m) = \bfF(\bbZ_m K_3)$ of a finite cyclic group.  If a $k$-net in $\bbC\bbP^2$ represents $\bfF(\fA K_3)$ for a finite abelian group $\fA$, then $\fA$ is a subgroup of a 2-torus (Thm.\ 4.4) or has small invariant factors (Thm.\ 5.4); in particular it cannot be $\bbZ_2^3$ (Thm.\ 4.2).  The author conjectures more definitive characterizations.
\par
\codes{\Cgg:~\CGeom}%
\itemspace


\label{LETTER Z} \hypertarget{LETTER Z}{}	% LETTER  Z


\author{C.T.\ Zahn, Jr.}						%ZAHN
\authorspace
\authref  See also \hyperlink{C.\ Witzgall}{C.~Witzgall}.
\authrefspace

\bsgitem{1964a}{}  Approximating symmetric relations by equivalence relations.  \emph{J.\ Soc.\ Industrial Appl.\ Math.}\ 12 (1964), no.\ 4, 840--847.
\MR{0172276} (30 \#2496).  \Zbl{129.16003} (129, 160c).				
\annot{} Let $l_{\clu}(K_n,\sigma):=\min_{\sigma^*} \rho(\sigma,\sigma^*)$ over all clusterable $\sigma^*$, where $\rho(\sigma,\sigma^*) := \#(E^+(\sigma^*)\oplus E^+(\sigma))$.  \S3 finds a minimizing $\sigma^*$ when $\Sigma^+$ is a graph with a nontrivial clique attached to each vertex.  \S4 attaches cliques to the preceding clique vertices in certain cases.  
Dictionary:  ``symmetric relation'' = graph $G$ = $\Sigma^+$, ``equivalence relation'' = $\sigma^*$, ``approximating equivalence relation'' = ``optimal partition'' = $\sigma^*$ minimizing $\rho$.  
[Sequel: \xref{Moon}{1966a}{J.W.\ Moon}.]  
[This appears to be the first paper that is implicitly on clustering a signed complete graph, before \xref{Davis}{1967a}{James A.\ Davis}.]
\dated{10 Nov 2017} 
\codes{\Csg:~\CClu}%
\itemspace

\bsgitem{1973a}{}  Alternating Euler paths for packings and covers.  \emph{Amer.\ Math.\ Monthly} 80 (1973), 395--403.  
\MR{0373937} (51 \#10137).  \Zbl{274.05112}.  
\codes{\Cpar:~\Cori}%
\itemspace

\author{Katarzyna Zaj\c{a}c}						%ZAJAC
\authorx{Katarzyna Zajac}
\authorspace
\authref  See also \hyperlink{Marcin Gasiorek}{M.~G\c{a}siorek}, \hyperlink{Grzegorz Marczak}{G.~Marczak}, and \hyperlink{Daniel Simson}{D.~Simson}.
\authrefspace

\bsgitem{2017a}{}  Numeric algorithms for corank two edge-bipartite graphs and their mesh geometries of roots.  \emph{Fundamenta Inform.}\ 152 (2017), no.\ 2, 185--222.
\MR{3636158}.  \Zbl{1375.05167}.  
\codes{\CSG:~\CAlgor}%
\itemspace

\bsgitem{2019a}{}  On the structure of loop-free non-negative edge-bipartite graphs.  \emph{Linear Algebra Appl.}\ 579 (2019), 262--283.
\MR{3959735}.  \Zbl{1419.05101}.  
\codes{\CSG}%
\itemspace

\bsgitem{2020a}{}  On polynomial time inflation algorithm for loop-free non-negative edge-bipartite graphs.  \emph{Discrete Appl.\ Math.}\ 283 (2020), 28--43.
\MR{4114879}.  \Zbl{1442.05228}.  
\par
\codes{\CSG:~\CAlgor}%
\itemspace

\author{Mariusz Zaj\c{a}c}						%ZAJAC
\authorx{Mariusz Zajac}
\authorspace
\bsgitem{20xxa}{}  A short proof of Brooks' theorem.  Submitted.  \emph{}\ 
\arXiv{1805.11176}.  
\annot{} Simple proof and algorithm, also applied to signed graphs and list coloring, generalizing \xref{M\'a\v{c}ajov\'a--Raspaud--\v{S}koviera}{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}, \xref{Fleiner--Wiener}{2016a}{T.\ Fleiner \& G.\ Wiener}, \xref{Dvo\v{r}\'ak--Postle}{2018a}{Zdenek Dvorak \& Luke Postle}.
\dated{2 Jul 2020} 
\codes{\CSG:~\CCol,~\CAlgor}%
\itemspace

\author{Robert B.\ Zajonc} 						%ZAJONC
\authorspace
\bsgitem{1968a}{}  Cognitive theories in social psychology.  
In:  Gardner Lindzey and Elliot Aronson, eds., \emph{The Handbook of Social Psychology}, second ed., Vol.~1, Ch.~5, pp.\ 320--411.  Addison-Wesley, Reading, Mass., 1968.
\annot{} ``Structural balance,'' pp.\ 338--353.  ``The congruity principle,'' pp.\ 353--359.  
\codes{\CPsS:~\CSD,~\CSG,~\CBal:~\CExp,~\CRef}%
\itemspace

\author{Robert B.\ Zajonc \& Eugene Burnstein}
\authorspace
\bsgitem{1965a}{}  Structural balance, reciprocity, and positivity as sources of cognitive bias.  \emph{J.\ Personality} 33 (1965), no.\ 4, 570--583.
\annot{} Test of the relative importance of balance, reciprocity (= digon balance), and number of positive arcs on experimental subjects memorizing a [simple] signed digraph (represented by a sociological story).  [The question raised is mathematically intriguing, but thus far undeveloped.]
\dated{24 Nov 2012} 
\codes{\CPsS:~\CSD:~\CBal}%
\itemspace

\author{Robert B.\ Zajonc \& Steven J.\ Sherman}
\authorspace
\bsgitem{1967a}{}  Structural balance and the induction of relations  \emph{J.\ Personality} 35 (1967), no.\ 4, 635--650.
\annot{} Experimental test of importance of balance in subjects' attitudes towards signed graphs of order 3 suggests that balance is a weak criterion.  Also, concise survey of several previous experiments.
\dated{24 Nov 2012} 
\codes{\CPsS:~\CSG:~\CBal}\codesctd{\CPsS:~\CSG,~\CSD:~\CBal:~\CExp}%
\itemspace

\author{[Kh.\ Zakhs]}							%ZAKHS
\authorspace
\authref  See \hyperlink{Horst Sachs}{H.~Sachs}.
\itemspace

\authorx{Shahid Zaman}						%ZAMAN
\author{Shahid Zaman \& Xiaocong He}
\authorspace
\bsgitem{2022a}{}  Relation between the inertia indices of a complex unit gain graph and those of its underlying graph.  \emph{Linear Multilinear Algebra} 70 (2022), no.\ 5, 843--877.
\MR{4395818}.  \Zbl{1485.05034}.  
\codes{\CGG:~\CAdj:~\CEig}%
\itemspace

\author{Giacomo Zambelli}						%ZAMBELLI
\authorspace
\authref  See also \hyperlink{Alberto Del Pia}{A.~Del Pia}.
\authrefspace

\bsgitem{2005a}{}  A polynomial recognition algorithm for balanced matrices.  \emph{J.\ Combin.\ Theory Ser.\ B} 95 (2005), 49--67.
\MR{2154177} (2006g:05041).  \Zbl{1071.05019}.  
\codes{\CSG:~\CBal,~\CAlgor}%
\itemspace

\author{Wenan Zang}						%ZANG WN
\authorspace
\authref  See also \hyperlink{Zhibin Chen}{Z.B.~Chen} and \hyperlink{Yezhou Wu}{Y.-Z.~Wu}.
\authrefspace

\bsgitem{1998a}{}  Proof of Toft's conjecture: Every graph containing no fully odd $K_4$ is 3-colorable (extended abstract).  In: Wen-Lian Hsu and Ming-Yang Kao, eds., \emph{Computing and Combinatorics} (Proc., 4th Ann.\ Int.\ Conf., COCOON'98, Taipei, 1998), pp.\ 261--268.  \LNCS 1449.  Springer-Verlag, Berlin, 1998.
\MR{1683343} (no rev).  \Zbl{914.05030}.  
\annot{} Summary of \xrefd{1998b}{Wenan Zang}.
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\bsgitem{1998b}{}  Proof of Toft's conjecture: Every graph containing no fully odd $K_4$ is 3-colorable.  \emph{J.\ Combin.\ Optim.}\ 2 (1998), 117--188.
\MR{1631313}.  \Zbl{914.05031}.  
\annot{} Proves \xref{Toft's}{1975a}{Bjarne Toft} conjecture:  For every 4-chromatic graph $\Gamma$, $-\Gamma$ contains a subdivided $-K_4$.  [\Cf\ \xref{Thomassen}{2001b}{Carsten Thomassen}.]  [\Question.  What is the signed-graph generalization?]
\dated{29 Oct 2017} 
\codes{\Csg:~\Cpar:~\CCol}%
\itemspace

\bsgitem{1998c}{}  Coloring graphs with no odd-$K_4$.  \emph{Discrete Math.}\ 184 (1998), 205--212.
\MR{1609310} (99e:05056).  \Zbl{957.05046}.  
\annot{} An algorithm, based in part on \xref{Gerards}{1994a}{A.M.H.\ Gerards}, that, given $\Gamma$, finds a subdivided $[-K_4]$ in $\Gamma$ or a $3$-coloring of $\Gamma$.  
	
[\Question.  Is there a generalization to all signed graphs?]
\dated{rev.\ 29 Oct 2017} 
\par
\codes{\Csg:~\Cpar:~\CCol,~\CAlgor,~\CRef}%
\itemspace

\author{Giovanni Zappella}						%ZAPPELLA
\authorspace
\authref  See \hyperlink{Nicolo Cesa-Bianchi}{N.~Cesa-Bianchi}.
\itemspace

\author{Thomas Zaslavsky} 						%ZASLAVSKY
\authorspace
\authref   See also \hyperlink{Laura Anderson}{L.~Anderson}, \hyperlink{Mukti Acharya, Joseph Varghese Kureethara, \& Thomas Zaslavsky}{M.~Acharya}, \hyperlink{Matthias Beck}{M.~Beck}, \hyperlink{Francesco Belardo}{F.~Belardo}, \hyperlink{Pascal Berthome}{P.~Berthom\'e}, \hyperlink{Ouahiba Bessouf}{O.~Bessouf}, \hyperlink{Ethan D.\ Bolker}{E.D.~Bolker}, \hyperlink{Seth Chaiken}{S.~Chaiken}, 
\hyperlink{Rigoberto Florez}{R.~Fl\'orez}, \hyperlink{David Forge}{D.~Forge}, \hyperlink{K.A.\ Germina}{K.A.~Germina}, \hyperlink{Curtis Greene}{C.~Greene}, \hyperlink{Shahul Hameed K}{S.~Ha\-meed}, \hyperlink{Phil Hanlon}{P.~Hanlon}, 
\hyperlink{Devlin Mallory, Abigail Raz, Christino Tamon, \& Thomas Zaslavsky}{D.~Mallory}, \hyperlink{A.M.\ Mathai}{A.M.~Mathai}, \hyperlink{Reza Naserasr}{R.~Naserasr}, \hyperlink{Roshni T Roy}{R.T.~Roy}, \hyperlink{Konstantin Rybnikov}{K.~Rybnikov}, \hyperlink{E.\ Sampathkumar, M.A.\ Sriraj, \& Thomas Zaslavsky}{E.~Sampathkumar}, 
\hyperlink{Alex Schaefer}{A.~Schaefer}, \hyperlink{Vaidy Sivaraman}{V.~Sivaraman}, \hyperlink{Daniel C.\ Slilaty}{D.C.~Slilaty}, and \hyperlink{Patrick Sole}{P.~Sol\'e}.
\authrefspace

\bsgitem{1977a}{}  Biased graphs.  Unpublished manuscript, 1977.
\annot{} Published, greatly expanded, as \xrefd{1989a}{Thomas Zaslavsky}, \xrefd{1991a}{Thomas Zaslavsky}, \xrefd{1995b}{Thomas Zaslavsky} and more; as well as (but restricted to signed graphs) \xrefd{1982a}{Thomas Zaslavsky}, \xrefd{1982b}{Thomas Zaslavsky}.
\par
\codes{\CGG:~\CBal,~\CMatrd}%
\itemspace

\bsgitem{1979a}{}  Line graphs of digraphs.  Abstract 768-05-3, \emph{Notices Amer.\ Math.\ Soc.}\ 26 (August, 1979), no.~5, A-448.
\codes{\CSG:~\CLG:~\COri,~\CInc,~\CEig(\CLG).~\CSw}
\itemspace

\bsgitem{1980a}{}  Voltage-graphic geometry and the forest lattice.  
In:  \emph{Report on the XVth Denison-O.S.U.\ Math.\ Conf.}\ (Granville, Ohio, 1980), pp.\ 85--89.  Dept.\ of Math., Ohio State University, Columbus, Ohio, 1980.
\codes{\CGG:~\CMatrd,~\CBic}%
\itemspace

\bsgitem{1981a}{}  The geometry of root systems and signed graphs.  \emph{Amer.\ Math.\ Monthly} 88 (1981), 88--105.  
\MR{0606249} (82g:05012).  \Zbl{466.05058}.					
\annot{} Signed graphs correspond to arrangements of hyperplanes in $\bbR^n$ of the forms $x_i = x_j$, $x_i = -x_j$, and $x_i = 0$.  Consequently, one can compute the number of regions of the arrangement from graph theory, esp.\ for arrangements corresponding to ``sign-symmetric'' graphs, i.e., having both or none of each pair $x_i = \pm x_j$.  Simplified account of parts of \xrefd{1982a}{Thomas Zaslavsky}, \xrefd{1982b}{Thomas Zaslavsky}, \xrefd{1982c}{Thomas Zaslavsky}, emphasizing geometry.
\codes{\CSG:~\CMatrd,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{1981b}{}  Characterizations of signed graphs.  \emph{J.\ Graph Theory} 5 (1981), 401--406.  
\MR{0635702} (83a:05122).  \Zbl{471.05035}.					
\annot{} Notably, Thm.\ 6:  A set $\cB$ of circles in $\Gamma$ is the set of positive circles in some signing of $\Gamma$ iff every theta subgraph contains an even number of circles in $\cB$.  [\Cf\ \xref{Soza\'nski}{1980a}{Tadeusz Sozanski}.]
\dated{rev.\ 22 Oct 2015, 26 Mar 2022}
\codes{\CSG:~\CBal}%
\itemspace

\bsgitem{1981c}{}  Is there a theory of signed graph embedding?  
In: \emph{Report on the XVIth Denison-O.S.U.\ Math.\ Conf.}\ (Granville, Ohio, 1981), pp.\ 79--82.  Dept.\ of Math., Ohio State University, Columbus, Ohio, 1981.
\annot{} See \xrefd{1997a}{Thomas Zaslavsky}.
\codes{\CSG:~\CTop,~\CMatrd}%
\itemspace

\bsgitem{1982a}{\dag\dag}  Signed graphs.  \emph{Discrete Appl.\ Math.}\ 4 (1982), no.\ 1, 47--74.  
\MR{0676405} (84e:05095a).  \Zbl{476.05080}.					
Erratum.  \emph{Discrete Appl.\ Math.}\ 5 (1983), no.\ 2, 248.  
\MR{0683518} (84e:05095b).  \Zbl{503.05060}.
\annot{$\bfF(\Sigma)$} Basic results on:  
Switching (\S3) (based on graph switching of \xref{Seidel}{1976a}{J.J.\ Seidel} \etal).  Prop.\ 3.2:  $\Sigma_1\sim\Sigma_2$ iff $\cB(\Sigma_1)=\cB(\Sigma_2)$, i.e., signed graphs are switching equivalent iff they have the same circle signs.  [\Cf\ \xref{Soza\'nski}{1980a}{Tadeusz Sozanski}.]
Minors (\S4).  
The frame matroid $\bfF(\Sigma)$ in many cryptomorphisms (\S5) (some erroneous: Thm.\ 5.1(f,g); partly corrected in the Erratum and fully in \xrefd{1991a}{Thomas Zaslavsky}), consistency of matroid with signed-graph minors; separators of $\bfF(\Sigma)$.  The signed covering graph $\tilde\Sigma$ (\S6).
\annotctd{} In \S8A, incidence and Laplacian matrices $\incid(\Sigma)$ and $\Lap(\Sigma)$, and matrix-tree theorem [different from that of \xref{Murasugi}{1989a}{Kunio Murasugi}] [generalized by \xref{Chaiken}{1982a}{Seth Chaiken} to a weighted, all-minors version, both directed and undirected].  [\xref{Grossman, Kulkarni, and Schochetman}{1995a}{Jerrold W.\ Grossman, Devadatta M.\ Kulkarni, \& Irwin E.\ Schochetman} improve Lemma 8A.2 (for special case $-\Gamma$) to \emph{Thm.}:  The nonzero minors of $\incid(\Sigma)$ are $\pm2^k$ for all $0\leq k \leq l_0(\Sigma)$, $l_0 :=$ frustration number = max \# disjoint negative circles.]
In \S8B, vector representation of the matroid $\bfF(\Sigma)$ by the incidence matrix [as multisubsets of root systems $B_n \cup C_n$].
\annotctd{} Conjectures about the interrelation between representability in characteristic $2$ and unique representability in characteristic $0$ [since answered by Geoff Whittle, A characterisation of the matroids representable over $\GF(3)$ and the rationals.  \emph{J.\ Combin.\ Theory Ser.~B} 65 (1995), 222--261.  \MR{1358987} (96m:05046).  \Zbl{0835.05015}) as developed by \xref{Pagano}{1998a}{Steven R.\ Pagano}, \xrefd{1999c}{Steven R.\ Pagano}].
\annotctd{} Examples (\S7) include:  Sign-symmetric graphs and signed expansions $\pm\Gamma$.  The all-negative graph $-\Gamma$, whose matroid (Cor.~7D.3; partially corrected in the Erratum) is the even-circle matroid (see \xref{Doob}{1973a}{Michael Doob}) and whose incidence matrices include the unoriented incidence matrix of $\Gamma$.  Signed complete graphs.
\annotctd{} Generalizations to gain graphs (called ``voltage graphs'') mentioned in \S9.
\annotctd{} Dictionary:  ``matroid $G(\ )$'' = $\bfF(\ )$, the frame matroid.
\dated{rev.\ 22 Oct 2015, 11 Mar 2022}
\par
\codes{\CSG,~\CGG:~\CMatrd,~\CBal,~\CSw,~\CCov,~\CInc,~\CGeom;~\CECyc,~\CKG}%
\itemspace

\bsgitem{1982b}{\dag}  Signed graph coloring.  \emph{Discrete Math.}\ 39 (1982), 215--228.
\MR{0675866} (84h:\-05050a).  \Zbl{487.05027}.					
\annot{$\chi_\Sigma(\lambda)$} A ``proper $k$-coloring'' of $\Sigma$ partitions $V$ into a special ``zero'' part, possibly void, that induces a stable subgraph, and up to $k$ other parts (labelled from a set of $k$ colors), each of which induces an antibalanced subgraph.  A ``zero-free proper $k$-coloring'' is similar but without the ``zero'' part.  [The suggestion is that a signed analog of a stable vertex set is one that induces an antibalanced subgraph.  \Problem.  Use this insight to develop generalizations of stable-set notions, such as cliques and perfection.  \emph{Example}.  Let $\alpha(\Sigma)$, the ``antibalanced vertex set number'', be the largest size of an antibalance-inducing vertex set.  Then $\alpha(\Gamma) = \alpha(+\Gamma \cup -K_n)$.]  
\S2, ``Counting the coloring ways'':  One gets two related chromatic polynomials.  The chromatic polynomial, $\chi_\Sigma(2k+1)$, counts all proper $k$-colorings; it is essentially the characteristic polynomial of the frame matroid.  It can often be most easily computed via the zero-free chromatic polynomial, $\chi^*_\Sigma(2k)$, which counts proper zero-free colorings: see \xrefd{1982c}{Thomas Zaslavsky}.  Contraction-deletion formulas; subset expansions, where the zero-free polynomial sums only over balanced edge sets.  
\S3, ``Pairs of colorings and orientations'':  Compatible and proper pairs.  Contraction and improper pairs.  Counting formulas.  (Generalizing R.P.\ Stanley, Acyclic orientations of graphs, \emph{Discrete Math.}\ 5 (1973), 171--178.  \MR{0317988} (47 \#6537).  \Zbl{258.05113}.)
\annotctd{} Continued in \xrefd{1982c}{Thomas Zaslavsky}.
\annotctd{} [$k$-coloring and zero-free $k$-coloring are usually better called $(2k+1)$- and $2k$-coloring, as in \xref{Macajova--Raspaud--Skoviera}{2014a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}.]
\dated{rev.\ 26 Aug 2018}
\codes{\CSG,~\CGG:~\CMatrd,~\CCol,~\CInvar,~\CCov,~\COri,~\CGeom}%
\itemspace

\bsgitem{1982c}{}  Chromatic invariants of signed graphs.  \emph{Discrete Math.}\ 42 (1982), 287--312. 
\MR{0677061} (84h:05050b).  \Zbl{498.05030}.						
\annot{} Continuation of \xrefd{1982b}{Thomas Zaslavsky}.  
\S1, ``Balanced expansion formulas'':  The fundamental balanced expansion formulas, that express the chromatic polynomial in terms of the zero-free chromatic polynomial.  
\S2, ``Counting by color magnitudes and signs''.  More complicated expansion formulas.
\S\S2--7:  Many special cases, treated in great detail: antibalanced graphs, signed graphs that contain $+K_n$ or $-K_n$, signed $K_n$'s (a.k.a.\ two-graphs), etc.  
\S3, ``Sign-symmetric graphs'': i.e., signed expansions, optionally with added half edges and loops.
\S4, ``Addition and deletion formulas''.  
\S5, ``All-negative graphs; the even-circle chromatic polynomial''.
\S6, ``Partial matching numbers and ordinary chromatic coefficients''.  
\S7, ``Signed complete graphs''.  
\S8, ``Orientations'': formulas for numbers of acyclic orientations in the examples (\cf\ \xrefd{1991b}{Thomas Zaslavsky}).
\dated{rev.\ 26 Feb 2012}
\codes{\CSG,~\CGG:~\CMatrd,~\CInvar,~\CCol,~\CCov,~\COri,~\CGeom;~\CECyc,~\CKG}%
\itemspace

\bsgitem{1982d}{}  Bicircular geometry and the lattice of forests of a graph.  \emph{Quart.\ J.\ Math.\ Oxford} (2) 33 (1982), 493--511.  
\MR{0679818} (84h:05050c).  \Zbl{519.05020}.					
\annot{} The set of all forests in a graph forms a geometric lattice.  The set of spanning forests forms a geometric semilattice.  The characteristic polynomials count (spanning) forests.
\codes{\CGG:~\CMatrd,~\CBic,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{1982e}{}  Voltage-graphic matroids.  
In:  Adriano Barlotti, ed., \emph{Matroid Theory and Its Applications} (Proc.\ Session of C.I.M.E., Varenna, Italy, 1980), pp.\ 417--423.  Liguore Editore, Naples, 1982.  
\MR{0863015}.  \Zbl{1225.05002}.  
Repr.: C.I.M.E.\ Summer Schools, Vol.\ 83, Springer, Heidelberg, and Fondazione C.I.M.E., Florence, 2010.
\MR{2768789}.  \Zbl{1225.05001}.
\annot{} The frame matroid of a gain graph.
\codes{\CGG:~\CMatrd,~\CECyc,~\CBic,~\CInvar:~\CExp}%
\itemspace

\bsgitem{1984a}{}  How colorful the signed graph?  \emph{Discrete Math.}\ 52 (1984), 279--284.  
\MR{0772289} (86m:05045).  \Zbl{554.05026}.					
\annot{} Studies zero-free [or balanced] chromatic number $\chi^* := \min(k: \exists$ proper coloring to $\{\pm1,\ldots,\pm k\}\}$, and in particular that of a complete signed graph (which may have parallel edges).  The signed graphs whose $\chi^*$ is largest or smallest. 
\annotctd{} \ [Updated notation: unbalanced chromatic number $\chi^0 :=$ similar for $\{0,\pm1,\ldots,\pm k\}$, chromatic number (from \xref{M\'a\v{c}ajov\'a--Raspaud--\v{S}koviera}{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}) $\chi := \min(2\chi^0+1,\ 2\chi^*) =$ min size of color set.]
\dated{rev 2 Jul 2020}
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{1984b}{}  Multipartite togs (analogs of two-graphs) and regular bitogs.  
In:  Proc.\ Fifteenth Southeastern Conf.\ on Combinatorics, Graph Theory and Computing (Baton Rouge, 1984), Vol.~III.  \emph{Congressus Numer.}\ 45 (1984), 281--293. 
\MR{0777728} (86d:05109).  \Zbl{625.05044}.					
\annot{} A modestly successful attempt to generalize two-graphs along the cohomological lines of \xref{Cameron and Wells}{1986a}{P.J.\ Cameron \& Albert L.\ Wells, Jr.}
\dated{6 July 2011}
\par
\codes{\CSG:~\CTG:~\CGen:~\CAdj,~\CSw}%
\itemspace

\bsgitem{1984c}{}  Line graphs of switching classes.  
In: \emph{Report of the XVIIIth O.S.U.\ Denison Maths Conference} (Granville, Ohio, 1984), pp.\ 2--4.  Dept.\ of Math., Ohio State University, Columbus, Ohio, 1984.
\annot{$\Lambda(\Sigma)$} The line graph of a switching class $[\Sigma]$ of signed graphs is a switching class of signed graphs; call it $[\Lambda'(\Sigma)]$.  The reduced line graph $\Lambda$ is formed from $\Lambda'$ by deleting parallel pairs of oppositely signed edges.  Then $A(\Lambda) = A(\Lambda') = 2I - \incid\trans\, \incid$, where $\incid$ is an incidence matrix of $\Sigma$.  Thm.\ 1:  $A(\Lambda)$ has all eigenvalues $\leq 2$.  Examples:  For an ordinary graph $\Gamma$, $\Lambda(-\Gamma) = - \Lambda(\Gamma)$.  Example: taking $-\Gamma$ and attaching any number of pendant negative digons to each vertex yields (the negative of) Hoffman's generalized line graph.  Additional results are claimed but there are no proofs.  [Also see \xrefd{20xxa}{Thomas Zaslavsky}.]  [This work is intimately related to that of \hyperlink{G.R.\ Vijayakumar}{Vijayakumar} \etal, which was then unknown to the author, and to \xref{Cameron}{1980a}{Peter J.\ Cameron} and \xref{Cameron, Goethals, Seidel, and Shult}{1976a}{P.J.\ Cameron, J.M.\ Goethals, J.J.\ Seidel, \& E.E.\ Shult}.]
\par
\codes{\CSG:~\CLG:~\CSw,~\CEig,~\CInc}%
\itemspace

\bsgitem{1987a}{}  The biased graphs whose matroids are binary.  \emph{J.\ Combin.\ Theory Ser.~B} 42 (1987), 337--347.  
\MR{0888686} (88h:05082).  \Zbl{667.05015}.					
\annot{} For the frame (bias), lift, and extended lift matroids: forbidden-minor and structural characterizations.  The latter for signed-graphiic frame matroids is superseded by a result of \xref{Pagano}{1998a}{Steven R.\ Pagano}.  
\annotctd{} [Error in Cor.\ 4.3:  In the last statement, omit ``$\bfF(\Omega) = \bfL(\Omega)$.''  That is true when $\Omega$ has no loops, but may not be if $\Omega$ has a loop $e$ (because Theorem 3(3) applies with unbalanced block $e$, but $(E \setminus e,e)$ is not a 2-separation).]
\codes{\CGG:~\CMatrd:~\CStr}%
\itemspace

\bsgitem{1987b}{}  Balanced decompositions of a signed graph.  \emph{J.\ Combin.\ Theory Ser.~B} 43 (1987), 1--13.  
\MR{0897236} (89c:05058).  \Zbl{624.05056}.					
\annot{} Decompose $E(\Sigma)$ into the fewest balanced subsets (generalizing the decomposition biparticity of an unsigned graph), or the fewest balanced connected subsets.  These minimum numbers are $\delta_0$ and $\delta_1$.  Thm.\ 1:  $\delta_0 = \lceil \log_2 \chi^*(-\Sigma) \rceil + 1$, where $\chi^*$ is the zero-free chromatic number.  Thm.\ 2:  $\delta_0 = \delta_1$ if $\Sigma$ is complete.  \Conjecture\ 1.  $\Sigma$ partitions into $\delta_0$ balanced, connected, and spanning edge sets (whence $\delta_0 = \delta_1$) if it has $\delta_0$ edge-disjoint spanning trees.  [Solved and generalized to basepointed matroids by \hyperlink{Daniel C.\ Slilaty}{D.~Slilaty} (unpublished).]  \Conjecture\ 2\ is a formula for $\delta_1$ in terms of $\delta_0$ of subgraphs.  [Thoroughly disproved by Slilaty (unpublished).] 
\par
\codes{\CSG:~\CFr}%
\itemspace

\bsgitem{1987c}{}  Vertices of localized imbalance in a biased graph.  \emph{Proc.\ Amer.\ Math.\ Soc.}\ 101 (1987), 199--204.  
\MR{0897095} (88f:05103).  \Zbl{622.05054}.					

\annot{} Such a vertex $u$ (also, a ``balancing vertex'') is a vertex of an unbalanced graph $\Omega$ whose removal leaves a balanced graph [i.e., frustration number $l_0=$].  Some elementary results, e.g., $\Omega = \Omega'/e$ where $\Omega\setm e$ is balanced and $e$ contracts to $u$.
\dated{rev.\ 19 Dec 2014}
\codes{\CGG:~\CFr}%
\itemspace

\bsgitem{1987d}{}  The M\"obius function and the characteristic polynomial.  
In: Neil White, ed., \emph{Combinatorial Geometries}, Ch.\ 7, pp.\ 114--138.  Encycl.\ Math.\  Appl., Vol.\ 29.  Cambridge Univ.\ Press, Cambridge, 1987.  
\MR{0921064} (88g:05048) (book).  \Zbl{632.05017}.						
\annot{} Pp.\ 134--135 expound the geometrical version of Dowling lattices as in \xref{Dowling}{1973a}{Thomas A.\ Dowling}.
\codes{\Cgg:~\CGeom,~\Cmatrd,~\CInvar:~\CExp}%
\itemspace

\bsgitem{1988a}{}  Togs (generalizations of two-graphs).  
In:  M.N.~Gopalan and G.A.~Patwardhan, eds., \emph{Optimization, Design of Experiments and Graph Theory} (Proc.\ Sympos.\ in Honour of Prof.\ M.N.\ 
Vartak, Bombay, 1986), pp.\ 314--334.  Indian Inst.\ of Technology, Bombay, 1988.  
\MR{0998807} (90h:05112).  \Zbl{689.05035}.						
\annot{} An attempt to generalize two-graphs (here [alas?] called ``unitogs'') in a way similar to that of \xref{Cameron and Wells}{1986a}{P.J.\ Cameron \& Albert L.\ Wells, Jr.} although largely independently.  The notable new example is ``Johnson togs'', based on the Johnson graph of $k$-subsets of a set.  ``Hamming togs'' are based on a Hamming graph (that is, a Cartesian product of complete graphs) and generalize examples of Cameron and Wells.  Other examples are as in \xrefd{1984b}{Thomas Zaslavsky}.
\codes{\CSG:~\CTG:~\CGen}%
\itemspace

\bsgitem{1988b}{}  The demigenus of a signed graph.  
In:  \emph{Report on the XXth Ohio State-Denison Mathematics Conference} (Granville, Ohio, 1988).  Dept.\ of Math., Ohio State University, Columbus, Ohio, 1988.  
\codes{\CSG:~\CTop,~\CMatrd}%
\itemspace

\bsgitem{1989a}{\dag\dag}  Biased graphs.  I.\  Bias, balance, and gains.  \emph{J.\ Combin.\ Theory Ser.~B} 47 (1989), 32--52.  
\MR{1007712} (90k:05138).  \Zbl{714.05057}.						
\annot{$\Omega, \Phi$} Fundamental concepts and lemmas of biased graphs.  Bias from gains; switching of gains; characterization of balance [for which see also \xref{Harary, Lindstr\"om, and Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom}].
\codes{\CGG:~\CBal,~\CSw}%
\itemspace

\bsgitem{1990a}{}  Biased graphs whose matroids are special binary matroids.  \emph{Graphs Combin.}\ 6 (1990), 77--93.  
\MR{1058551} (91f:05097).  \Zbl{786.05020}.					
\annot{} A complete list of the biased graphs $\Omega$ such that $\bfF(\Omega)$, $\bfL_0(\Omega)$, or $\bfL(\Omega)$ is one of the traditional special binary matroids, $\bfm(K_5), \bfM(K_{33}), F_7$, their duals, and $\bfM(K_m)$ (for $m \geq 4$) and $R_{10}$.  [Unfortunately omitted are nonbinary matroids like the non-Fano plane and its dual.]
\annotctd{} [Error:  The graphs $\langle +K_n^\circ \rangle$ were overlooked in the last statement of Lemma 1H---due to an oversight in \xrefd{1987a}{Thomas Zaslavsky} Cor.\ 4.3---and thus in Props.\ 2A and 5A.  A corrected last statement of Lemma 1H: ``If $\Omega$ has no two vertex-disjoint negative circles, then $\bfF(\Omega) = M \iff \bfL(\Omega) = M$.''  In Prop.\ 2A, add $\Omega = \langle +K_3^\circ \rangle$ to the list for $\bfM(K_4)$.  In Prop.\ 5A, add $\Omega = \langle +K_{m-1}^\circ \rangle$ to the list for $\bfM(K_m)$.  Thanks to Stefan van Zwam (25 July 2007).]
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{1991a}{\dag\dag}  Biased graphs.  II.\  The three matroids.  \emph{J.\ Combin.\ Theory Ser.~B} 51 (1991), 46--72.  
\MR{1088626} (91m:05056).  \Zbl{763.05096}.						
\annot{$\bfF, \bfL, \bfL_0$} Basic theory of the bias [or better, ``frame''] matroid $G$ (now called $\bfF$) (\S2) and the lift and complete lift matroids, $\bfL$ and $\bfL_0$ (\S3), of a gain graph or biased graph.  Infinite graphs.  Matroids that are intermediate between the bias and lift matroids.  Several questions and conjectures.	
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{1991b}{\dag}  Orientation of signed graphs.  \emph{European J.\ Combin.}\ 12 (1991), 361--375.  
\MR{1120422} (93a:05065).  \Zbl{761.05095}.						
\annot{} Oriented signed graph = bidirected graph.  The oriented matroid of an oriented signed graph.  A ``cycle'' in a bidirected graph is a bias circuit (a balanced circle, or a handcuff with both circles negative) oriented to have no source or sink.  Cycles in $\Sigma$ are compared with those in its signed (i.e., derived) covering graph $\tilde \Sigma$.  The correspondences among acyclic orientations of $\Sigma$ and regions of the hyperplane arrangements of $\Sigma$ and $\tilde \Sigma$, and dually the faces of the acyclotope of $\Sigma$.  Thm.\ 4.1: the net degree vector $d(\tau)$ of an orientation $\tau$ belongs to the face of the acyclotope that is determined by the union of all cycles.  Cor.~5.3 (easy): a finite bidirected graph has a source or sink.
\par
\codes{\CSG:~\COri,~\CMatrd,~\CCov,~\CGeom}\codesctd{\CSGw:~\CInvar}%
\itemspace

\bsgitem{1992a}{}  Orientation embedding of signed graphs.  \emph{J.\ Graph Theory} 16 (1992), 399--422.  
\MR{1185006} (93i:05056).  \Zbl{778.05033}.						
\annot{} Positive circles preserve orientation, negative ones reverse it.  The minimal embedding surface of a one-point amalgamation of signed graphs.  The formula is almost additive.  \Cf\ \xref{Lins}{1982a}{Sostenes Lins}, \xrefd{1985a}{Sostenes Lins} for related work in different formalism.
	
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{1992b}{}  Strong Tutte functions of matroids and graphs.  \emph{Trans.\ Amer.\ Math.\ Soc.}\ 334 (1992), 317--347.  
\MR{1080738} (93a:05047).  \Zbl{781.05012}.						
\annot{} Suppose that a function of matroids with labelled points is defined that is multiplicative on direct sums and satisfies a Tutte--Grothendieck recurrence with coefficients (the ``parameters'') that depend on the element being deleted and contracted, but not on the particular minor from which it is deleted and contracted: specifically, $F(M) = a_e F(M \setminus e) + b_e F(M/e)$ if $e$ is not a loop or coloop in $M$.  
Thm.\ 2.1 completely characterizes such ``strong Tutte functions'' for each possible choice of parameters: there is one general type, defined by a rank generating polynomial $R_M(a,b; u,v)$ (the ``parametrized rank generating polynomial'') involving the parameters $a = (a_e)$, $b = (b_e)$ and the variables $u,v$, and there are a few special types that exist only for degenerate parameters.  All have a Tutte-style basis expansion; indeed, a function has such an expansion iff it is a strong Tutte function (Thms.~7.1, 7.2).  The Tutte expansion is a polynomial within each type.
If the points are colored and the parameters of a point depend only on the color, one has a multicolored matroid generalization of \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} Tutte polynomial of a sign-colored graph.  Kauffman's particular choices of parameters are shown to be related to matroid and color duality.  
\annotctd{} For a graph, ``parametrized dichromatic polynomial'' $Q_\Gamma = u^{c(\Gamma)} R_{\bfM(\Gamma)}$, where $\bfM =$ graphic matroid.  A ``portable strong Tutte function'' of graphs is multiplicative on disjoint unions, satisfies the parametrized Tutte--Grothendieck recurrence, and has value independent of the vertex set.  Thm.\ 10.1: Such a function either equals $Q_\Gamma$ or is one of two degenerate exceptions.  Prop.\ 11.1:  \xref{Kauffman's}{1989a}{Louis H.\ Kauffman} polynomial of a sign-colored graph equals $R_{\bfM(|\Sigma|),\sigma}(a,b; d,d)$ for connected $\Sigma$, where $a_+ = b_- = B$ and $a_- = b_+ = A$.  [\Cf\ \xref{Traldi}{1989a}{Lorenzo Traldi}.]
\annotctd{} [This paper differs from other generalizations of Kauffman's polynomial, by \xref{Przytycka and Przytycki}{1988a}{Teresa M.\ Przytycka \& Jozef H.\ Przytycki} and \xref{Traldi}{1989a}{Lorenzo Traldi} (and partially anticipated by \xref{Fortuin and Kasteleyn}{1972a}{C.M.\ Fortuin \& P.W.\ Kasteleyn}), who also develop the parametrized dichromatic polynomial of a graph, principally in that it characterizes \emph{all} strong Tutte functions; also in generalizing to matroids and in having little to say about knots.  \xref{Schw\"arzler and Welsh}{1993a}{W.\ Schwarzler \& D.J.A.\ Welsh} generalize to signed matroids (and characterize their strong Tutte functions) but not to arbitrary colors.  \xref{Bollob\'as and Riordan}{1999a}{Bela Bollobas \& Oliver Riordan} initiate the study of the underlying commutative algebra.]
\par
\codes{\CSc(\CMatrd),~\CSGc:~\CGen:~\CInvar,~\CDual,~\CKnot}%
\itemspace

\bsgitem{1993a}{\dag}  The projective-planar signed graphs.  \emph{Discrete Math.}\ 113 (1993), 223--247.  
\MR{1212880} (94d:05047).  \Zbl{779.05018}.						
\annot{$\bbP^2$} Characterized by six forbidden minors or eight forbidden topological subgraphs, all small.  A close analog of Kuratowski's theorem; the proof even has much of the spirit of the Dirac--Schuster proof of the latter, and all but one of the forbidden graphs are simply derived from the Kuratowski graphs.  [Paul Seymour showed me an alternative proof from Kuratowski's theorem that explains this; but it uses sophisticated results, as yet unpublished, of Robertson, Seymour, and Shih.]  [\Problem.  Find a proof using the signed covering graph and Kuratowski's theorem.]
\par
\codes{\CSG:~\CTop}%
\annotctd{} [Related: ``projective outer-planarity'' (POP): embeddable in the projective plane with all vertices on a common face.  I found most of the 40 or so forbidden topological subgraphs for POP of signed graphs (finding the rest would be routine); the proof is long and tedious and will not be published.  \Problem.  Find a reasonable proof.]
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{1994a}{}  Frame matroids and biased graphs.  \emph{European J.\ Combin.}\ 15 (1994), 303--307.  
\MR{1273951} (95a:05021).  \Zbl{797.05027}.						
\annot{} A simple matroidal characterization of the frame (or ``bias'') matroids of biased graphs.
\codes{\CGG:~\CMatrd}%
\itemspace

\bsgitem{1995a}{}  The signed chromatic number of the projective plane and Klein bottle and antipodal graph coloring.  \emph{J.\ Combin.\ Theory Ser.~B} 63 (1995), 136--145. 
\MR{1309361} (95j:05099).  \Zbl{822.05028}.					
\annot{} Introducing the signed Heawood problem: what is the largest signed, or zero-free signed, chromatic number of any signed graph that orientation embeds in the sphere with $h$ crosscaps?  Solved for $h = 1, 2$.
\par
\codes{\CSG:~\CTop,~\CCol}%
\itemspace

\bsgitem{1995b}{\dag\dag}  Biased graphs.  III.\  Chromatic and dichromatic invariants.  \emph{J.\ Combin.\ Theory Ser.\ B} 64 (1995), 17--88.
\MR{1328292} (96g:05139).  \Zbl{857.05088}.					
\annot{} Polynomials of gain and biased graphs: the fundamental object is a four-variable polynomial, the ``polychromial'' (``polychromatic polynomial''), that specializes to the chromatic, dichromatic, and Whitney-number polynomials.  The polynomials come in two flavors: unrestricted and balanced, depending on the edge sets that appear in their defining sums.  (They can be defined in the even greater abstraction of ``two-ideal graphs'', which clarifies the most basic properties.)  
\annotctd{} \S4: ``Gain-graph coloring''.  In $\Phi = (\Gamma,\phi,\fG)$, a ``zero-free $k$-coloring'' is a mapping $f: V \to [k]\times\fG$; it is ``proper'' if, when $e\ind vw$ is a link or loop and $f(v) = (i,g), f(w) = (i, h)$, then $h \neq g\phi(e;v,w)$.  A ``$k$-coloring'' is similar but the color set is enlarged by inclusion of a color $0$; propriety requires the additional restriction that $f(v)$ and $f(w)$ are not both $0$ (and $f(v) \neq 0$ if $v$ supports a half edge).
In particular, a ``group-coloring'' of $\Phi$ is a zero-free $1$-coloring (ignoring the irrelevant numerical part of the color).  A ``partial group-coloring'' is a group-coloring of an induced subgraph [which can only be proper if the uncolored vertices form a stable set].
The unrestricted and balanced chromatic polynomials count, respectively, unrestricted and zero-free proper $k$-colorings; the two Whitney-number polynomials count all colorings, proper and improper, by their improper edge sets.  
\annotctd{} \S5: ``The matroid connection''.  The various polynomials are, in essence, frame matroid invariants and closely related to corresponding lift matroid and extended lift matroid invariants.  
\annotctd{} Almost infinitely many identities, some of them (esp., the balanced expansion formulas in \S6) essential.  Innumerable examples worked in detail.  [The first half, to the middle of \S6, is fundamental.  The rest is more or less ornamental.  Most of the results are, intentionally, generalizations of properties of ordinary graphs.]
\codes{\CGG:~\CInvar,~\CMatrd,~\CCol}%
\itemspace

\bsgitem{1996a}{}  The order upper bound on parity embedding of a graph.  \emph{J.\ Combin.\ Theory Ser.\ B} 68 (1996), 149--160.  
\MR{1405709} (98f:05055).  \Zbl{856.05030}.					
\annot{} The smallest surface that holds $K_n$ with loops, if odd circles reverse orientation, even ones preserve it (this is parity embedding).  I.e., the demigenus $d(-K_n^{\circ})$.
\codes{\CPar:~\CTop}%
\itemspace

\bsgitem{1997a}{}  Is there a matroid theory of signed graph embedding?  \emph{Ars Combin.}\ 45 (1997), 129--141.
\MR{1447764} (97m:05084).  \Zbl{933.05067}.					
\codes{\CSG:~\CMatrd,~\CTop}%
\itemspace

\bsgitem{1997b}{}  The largest parity demigenus of a simple graph.  \emph{J.\ Combin.\ Theory Ser.~B} 70 (1997), 325--345.
\MR{1459877} (99e:05043).  \Zbl{970.37744}.					
\annot{} Like \xrefd{1996a}{Thomas Zaslavsky}, but without loops.  \Conjecture\ 1.  The minimal surface for parity embedding $K_n$ is sufficient for orientation embedding of any signed $K_n$.  \Conjectures\ 3--4.  The minimal surfaces of $\pm K_n^{\circ}$ and $\pm K_n$ are the smallest permitted by the lower bound obtained from Euler's polyhedral formula.
\codes{\CPar:~\CKG:~\CTop}%
\itemspace

\bsgitem{1997c}{}  Avoiding the identity.  Problem 10606, \emph{Amer.\ Math.\ Monthly} 104 (Aug.--Sept., 1997), no.\ 7, 664.
\annot{} Find an upper bound on $f(m) =$ largest $r$ such that any group of order $\geq r$ has $m$ elements such that no product of any subset, possibly with inverted elements, equals the identity.  Solution by \xref{Gagola}{1999a}{Stephen M.\ Gagola}.  
\annotctd{} [The solution implies that $(*)$ $f_1(m) \leq \lceil 2^{m-1}(m-1)!\surd e \rceil$, where $f_1(m) =$ smallest $r$ such that every group of order $\geq r$ is a possible gain group for every contrabalanced gain graph of cyclomatic number $m$.  \Problem\ 1.  Find a good upper bound on $f_1$.  $(*)$ is probably weak.   \Problem\ 2.  Find a good lower bound.  \Problem\ 3.  Estimate $f_1$ asymptotically.]
\codes{\Cgg:~\Cbic}%
\itemspace

\bsgitem{1998a}{}  Signed analogs of bipartite graphs.  \emph{Discrete Math.}\ 179 (1998), 205--216.
\MR{1489083} (2000b:05067).  \Zbl{980.06737}.					
\annot{} Basically, they are the antibalanced and bipartite signed graphs; but the exact description depends on the characterization one chooses for biparticity: whether it is evenness of circles, closed walks, face boundaries in surface embeddings, etc.  Characterization by chromatic number leads to a slightly more different list of analogs.
\codes{\CSG:~\CStr,~\CTop}%
\itemspace

\bsgitem{1998b}{}  A mathematical bibliography of signed and gain graphs and allied areas.  \emph{Electronic J.\ Combin.}, Dynamic Surveys in Combinatorics (1998), No.~DS8.  \\ 
URL \url{http://www.combinatorics.org/issue/view/Surveys}
\\ \MR{1744869} (2000m:05001a).  \Zbl{898.05001}.					
\annot{} Complete and annotated---or as nearly so as I can make it (est.\ 50\
\annotctd{} Published edns.:  
Edn.\ 6a (Edition 6, Revision a), 20 July 1998 (iv + 124 pp.).  
Edn.\ 7, 22--26 Sept.\ 1999 (vi + 151 pp.).  
Edn.\ 8,  8 Sept.\ 2012 (vi + 341 pp.).
Edn.\ 9, 21 Dec.\ 2018 (vi + 518 pp.).
\par
\codes{\CSG,~\COri,~\CGG,~\CGN,~\CSD,~\CVS,~\CTG,~\dots,~\CChem,~\CPhys,~\CBiol,~\CPsS,~\CAppl}%
\itemspace

\bsgitem{1998c}{}  Glossary of signed and gain graphs and allied areas.  \emph{Electronic J.\ Combin.}, Dynamic Surveys in Combinatorics (1998), No.~DS9. \\  
URL \url{http://www.combinatorics.org/issue/view/Surveys}
\\ \MR{1744870} (2000m:05001b).  \Zbl{898.05002}.					
\annot{} A complete (or so it is intended) terminological dictionary of signed, gain, and biased graphs and related topics; including necessary special terminology from ordinary graph theory and mathematical interpretations of the special terminology of applications.
\annotctd{} Published edns.:  
21 July 1998 (25 pp.).  
Second ed.\ 18 September 1998 (41 pp.).
\par
\codes{\CSG,~\COri,~\CGG,~\CGN,~\CSD,~\CVS,~\CTG,~\dots,~\CChem,~\CPhys,~\CPsS,~\CAppl}%
\itemspace

\bsgitem{2001a}{}  Supersolvable frame-matroid and graphic-lift lattices.  \emph{European J.\ Combin.}\ 22 (2001), 119--133.
\MR{1808091} (2001k:05051).  \Zbl{966.05013}.					
\annot{} Biased graphs whose bias and lift matroids are supersolvable are characterized by a form of simplicial vertex ordering---with a few exceptions.  As preliminary results, modular copoints are characterized [but incompletely in the bias-matroid case, as observed by \xref{Koban}{2004a}{Lori Koban}].  
\S4: ``Examples'':  4a: ``Group expansions and biased expansions''; 4b: ``Near-Dowling and Dowling lift lattices''; 4c: ``An extension of Edelman and Reiner's theorem'' to general gain groups (see \xref{Edelman and Reiner}{1994a}{Paul H.\ Edelman \& Victor Reiner}); 4d: ``Bicircular matroids''.  
[Written in 1992 and long delayed.  Correction in \xref{Koban}{2004a}{Lori Koban}.  Independently, \xref{Yoon}{1997a}{Young-Jin Yoon} incorrectly attempted the case of $\bfF(\Sigma)$.  \xref{Jiang and Yu}{2004a}{Guangfeng Jiang \& Jianming Yu} rediscovered the case of $\bfF(K_n,\sigma)$.]
\dated{rev.\ 20 Oct 2012}
\codes{\CGG,~\CSG:~\CMatrd,~\CGeom}%
\itemspace

\bsgitem{2001b}{}  The largest demigenus of a bipartite signed graph.  \emph{Discrete Math.}\ 232 (2001), 189--193.
\MR{1823637} (2001m:05100).  \Zbl{982.05041}.					
\annot{} The smallest surface for orientation embedding of $\pm K_{r,s}$.  
\codes{\CSG:~\CTop}%
\itemspace

\bsgitem{2002a}{}  Perpendicular dissections of space.  \emph{Discrete Comput.\ Geom.}\ 27 (2002), 303--351.
\MR{1921558} (2003i:52026).  \Zbl{1001.52011}.					
\annot{} Given an additive real gain graph $\Phi$ on $n$ vertices and $n$ reference points $Q_i$ in $\bbE^d$, use $\Phi$ to specify perpendicular hyperplanes to each of the lines $Q_iQ_j$ by means of the ``Pythagorean coordinate'' along $Q_iQ_j$.  For generic points, the number of regions is computable based on the fact that the generic hyperplane intersection lattice is $\Latb \Phi$.  Modifications of Pythagorean coordinates give intersection lattice $\Latb(\|\Phi\|,\eset)$ or a slightly more complex variant, still for generic reference points.
\par
\codes{\CGG:~\CGeom,~\CMatrd,~\CInvar}%
\itemspace

\bsgitem{2003a}{}  Faces of a hyperplane arrangement enumerated by ideal dimension, with application to plane, plaids, and Shi.  \emph{Geom.\ Dedicata} 98 (2003), 63--80.
\MR{1988424} (2004f:52025).  \Zbl{1041.52021}.					
\annot{} \S6, ``Affinographic arrangements'': hyperplane arrangements that represent the extended lift matroid $\bfL_0(\Phi)$ where $\Phi$ is an additive real gain graph.  Examples: the weakly-composed-partition, extended Shi, and extended Linial arrangements.  The faces are counted in terms of dimension and dimension of the infinite part.  \xref{Ehrenborg}{2019a}{Richard Ehrenborg} has more explicit formulas for Shi.
\codes{\CGG:~\Cmatrd,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2003b}{\dag\dag}  Biased graphs  IV:  Geometrical realizations.  \emph{J.\ Combin.\ Theory Ser.\ B} 89 (2003), no.\ 2, 231--297.
\MR{2017726} (2005b:05057).  \Zbl{1031.05034}.					
\annot{} \S\S2--4: Various ways in which to represent the bias and lift matroids of a gain or biased graph over a skew field $F$.  Bias matroid: canonical vector and hyperplanar representations (generalizing those of a graph) based on a gain group $\subseteq F^\times$, Menel{\ae}an and Cevian representations (generalizations of theorems of Menelaus and Ceva), switching vs.\ change of ideal hyperplane, equational logic.  Lift matroid: canonical vector and hyperplanar representations (the latter generalizing the Shi and Linial arrangements among others) based on a gain group $\subseteq F^+$, orthographic representation (an affine variation on canonical representation), Pythagorean representation (\xrefd{2002a}{Thomas Zaslavsky}).  Both: effect of switching, nonunique gain-group embedding.  
\S5: Effect of Whitney operations, separating vertex.  
\S6: Matroids characterized by restricted general position.  
\S7, ``Thick graphs'': A partial unique-representation theorem for biased graphs with sufficient edge multiplicity.  
\S8: The 7 biased $K_4$'s.
\par
\codes{\CGG:~\CMatrd,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2006a}{}  Quasigroup associativity and biased expansion graphs.  \emph{Electronic Res.\ Announc.\ Amer.\ Math.\ Soc.}\ 12 (2006), 13--18.
\MR{2200950} (2006i:20081).  \Zbl{1113.05044}.					
\annot{} Summary of \xrefd{2012a}{Thomas Zaslavsky}.
\codes{\CGG:~\CStr}%
\itemspace

\bsgitem{2007a}{}  Biased graphs.  VII.\  Contrabalance and antivoltages.  \emph{J.\ Combin.\ Theory Ser.\ B} 97 (2007), no.\ 6, 1019--1040.
\MR{2354716} (2008h:05025).  \Zbl{1125.05048}.					
\annot{} Contrabalanced graphs, whose gains are called antivoltages.  Emphasis on the existence of antivoltages in groups $\fG = \bbZ_\mu$, $\bbZ$, and $\bbZ_p^k$ for application to canonical representation of the contrabalanced bias and lift matroids.  The number of such antivoltages is a polynomial function of the group order or (for $\bbZ$) the bound on circle gains.  [\Cf\ \xref{Sivaraman}{2014a}{Vaidy Sivaraman}, \xref{Chun, Moss, Slilaty, and Zhou}{2016a}{Deborah Chun, Tyler Moss, Daniel Slilaty, \& Xiangqian Zhou}, and especially \xref{Sivaraman--Slilaty}{2019a}{Vaidy Sivaraman \& Daniel Slilaty}.]
\dated{rev 11 Jun 2019}
\par
\codes{\CGG:~\CMatrd,~\Cbic,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{2009a}{}  Totally frustrated states in the chromatic theory of gain graphs.  \emph{European J.\ Combinatorics} 30 (2009), 133--156.  
\MR{2460223} (2009k:05100).  \Zbl{1125.05048}.					
\annot{} Given a set $Q$ of ``spins'', a state is $s: V \to Q$.  The gain group $\fG$ acts on the spin set.  In a permutation gain graph $\Phi$ with gain group $\fG$, edge $e\ind vw$ is ``satisfied'' if $s(w) = s(v)\phi(e)$, otherwise ``frustrated''.  A totally frustrated state (every edge is frustrated) generalizes a proper coloring.  Enumerative theory, including deletion/contraction, a monodromy formula for the number of totally frustrated states, and a multivariate chromatic polynomial.  An abstract partition function in the edge algebra.
\codes{\CGG:~\CCol:~\CGen:~\CInvar,~\CMatrd}%
\itemspace

\bsgitem{2010a}{}  Six signed Petersen graphs.  
In:  \emph{International Conference on Recent Trends in Graph Theory and Combinatorics (ICRTBC-2010)} (Cochin, 2010) [Summaries], pp.\ 75--76.  
Dept.\ of Math., Cochin Univ.\ of Science and Technology, 2010.
\annot{} Extended abstract of \xrefd{2012b}{Thomas Zaslavsky} [but not entirely correct].  There are six ways to sign the Petersen graph $P$ up to switching isomorphism.  Their frustration indices, automorphism and switching automorphism groups, chromatic numbers, and clusterability indices. 
\dated{30 Aug, 26 Dec 2010}
\codes{\CSG:~\CFr,~\CAut,~\CCol,~\CClu}%
\itemspace

\bsgitem{2010b}{}  Matrices in the theory of signed simple graphs.  In:  B.D.\ Acharya, G.O.H.\ Katona, and J.\ Nesetril, eds., \emph{Advances in Discrete Mathematics and Applications: Mysore, 2008} (Proc.\ Int.\ Conf.\ Discrete Math.\ 2008, ICDM-2008, Mysore, India, 2008), pp.\ 207--229.  Ramanujan Math.\ Soc.\ Lect.\ Notes Ser., No.\ 13.  Ramanujan Mathematical Soc., Mysore, India, 2010.
\MR{2766941} (2012d:05017).  \Zbl{1231.05120}.  \arXiv{1303.3083}.			
\annot{} The adjacency, incidence, and Laplacian matrices, along with the adjacency matrices of line graphs.  Balance, vertex degrees, eigenvalues, line graphs, strong regularity, etc.  A survey, emphasizing work of \hyperlink{J.J.\ Seidel}{Seidel}, \hyperlink{G.R.\ Vijayakumar}{Vijayakumar}, and \hyperlink{Thomas Zaslavsky}{Zaslavsky}.
\annotctd{} \xref{Abelson and Rosenberg's}{1958a}{Robert P.\ Abelson \& Milton J.\ Rosenberg} adjacency matrix is mentioned.
\par
\codes{\CSG:~\CAdj,~\CLap,~\CEig,~\CInc,~\CLG:~\CExp}%
\itemspace

\bsgitem{2012a}{}  Associativity in multiary quasigroups: The way of biased expansions.  \emph{Aequationes Math.}\ 83 (2012), no.\ 1, 1--66.
\MR{2885498}.  \Zbl{1235.05059}.  \arXiv{0411268}.					
\annot{} An $n$-ary quasigroup $(\fQ,f)$ is essentially equivalent, up to isotopy, to a biased expansion $m\cdot C_{n+1}$.  Factorizations of $f$ appear as chords in a maximal extension of $m\cdot C_{n+1}$.  Thm.:  A biased expansion of a 3-connected graph (of order $\geq 4$) is a group expansion.  Cor.:  If $n\geq3$ and the factorization graph of $(\fQ,f)$ is 3-connected, $(\fQ,f)$ is isotopic to an iterated group.  Thm.:  For a biased expansion of a 2-connected graph of order $\geq 4$, if all minors of order 4 are group expansions, so is the whole expansion.  Cor.:  If in $(\fQ,f)$ ($n\geq 3$) all ternary residual quasigroups are iterated group isotopes, so is $(\fQ,f)$.  Cor.:  $(\fQ,f)$ is an iterated group isotope if $\#\fQ=3$.  
\annotctd{} Other results: complete structural decomposition of nongroup biased expansions, or equivalently partially reducible multiary quasigroups, in terms of groups and, respectively, irreducible expansions or irreducible multiary quasigroups.
\annotctd{} The matroids of maximal nongroup biased expansions are the nearest generalization of \xref{Dowling's}{1973b}{Thomas A.\ Dowling} geometries.
\codes{\CGG:~\CStr,~\CM}%
\itemspace

\bsgitem{2012b}{}  Six signed Petersen graphs, and their automorphisms.  Recent Trends in Graph Theory and Combinatorics (Cochin, 2010).  \emph{Discrete Math.}\ 312 (2012), no.\ 9, 1558--1583.  
\MR{2899889}.  \Zbl{1239.05086}.  \arXiv{1303.3347}.				
\annot{} There are six ways to sign the Petersen graph $P$ up to switching isomorphism.  The frustration indices and numbers, automorphism and switching automorphism groups (in extensive detail), chromatic numbers, and clusterability indices of them and their negatives.  All but automorphisms and clusterability are switching invariant, thus are solved for all signed Petersens.  
\dated{26 Dec 2010}
\codes{\CSG:~\CFr,~\CAut,~\CCol,~\CClu}%
\itemspace

\bsgitem{2012c}{}  Signed graphs and geometry.  Set Valuations, Signed Graphs and Geometry (IWSSG-2011, Int.\ Workshop, Mananthavady, Kerala, 2011).  \emph{J.\ Combin.\ Inform.\ System Sci.}\ 37 (2012), no.\ 2-4, 95--143.
\Zbl{1301.05162}.  \arXiv{1303.2770}.  
\par
\codes{\CSG:~\CBal,~\CFr,~\CGeom,~\CInc,~\CAdj,~\CEig,~\CMatrd,~\CExp}%
\itemspace

\bsgitem{2016a}{}  Consistency in the naturally vertex-signed line graph of a signed graph.  \emph{Bull.\ Malaysian Math.\ Sci.\ Soc.}\ 39 (2016), no.\ 1, suppl., S307--S314.
\MR{3509082}.  \Zbl{1339.05174}.  \arXiv{1404.1652}.				
\annot{} Constructive answers to the question about consistency of the vertex-signed line graph treated in \xref{Acharya, Acharya, and Sinha}{2009a}{Belmannu Devadas Acharya, Mukti Acharya, \& Deepa Sinha} and \xref{Slilaty and Zaslavsky}{2015a}{Daniel C.\ Slilaty \& Thomas Zaslavsky}.
\dated{3 Nov 2013} 
\par
\codes{\CSG:~\CLG:~\CVS:~\CBal}%
\itemspace

\bsgitem{2017a}{}  Negative circles in signed graphs: A problem collection.  Int.\ Conf.\ Current Trends in Graph Theory and Computation (CTGTC-2016, New Delhi).  \emph{Electronic Notes Discrete Math.}\ 63 (2017), 41--47.
\MR{3754789}.  \Zbl{1383.05152}.  \arXiv{1610.04691}.				
\annot{} Short version of \xrefd{2018a}{Thomas Zaslavsky}.
\codes{\CSG:~\CCircles,~\CBal,~\CFr}%
\itemspace

\bsgitem{2018a}{}  Negative (and positive) circles in signed graphs:  A problem collection.  \emph{AKCE Int.\ J.\ Graphs Combin.}\ 15 (2018), no.\ 1, 31--48.
\MR{3803228}.  \Zbl{1390.05085}.  \arXiv{1701.07963}.				
\annot{} A collection of mostly open questions, some with solutions, about circles of specified sign and their interactions: e.g., detection, uniqueness, intersection, packing, covering.
\dated{15 Jan 2018}
\par
\codes{\CSG:~\CCircles,~\CBal,~\CFr}%
\itemspace

								
\bsgitem{20xxa}{}  Line graphs of signed graphs and digraphs.  In preparation.
\annot{$\Lambda$} Line graphs of signed graphs are, fundamentally, (bidirected) line graphs of bidirected graphs, $\Lambda(\Beta)$.  Then the line graph $\Lambda(\Sigma)$ of a signed graph is a polar graph, i.e., a switching class of bidirected graphs; the line graph of a polar graph is a signed graph; and the line graph of a sign-biased graph, i.e., of a switching class of signed graphs, is a sign-biased graph, $[\Lambda[\Sigma]]$.  
In particular, the line graph of an antibalanced switching class is an antibalanced switching class.  (Partly for this reason, ordinary graphs should usually be regarded as antibalanced, i.e., all negative, in line graph theory.)  
Since a digraph is an oriented all-positive signed graph, its line graph is a bidirected graph whose positive part is the Harary--Norman line digraph.  
Among the line graphs of signed graphs, some reduce by cancellation of parallel but oppositely signed edges to all-negative graphs; these are precisely Hoffman's generalized line graphs of ordinary graphs, a fact which explains their line-graph-like behavior.  [See \xrefd{1984c}{Thomas Zaslavsky}, \xrefd{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}.  Also see \xref{Zelinka}{1976a}{Bohdan Zelinka} \etal]
\dated{$\leq$ 1998; rev}
\annotctd{} [Attempts at a completely descriptive line graph $\Lambda(\vec\Gamma)$ of a digraph have been \xref{Muracchini and Ghirlanda}{1965a}{Luigi Muracchini \& Anna Maria Ghirlanda} and \xref{Hemminger and Klerlein}{1979a}{Robert L.\ Hemminger \& Joseph B.\ Klerlein}.]  
\dated{1999}
\annotctd{} [The geometry of line graphs and signed graphs has been developed by \hyperlink{G.R.\ Vijayakumar}{Vijayakumar} \etal\ (\qv).]
\dated{1999 \emph{et seq}}
\annotctd{} The competition graph of a digraph $\vec\Gamma$ is the extraverted part of $\Lambda(\vec\Gamma)$.
\dated{16 Aug 2016}
\codes{\CSG:~\CLG:~\COri,~\CInc,~\CEig(\CLG),~\CSw}%
\itemspace

\bsgitem{20xxb}{}  Whitney numbers of partial Dowling lattices.  \emph{Trans.\ Combin.}\ (to appear).
\annot{} The Whitney numbers of the first kind of group expansions $\fG\Gamma^{(X)}$, also biased expansions $\gamma{\cdot}\Gamma^{(X)}$ ($\gamma \in \bbZ_{>0}$), are polynomial functions of $\#\fG$, or $\gamma$.  Very detailed.
\dated{6 Sept 2022} 
\codes{\CGG:~\CMatrd:~\CInvar}%
\itemspace


\bsgitem{20xxd}{}  Geometric lattices of structured partitions:  I.\ Gain-graphic matroids and group-valued partitions.  Manuscript, 1985 \etseq
\codes{\CGG:~\CMatrd,~\CInvar,~\Ccol}%
\itemspace

\bsgitem{20xxe}{}  Geometric lattices of structured partitions:  II.\ Lattices of group-valued partitions based on graphs and sets.  Manuscript, 1985 \etseq
\codes{\CGG:~\CMatrd,~\CInvar,~\Ccol}%
\itemspace


\bsgitem{20xxg}{}  Universal and topological gains for biased graphs.  In preparation.
\codes{\CGG:~\CTop}%
\itemspace

\bsgitem{20xxi}{}  Big flats in a box.  In preparation.  
\annot{} The naive approach to characteristic polynomials via lattice point counting (in characteristic $0$) and M\"obius inversion (as in \xref{Blass--Sagan}{1998a}{Andreas Blass \& Bruce Sagan}) can only work when one expects it to.  (This is a theorem.)
\par
\codes{\CGG:~\CGeom,~\CMatrd,~\CInvar,~\Ccol}%
\itemspace

\bsgitem{20xxj}{}  Biased graphs.  V.\  Group and biased expansions.  In preparation.  
\par
\codes{\CGG:~\CMatrd,~\CGeom,~\CInvar}%
\itemspace

\bsgitem{20xxk}{}  Petersen signed graphs.  In preparation.  
\annot{} There are 6 signatures of the Petersen graph $P$, up to switching isomorphism; \cf\ \xrefd{2012b}{Thomas Zaslavsky}.  For four of them ($+P$, $-P$, $P_I =$ the antipodal quotient of the icosahedral graph, $P_1$ with one negative edge), many facets are examined closely.  Other examples:  $\pm P$, $\tilde(P,\sigma) =$ the double cover of a signed $P$.
	
\codes{\CSG:~\CSw,~\CBal,~\CFr,~\CClu,~\CCov,~\CTop,~\CCol,~\CMatrd:~\CExp}%
\itemspace

\bsgitem{20xxm}{}  Biased graphs.  VIII.\  A cornucopia of examples.  In preparation.  
\annot{} Numerous types of examples of biased graphs, many having particular theory of their own, e.g., Hamiltonian bias.  
	
\codes{\CGG:~\CMatrd,~\CGeom}%
\itemspace



\author{Morris Zelditch, Jr.}						%ZELDITCH
\authorspace
\authref   See \hyperlink{Joseph Berger}{J.~Berger}.
\itemspace

\author{Bohdan Zelinka} 						%ZELINKA*
\authorspace
\authref   See also \hyperlink{Robert L.\ Hemminger}{R.L.~Hemminger}.
\authrefspace

\bsgitem{1973a}{}  Polare und polarisierte Graphen.  
In:  \emph{XVIII.\ Int.\ Wiss.\ Kolloqu.}\ (Ilmenau, 1973), Vol.~2, Vortragsreihe ``Theorie der Graphen und Netzwerke'', pp.\ 27--28.  Technische Hochschule, Ilmenau, 1973.  
\Zbl{272.05102}.  
\annot{} See \xrefd{1976a}{Bohdan Zelinka}.  [This appears to be a very brief abstract of a lecture.]
\par
\codes{\Csg:~\COri,~\Csw}%
\itemspace


\bsgitem{1974a}{}  Polar graphs and railway traffic.  \emph{Aplikace Mat.}\ 19 (1974), 169--176.  
\MR{0347346} (49 \#12066).  \Zbl{283.05116}.					
\annot{} See \xrefd{1976a}{Bohdan Zelinka} for definitions.  Railway tracks and switches modeled by edges and vertices of a polar graph.  Forming its derived graph (see \xrefd{1976d}{Bohdan Zelinka}), thence a digraph obtained therefrom by splitting vertices into two copies and adjusting arcs, the time for a train to go from one segment to another is found by a shortest path calculation in the digraph.  A similar method is used to solve the problem for several trains. 
\par
\codes{\Csg:~\COri,~\Csw:~\CLG:~\CAppl}%
\itemspace

\bsgitem{1976a}{}  Isomorphisms of polar and polarized graphs.  \emph{Czechoslovak Math.\ J.}\ 26(101) (1976), 339--351.  
\MR{0498288} (58 \#16429).  \Zbl{341.05121}.					
\annot{} Basic definitions (\xref{Z\'{\i}tek}{1972a}{Frantisek Zitek}):  ``Polarized graph'' $\Beta$ = bidirected graph (with no negative loops and no parallel edges sharing the same bidirection).  ``Polar graph'' $P \cong$ switching class of bidirected graphs (that is, we forget which direction at a vertex is in and which is out---here called ``north'' and ``south'' poles---but we remember that they are different).
\annotctd{} Thms.~1--6.  Elementary results about automorphisms, including finding the automorphism groups of the ``complete polarized'' and polar graphs.  (The ``complete polarized graph'' has every possible bidirected link and positive loop, without repetition.)  Thm.\ 7:  With small exceptions, any (ordinary) graph can be made polar as, say, $P$ so that $\Aut P$ is trivial.
\annotctd{} Thms.~8--10.  Analogs of Whitney's theorem that the line graph almost always determines the graph.  The ``pole graph'' $\Beta^*$ of $\Beta$ or $[\Beta]$:  Split each vertex into an ``in'' copy and an ``out'' copy and connect the edges appropriately.  [Generalizes splitting a digraph into a bipartite graph.  It appears to be a ``twisted'' signed double covering graph.]  Thm.\ 8.  The pole graph is determined, with two exceptions, by the edge relation $e \sim_1 f$ if both enter or both leave a common vertex.  (A trivial consequence of Whitney's theorem.)  Thm.\ 9.  A polar graph $[\Beta]$ with enough edges going in and out at each vertex is determined by the edge relation $e \sim_2 f$ if one enters and the other exits a common vertex.  (Examples show that too few edges going in and out leave $[\Beta]$ undetermined.)  Thm.\ 10.  Knowing $\sim_1$, $\sim_2$, and which edges are parallel with the same sign, and if no component of the simplified underlying graph of $\Beta$ is one of twelve forbidden graphs, then $[\Beta]$ is determined.  [\Problem\ 1.  Improve Thm.\ 10 to a complete characterization of the bidirected graphs that are reconstructible from their line graphs (which are to be taken as bidirected; see \xref{Zaslavsky}{2010b}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}).  In connection with this, see results on characterizing line graphs of bidirected (or signed) graphs by \xref{Vijayakumar}{1987a}{G.R.\ Vijayakumar}.  \Problem\ 2.  It would be interesting to improve Thm.\ 9.]
\codes{\Csg:~\COri,~\Csw:~\CAut,~\Clg}%
\itemspace

\bsgitem{1976b}{}  Analoga of Menger's theorem for polar and polarized graphs.  \emph{Czechoslovak Math.\ J.}\ 26(101) (1976), 352--360.  
\MR{0498289} (58 \#16430).  \Zbl{341.05122}.					
\annot{} See \xrefd{1976a}{Bohdan Zelinka} for basic definitions.  Here is the framework of the 8 theorems.  Given a bidirected or polar graph, $\Beta$ or $P$, vertices $a$ and $b$, and a type $X$ of walk, let $s_X$ [$s'_X$] = the fewest vertices [edges] whose deletion eliminates all $(a,b)$ walks of type $X$, and let $d_X$ [$d'_X$] = maximum number of suitably pairwise internally vertex-disjoint [or, suitably pairwise edge-disjoint] walks of type $X$ from $a$ to $b$.  [My notation.]  By ``suitably'' I mean that a common internal vertex or edge is allowed in $P$ (but not in $\Beta$) if it is used oppositely by the two walks using it.  (See the paper for details.)  Thms.~1--4$_1$ (there are two Theorems 4) concern all-positive and all-introverted walks in a bidirected (``polarized'') graph, and are simply the vertex and edge Menger theorems applied to the positive and introverted subgraphs.  Thms.~4$_2$--7 concern polar graphs and have the form $s_X \leq d_X \leq 2s_X$ [$s'_X \leq d'_X \leq 2s'_X$], which is best possible.  Thms.~4$_2$--5 concern type ``heteropolar'' (equivalently, directed walks in a bidirected graph).  The proofs depend on Menger's theorems in the double covering graph of the polar graph.  [Since this has 2 vertices for each 1 in the polar graph, the range of $d_X$ [$d'_X$] is explained.]  Thms.~6--7 concern type ``homopolar'' (i.e., antidirected walks).  The proofs employ the pole graph (see \xrefd{1976a}{Bohdan Zelinka}).
\codes{\Csg:~\COri,~\Csw:~\CPaths}%
\itemspace

\bsgitem{1976c}{}  Eulerian polar graphs.  \emph{Czechoslovak Math.\ J.}\ 26(101) (1976), 361--364.  
\MR{0505895} (58 \#21869).  \Zbl{341.05123}.					
\annot{} See \xrefd{1976a}{Bohdan Zelinka} for basic definitions.  An Eulerian trail in a bidirected graph is a directed trail containing every edge.  [Equivalently, a heteropolar trail that contains all the edges in the corresponding polar graph.]  It is closed if the endpoints coincide and the trail enters at one end and departs at  the other.  The fewest directed trails needed to cover a connected bidirected graph is $\frac{1}{2}$ the total of the absolute differences between in-degrees and out-degrees at all vertices, or $1$ if in-degree = out-degree everywhere.
\codes{\Csg:~\COri,~\Csw:~\CPaths}%
\itemspace

\bsgitem{1976d}{}  Self-derived polar graphs.  \emph{Czechoslovak Math.\ J.}\ 26(101) (1976), 365--370.  
\MR{0498290} (58 \#16431).  \Zbl{341.05124}.					
\annot{} See \xrefd{1976a}{Bohdan Zelinka} for basic definitions.  The ``derived graph'' of a bidirected graph [this is equivalent to the author's terminology] is essentially the positive part of the bidirected line graph.  The theorem can be restated, somewhat simplified:  A finite connected bidirected graph $\Beta$ is isomorphic to its derived graph iff $\Beta$ is balanced and contains exactly one circle.
\par
\codes{\Csg:~\COri,~\Csw:~\CLG}%
\itemspace

\bsgitem{1976e}{}  Groups and polar graphs.  \emph{\v{C}asopis P\v{e}st.\ Mat.}\ 101 (1976), 2--6.  
\MR{0505793} (58 \#21790).  \Zbl{319.05118}.					
\annot{} See \xrefd{1976a}{Bohdan Zelinka} for basic definitions.  A polar graph $PG(\fG, A)$ of a group and a subset $A$ is defined.  [It is the Cayley digraph.]  In bidirected language: a (bi)directed graph is ``homogeneous'' if it has automorphisms that are transitive on vertices, both preserving and reversing the orientations of edges, and that induce an arbitrary permutation of the incoming edges at any given vertex, and similarly for outgoing edges.  It is shown that the Cayley digraph $PG(\fG, A)$, where $\fG$ is a group and $A$ is a set of generators, is homogeneous if $A$ is both arbitrarily permutable and invertible by $\Aut\fG$.  [Bidirection---i.e., the polarity---seems to play no part here.]
\codes{\Csg:~\COri,~\Csw:~\CAut}%
\itemspace

\bsgitem{1982a}{}  On double covers of graphs.  \emph{Math.\ Slovaca} 32 (1982), 49--54.  
\MR{0648219} (83b:05072).  \Zbl{483.05057}.					
\annot{} Is a simple graph $\Gamma$ a double cover of some signing of a simple graph?  An elementary answer in terms of involutions of $\Gamma$.  Further: if there are two such involutions $\alpha_0, \alpha_1$ that commute, then $\Gamma/\alpha_i$ has involution induced by $\alpha_{1-i}$, so is a double cover of $\Gamma/\langle\alpha_0,\alpha_1\rangle$, which is not necessarily simple.  [No properties of particular interest for signed covering are treated.]
\par
\codes{\Csg:~\CCov}%
\itemspace

\bsgitem{1983a}{}  Double covers and logics of graphs.  \emph{Czechoslovak Math.\ J.}\ 33(108) (1983), 354--360.  
\MR{0718920} (85k:05098a).  \Zbl{537.05070}.					
\annot{} The double covers here are those of all-negative simple graphs (hence are bipartite).  Some properties of these double covers are proved, then connections with a certain lattice (the ``logic'') of a graph.
\codes{\Cpar:~\CCov:~\CAut}%
\itemspace

\bsgitem{1983b}{}   Double covers and logics of graphs II.  \emph{Math.\ Slovaca} 33 (1983), 329--334.  
\MR{0720501} (85k:05098b).  \Zbl{524.05058}.					
\annot{} The second half of \xrefd{1983a}{Bohdan Zelinka}.
\codes{\Cpar:~\CCov:~\CAut}%
\itemspace

\bsgitem{1988a}{}  A remark on signed posets and signed graphs.  \emph{Czechoslovak Math.\ J.}\ 38(113) (1988), 673--676.  
\MR{0962910} (90g:05157).  \Zbl{679.05067} (\qv).					
\annot{} \xref{Harary and Sagan}{1984a}{Frank Harary \& Bruce Sagan} asked: which signed graphs have the form $S(P)$ for some poset $P$?  Zelinka gives a rather complicated answer for all-negative signed graphs, which has interesting corollaries.  For instance, Cor.~3:  If $S(P)$ is all negative, and $P$ has $\hat 0$ or $\hat 1$, then $S(P)$ is a tree. 
\par
\codes{\CSG,~\CS}%
\itemspace

\author{Hans-Olov Zetterstr\"om} 					%ZETTERSTROM
\authorx{Hans-Olov Zetterstrom}
\authorspace
\authref   See \xref{Harary, Lindstr\"om, and Zetterstr\"om}{1982a}{Frank Harary, Bernt Lindstrom, \& Hans-Olov Zetterstrom}.
\itemspace

\author{Ahmed A.\ Zewail}						%ZEWAIL
\authorspace
\authref  See \hyperlink{Basak Guler}{B.~Guler}.
\itemspace

\author{Hongyuan Zha}						%ZHA HY
\authorspace
\authref  See \hyperlink{Shuang Hong Yang}{S.H.~Yang}.
\itemspace

\authorx{Mingqing Zhai}						%ZHAI MQ
\author{Mingqing Zhai, Ruifang Liu, \& Jinlong Shu}
\authorspace
\bsgitem{2011a}{}  An edge-grafting theorem on Laplacian spectra of graphs and its application.  \emph{Linear Multilinear Algebra} 59 (2011), no.\ 3, 303--315.
\MR{2774085} (2012c:\-05202).  \Zbl{1226.05175}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shidong Zhai}						%ZHAI SD
\authorspace
\bsgitem{2016a}{}  Modulus synchronization in a network of nonlinear systems with antagonistic interactions and switching topologies.  \emph{Commun.\ Nonlinear Sci.\ Numerical Simulation} 33 (2016), 184--193.
\MR{3417178} (no rev).  
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\author{Shidong Zhai, Tao Huang, Guoqiang Luo, Xin Wang, \& Jun Ma}
\authorspace
\bsgitem{2022a}{}  Pinning bipartite synchronization for coupled nonlinear systems with antagonistic interactions and time delay.  \emph{Physica A} 593 (2022), art.\ 126954, 13 pp.
\MR{4376488} (no rev).  
\codes{\CSG:~\CDyn,~\CBal}%
\itemspace

\author{Shidong Zhai \& Qingdu Li}
\authorspace
\bsgitem{2016a}{}  Pinning bipartite synchronization for coupled nonlinear systems with antagonistic interactions and switching topologies.  \emph{Systems Control Lett.}\ 94 (2016), 127--132.
\MR{3530606}.  \Zbl{1344.93011}.  
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\bsgitem{2016b}{}  Bipartite synchronization in a network of nonlinear systems: A contraction approach.  \emph{J.\ Franklin Inst.}\ 353 (2016), no.\ 17, 4602--4619.
\MR{3562993} (n rev).  \Zbl{1349.93026}.  
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\author{Shidong Zhai, Min Xiao, \& Qingdu Li}
\authorspace
\bsgitem{2017a}{}  Synchronization analysis of coupled identical linear systems with antagonistic interactions and time-varying topologies.  \emph{Neurocomputing} 244 (2017), 53--62.
\codes{\CSG:~\CBal:~\CDyn:~\CAlgor}%
\itemspace

\author{Shidong Zhai \& Wei Xing Zheng}
\authorspace
\bsgitem{2021a}{}  Dynamic behavior for social networks with state-dependent susceptibility and antagonistic interactions.  \emph{Automatica} 129 (2021), art.\ 109652, 14 pp.
\MR{4255370}.  \Zbl{1478.91151}.  
\codes{\CSG:~\CDyn}%
\itemspace

\bsgitem{2021a}{}   Generalized dynamics in social networks with antagonistic interactions.  \emph{Automatica} 129 (2021), art.\ 109652, 14 pp.
\Zbl{1478.91151}.  \arXiv{1801.08713}.  
\codes{\CSG:~\CDyn}%
\itemspace

\author{H.\ Zhan}							%ZHAN H
\authorspace
\authref  See \hyperlink{G.\ Coutinho}{G.~Coutinho}.
\itemspace

\author{Bingyan Zhang}						%ZHANG BY
\authorspace
\authref  See \hyperlink{Yuanping Zhang}{Y.P.~Zhang}.
\itemspace

\author{Chao Zhan}							%ZHANG C
\authorspace
\authref  See \hyperlink{Chao Wen}{C.~Wen}.
\itemspace

\author{Cun-Quan Zhang}						%ZHANG CQ
\authorspace
\authref  See also \hyperlink{Jian Cheng}{J.-A.~Cheng}, \hyperlink{Matt DeVos}{M.~DeVos}, \hyperlink{You Lu}{Y.~Lu}, \hyperlink{Xingqin Qi}{X.Q.~Qi}, \hyperlink{Rui Xu}{R.~Xu}, \hyperlink{Xiao Wang}{X.~Wang}, and \hyperlink{Yezhou Wu}{Y.-Z.~Wu}.
\authrefspace

\bsgitem{1993a}{}  Even-cycle decomposition.  Problem 4.2, p.\ 681, in Nathaniel Dean, Open problems.  In: Neil Robertson and Paul Seymour, eds., \emph{Graph Structure Theory} (Proc., Seattle, 1991), pp.\ 677--688.  
Contemp.\ Math., Vol.\ 147.  Amer.\ Math.\ Soc., Providence, R.I., 1993.
\MR{1224693} (93m:05004) (book).  \Zbl{789.05080}.  
\annot{} Conj.\ 12 is a sufficient condition for $-\Gamma$ to decompose into balanced circles.  [\Problem.  Solve the obvious generalization to signed graphs.  Is that easier because minors exist?]
\dated{11 Jun 2012} 
\codes{\Csg:~\Cpar:~\CStr}%
\itemspace

\bsgitem{1994a}{}  On even circuit decompositions of Eulerian graphs.  \emph{J.\ Graph Theory} 18 (1994), no.\ 1, 51--57.
\MR{1248485} (95e:05080).  \Zbl{804.05053}.  
\annot{} On decomposing $E(-\Gamma)$ into positive circles.  Thm.:  It is possible if $\Gamma$ is 2-connected and Eulerian and has no $K_5$ minor.  [\Problem.  The same, for any signed graph.  Is a $-K_5$ minor an obstruction?  Also see \xref{R.~Rizzi}{2001a}{Romeo Rizzi} and \xref{K.~Markstr\"om}{2012a}{Klas Markstrom}.]
\dated{13 Aug 2013} 
\codes{\CPar:~\CStr:~\CCircles}%
\itemspace

\authorx{De Long Zhang}						%ZHANG DL
\author{De Long Zhang \& Shang Wang Tan}
\authorspace
\bsgitem{2003a}{}  On the strongly regular graphs and the Seidel switching.  (In Chinese.)  \emph{Math.\ Appl.\ (Wuhan)}\ 16 (2003), no.\ 2, 145--148.
\MR{1979481} (no rev).  \Zbl{1030.05076} (no rev).  
\codes{\Ctg,~\CSw}%
\itemspace

\author{Dong Zhang}							%ZHANG D
\authorspace
\authref  See \hyperlink{Aida Abiad}{A.~Abiad} and \hyperlink{Raffaella Mulas}{R.~Mulas}.
\itemspace

\author{Fuji Zhang}							%ZHANG FJ
\authorspace
\authref  See \hyperlink{Xian'an Jin}{X.A.~Jin} and \hyperlink{Weiling Yang}{W.~Yang}.
\itemspace

\authorx{Guang-Jun Zhang}						%ZHANG GJ
\author{Guang-Jun Zhang \& Xiao-Dong Zhang}
\authorspace
\bsgitem{2011a}{}  The $p$-Laplacian spectral radius of weighted trees with a degree sequence and a weight set.  \emph{Electronic J.\ Linear Algebra}\ 22 (2011), 267--276.
\MR{2788647} (2012i:05051).  \Zbl{1227.05190}.  
\annot{} Generalizes \xref{Biyiko{\u g}lu, Hellmuth, \& Leydold}{2009a}{Turker Biyikoglu, Marc Hellmuth, \& Josef Leydold} to positively edge-weighted graphs.
[\Problem.  Generalize to signed graphs.]
\dated{21 Jan 2012} 
\codes{\CPar:~\CEig:~\CGen}%
\itemspace

\author{Hongwei Zhang}						%ZHANG HW
\authorspace
\authref  See also \hyperlink{Huidi Jiang}{H.-D.~Jiang}, \hyperlink{Ye Jiang}{Y.~Jiang}, and \hyperlink{Zhenyu Sun}{Z.-Y.~Sun}.
\authrefspace

\bsgitem{2014a}{}  Dynamic output feedback control of multi-agent systems over signed graphs.  In:  \emph{Proceedings of the 33rd Chinese Control Conference} (Nanjing, 2014), pp.\ 1327--1332.  IEEE, 2014.
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\bsgitem{2015a}{}  Output feedback bipartite consensus and consensus of linear multi-agent systems.  In:  \emph{2015 IEEE 54th Annual Conference on Decision and Control} (CDC, Osaka, 2015), pp.\ 1731--1735.  IEEE, 2015.
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\author{Hongwei Zhang \& Jie Chen}
\authorspace
\bsgitem{2014a}{}  Bipartite consensus of general linear multi-agent systems.  In:  \emph{2014 American Control Conference} (ACC, Portland, Ore., 2014), pp.\ 808--812.  IEEE, 2014.
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\bsgitem{2014b}{}  Bipartite consensus of linear multi-agent systems over signed digraphs: An output feedback control approach.  In:  \emph{Proceedings of the 19th World Congress, The International Federation of Automatic Control} (Cape Town, 2014), pp.\ 4681--4686.  IEEE, 2014.
\codes{\CSD:~\CBal:~\CDyn}%
\itemspace

\bsgitem{2017a}{}  Bipartite consensus of multi-agent systems over signed graphs:  State feedback and output feedback control approaches.  \emph{Int.\ J.\ Robust Nonlinear Control} 27 (2017), no.\ 1, 3--14.
\MR{3589235}.  \Zbl{1353.93011}.  
\codes{\CSG:~\CBal:~\CDyn}%
\itemspace

\author{Jianbin Zhang}						%ZHANG JB
\authorspace
\authref  See \hyperlink{Xueliang Li}{X.L.~Li}.
\itemspace

\author{Jianghua Zhang}						%ZHANG JH
\authorspace
\authref  See \hyperlink{Guangfeng Jiang}{G.~Jiang}.
\itemspace

\authorx{Jie Zhang}							%ZHANG J
\author{Jie Zhang \& Xiao-Dong Zhang}	
\authorspace
\bsgitem{2013a}{}  The signless Laplacian coefficients and incidence energy of bicyclic graphs.  \emph{Linear Algebra Appl.}\ 439 (2013), 3859--3869.
\MR{3133462}.  \Zbl{1282.05163}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jing Zhang}							%ZHANG J
\authorspace
\authref  See \hyperlink{Li Su}{L.~Su}.
\itemspace

\authorx{Jingming Zhang}						%ZHANG JM
\author{Jingming Zhang \& Jiming Guo}
\authorspace
\bsgitem{2012a}{}  The signless Laplacian spectral radius of tricyclic graphs with $k$ pendant vertices.  \emph{J.\ Math.\ Res.\ Appl.}\ 32 (2012), no.\ 3, 281--287.
\MR{2985366}.  \Zbl{1265.05430}.  
\annot{} Given $n$ and $k$, the unique tricyclic graph with largest max eigenvalue of $\Lap(-\Gamma)$.  [\Question.  Is it the same for arbitrarily signed tricyclic graphs?]
\dated{7 Jul 2022} 
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jing-Ming Zhang, Ting-Zhu Huang, \& Ji-Ming Guo}
\authorspace
\bsgitem{2014a}{}  On the signless Laplacian spectral radius of bicyclic graphs with perfect matchings.  \emph{Sci.\ World J.}\ 2014 (2014), art.\ 374501, 6 pp.
\annot{} The graph maximizing $\lambda_1(\Lap(-\Gamma))$.
\dated{20 Jan 2015} 
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Jing-Yue Zhang}						%ZHANG JY
\authorspace
\authref  See \hyperlink{Ling Zhang}{L.~Zhang}.
\itemspace

\author{Junxia Zhang}						%ZHANG JX
\authorspace
\authref  See \hyperlink{Xiangyu Ren}{X.-Y.~Ren}.
\itemspace

\author{Kuan Zhang}							%ZHANG K
\authorspace
\authref  See \hyperlink{David Lo}{D.~Lo}.
\itemspace

\author{Li Zhang}							%ZHANG L
\authorspace
\authref  See also \hyperlink{Shuchao Li}{S.C.~Li}.
\authrefspace

\author{Li Zhang, You Lu, Rong Luo, Dong Ye, \& Shenggui Zhang}
\authorspace
\bsgitem{2020a}{}  Edge coloring of signed graphs.  \emph{Discrete Appl.\ Math.}\ 282 (2020), 234--242.
\MR{4107751}.  \Zbl{1442.05081}.  
\annot{} Partial proof of Vizing theorem $\chi' \leq \Delta+1$ for signed simple graphs.  [Independent full proof by \xref{Behr}{2020a}{Richard Behr}.]  $\chi' \leq \Delta$ for some planar graphs.  
Dictionary:  ``signed line graph'' = $-\Lambda(\Sigma)$ (\cf\ \xref{Zaslavsky}{20xxa}{Thomas Zaslavsky} \xrefd{2010b}{Thomas Zaslavsky}, \xrefd{2012c}{Thomas Zaslavsky}); ``$\chi'_\pm$'' = $\chi'$ (chromatic index).
\dated{2 Jul 2020}
\par
\codes{\CSG:~\CECol:~\CLG,~\COri}%
\itemspace

\author{Li Jun Zhang}						%ZHANG LJ
\authorspace
\authref  See \hyperlink{Xiao Hui Hao}{X.H.~Hao}.
\itemspace

\authorx{Ling Zhang}							%ZHANG L
\author{Ling Zhang, Ting-Zhu Huang, Zhongshan Li, \& Jing-Yue Zhang}
\authorspace
\bsgitem{2013a}{}  Several spectrally arbitrary ray patterns.  \emph{Linear Multilinear Algebra} 61 (2013), no.\ 4, 543--564.
\MR{3005636}.  \Zbl{1273.05093}.  
\codes{\CGG:~\CQM}%
\itemspace

\author{Long Zhang}							%ZHANG L
\authorspace
\authref  See \hyperlink{Jianshe Wu}{J.-S.~Wu}.
\itemspace

\author{Minjie Zhang}						%ZHANG MJ
\authorspace
\authref  See also \hyperlink{Chen Chen}{C.~Chen} and \hyperlink{Shuchao Li}{S.C.~Li}.
\authrefspace

\author{Minjie Zhang \& Shuchao Li}
\authorspace
\bsgitem{2015a}{}  On the signless Laplacian spectra of $k$-trees.  \emph{Linear Algebra Appl.}\ 467 (2015), 136--148.
\MR{3284805}.  \Zbl{1304.05098}.  
Corrigendum.  \emph{Ibid.}\ 485 (2015), 527--530.
\MR{3394162} (no rev).  \Zbl{1409.05134}.  \arXiv{1507.02536}.			
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Peikang Zhang}						%ZHANG PK
\author{Peikang Zhang, Baofeng Wu, \& Changxiang He}
\authorspace
\bsgitem{2021a}{}  Balancedness and spectra of signed graphs obtained by $\dot{H}$-join operation.  \emph{Comput.\ Appl.\ Math.}\ 40 (2021), no.\ 3, art.\ 90, 11 pp.
\MR{4233561}.  \Zbl{1476.05068}.  
\codes{\CSG:~\CBal,~\CEig:~\CAdj,~\CLap}%
\itemspace

\author{Ping Zhang}							%ZHANG P
\authorspace
\bsgitem{1997a}{}  The characteristic polynomials of subarrangements of Coxeter arrangements.  \emph{Discrete Math.}\ 177 (1997), 245--248.
\MR{1483448} (98i:52016).  \Zbl{980.06614}.					
\annot{} \xref{Blass and Sagan's}{1998a}{Andreas Blass \& Bruce Sagan} geometrical form of signed-graph coloring is used to calculate (I) characteristic polynomials of several versions of $k$-equal subspace arrangements (these are the main results) and (II) [also in \xrefd{2000a}{Ping Zhang}] the chromatic polynomials (in geometrical guise) of ordinary graphs extending $K_n$ by one vertex, signed graphs extending $\pm K_n^\circ$ by one vertex, and $\pm K_n$ with any number of negative loops adjoined.
\par
\codes{\Csg:~\CInvar,~\CGeom,~\Ccol}%
\itemspace

\bsgitem{2000a}{}  The characteristic polynomials of interpolations between Coxeter arrangements.  \emph{J.\ Combin.\ Math.\ Combin.\ Comput.}\ 34 (2000), 109--117.
\MR{1772789} (2001b:\-05220).  \Zbl{968.32017}.						
\annot{} Uses signed-graph coloring (in geometrical guise) to evaluate the chromatic polynomials (in geometrical guise) of all signed graphs interpolating between (1) $+K_n$ and $+K_{n+1}$ [i.e., ordinary graphs extending a complete graph by one vertex]; (2) $\pm K_{n-1}^\circ$ and $\pm K_{n}^\circ$; (3) $\pm K_n$ and $\pm K_n^\circ$ [known already by several methods, including this one]; (4a) $\pm K_{n-1}$ and $\pm K_{n-1} \cup +K_n$; (4b) $\pm K_{n-1} \cup +K_n$ and $\pm K_n$; and certain signed graphs interpolating (by adding negative edges one vertex at a time, or working down and removing them one vertex at a time) between (5) $+K_n$ and $\pm K_n^\circ$; (6) $+K_n$ and $\pm K_n$.  In cases (1)--(3) the chromatic polynomial depends only on how many edges are added [which is obvious from the coloring procedure, if it were not disguised by geometry].
\par
\codes{\Csg:~\CInvar,~\Ccol,~\CGeom}%
\itemspace

\author{Shenggui Zhang}						%ZHANG SG
\authorspace
\authref  See \hyperlink{Dan Hu}{D.~Hu}, \hyperlink{Xiao Wang}{X.~Wang}, and \hyperlink{Li Zhang}{L.~Zhang}.
\itemspace

\author{Shengping Zhang}						%ZHANG SP
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Shengtong Zhang}						%ZHANG ST
\authorspace
\authref  See \hyperlink{Zilin Jiang}{Z.-L.~Jiang}.
\itemspace

\author{Shunzhe Zhang}						%ZHANG SZ
\authorspace
\authref  See \hyperlink{Chao Wen}{C.~Wen}.
\itemspace

\author{Siqi Zhang}							%ZHANG SQ
\authorspace
\authref  See \hyperlink{Shuchao Li}{S.-C.~Li} and \hyperlink{Guangfu Wang}{G.-F.~Wang}.
\itemspace

\authorx{W.J.\ Zhang}						%ZHANG WJ
\author{W.J.\ Zhang \& A.M.\ Yu}
\authorspace
\bsgitem{2017a}{}  On the rank of weighted graphs.  \emph{Linear Multilinear Algebra} 65 (2017), no.\ 3, 635--652.
\MR{3589626}.  \Zbl{1356.05067}.  
\annot{} Characterizes signed graphs with adjacency rank 4, separately for bipartite and non-bipartite graphs.
\dated{27 Dec 2017} 
\codes{\CSG,~\CWG:~\CAdj}%
\itemspace

\author{Wanli Zhang}						%ZHANG WL
\authorspace
\authref  See \hyperlink{Shiju Yang}{S.-J.~Yang}.
\itemspace

\author{Xiao-Dong Zhang}						%ZHANG XD
\authorspace
\authref  See also \hyperlink{Ming-Zhu Chen}{M.-Z.~Chen}, \hyperlink{Ya-Hong Chen}{Y.H.~Chen}, \hyperlink{Bian He}{B.A.~He}, \hyperlink{Yuan Hong}{Y.~Hong}, \hyperlink{Guang-Jun Zhang}{G.J.~Zhang}, and \hyperlink{Jie Zhang}{J.~Zhang}.
\authrefspace

\bsgitem{2004a}{}  Two sharp upper bounds for the Laplacian eigenvalues.  \emph{Linear Algebra Appl.}\ 376 (2004), 207--213.
\MR{2015534} (2004m:05173).  \Zbl{1037.05032}.  
\annot{} \S4, Remark 2:  The main results extend to signed graphs (``mixed graphs").
\dated{23 Mar 2009}
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\bsgitem{2004b}{}  Bipartite graphs with small third Laplacian eigenvalue.  \emph{Discrete Math.}\ 278 (2004), no.\ 1-3, 241--253.
\MR{2035402} (2004m:05172).  \Zbl{1033.05073}.					
\annot{} 
	
[\Problem.  Explain in terms of signed graphs, generalizing to $\Lap(-\Gamma)$.]
\par
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\bsgitem{2009a}{}  The signless Laplacian spectral radius of graphs with given degree sequences.  \emph{Discrete Appl.\ Math.}\ 157 (2009), no.\ 13, 2928--2937.
\MR{2537494} (2011a:05210).  \Zbl{1213.05153}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xiao-Dong Zhang \& Jiong-Sheng Li}
\authorspace
\bsgitem{2002a}{}  The Laplacian spectrum of a mixed graph.  \emph{Linear Algebra Appl.}\ 353 (2002), 11--20.
\MR{1918746} (2003d:05138).  \Zbl{1003.05073}.					
\annot{} Spectrum and spectral radius of the Laplacian matrix of a signed simple graph.    [For this topic, orientation is irrelevant so the results apply to all signed simple graphs, although they are stated for oriented signed graphs in the guise of mixed graphs.]
Dictionary: ``mixed graph'' = bidirected graph where all negative edges are extraverted;  ``quasibipartite'' = balanced;  ``line graph'' = $-\Lambda(\Sigma)$ (the negative of the line graph of $\Sigma$).
\dated{23 Mar 2009}
\codes{\Csg:~\CEig:~\CLap,~\CLG}%
\itemspace

\author{Xiao-Dong Zhang \& Rong Luo}
\authorspace
\bsgitem{2003a}{}  The Laplacian eigenvalues of mixed graphs.  \emph{Linear Algebra Appl.}\ 362 (2003), 109--119.
\MR{1955457} (2003m:05128).  \Zbl{1017.05078}.					
\annot{} $\Sigma$ is a signed simple graph.  $\lambda_{\max}(\Lap(\Sigma)) \leq$ max edge degree + 2 (same as \xref{Hou, Li, and Pan}{2003a}{Yaoping Hou, Jiongsheng Li, \& Yongliang Pan}, Thm.\ 3.5(1)).  Also, other bounds on $\lambda_{\max}$.  
Thm.\ 2.5:  The second smallest eigenvalue is $\leq \kappa(|\Sigma|)$ if there exists a minimum separating vertex set $X$ such that $\Sigma \setminus X$ is balanced.
Dictionary:  See \xref{X.D.~Zhang and Li}{2002a}{Xiao-Dong Zhang \& Jiong-Sheng Li}.
\dated{23 Mar 2009}
\par
\codes{\Csg:~\CEig}%
\itemspace

\bsgitem{2006a}{}  Non-bipartite graphs with third largest Laplacian eigenvalue less than three.  \emph{Acta Math.\ Sinica (Engl.\ Ser.)}\ 22 (2006), no.\ 3, 917--934. 
\MR{2220184} (2007b:\-05141).  \Zbl{1102.05040}.					
\annot{} [\Problem.  Generalize to connected, unbalanced signed graphs.]
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xiao-Peng Zhang}						%ZHANG XP
\authorspace
\authref  See \hyperlink{Xiao-Jun Tian}{X.-J.~Tian}.
\itemspace

\author{Xin Zhang}							%ZHANG X
\authorspace
\authref  See \hyperlink{Weichan Liu}{W.-C.~Liu}.
\itemspace

\author{Y.\ Zhang}							%ZHANG Y
\authorspace
\authref  See \hyperlink{Bhaskar DasGupta}{B.~DasGupta}.
\itemspace

\author{Yan X Zhang}						%ZHANG YX
\authorspace
\authref  See \hyperlink{Takayuki Hibi}{T.~Hibi}.
\itemspace

\authorx{Yanan Zhang}						%ZHANG YN
\author{Yanan Zhang \& Haiyan Chen}
\authorspace
\bsgitem{2020a}{}  The average Laplacian polynomial of a graph.  \emph{Discrete Appl.\ Math.}\ 283 (2020), 737--743.
\MR{4114935}.  \Zbl{1442.05105}.  
\codes{\CSG:~\CLap}%
\itemspace

\author{Yanguo Zhang}						%ZHANG YG
\authorspace
\authref  See \hyperlink{Zhiyun Cheng}{Z.-Y.~Cheng}.
\itemspace

\author{Yingying Zhang}						%ZHANG YY
\authorspace
\authref  See \hyperlink{Xiaolin Chen}{X.-L.~Chen}.
\itemspace

\author{Yuanping Zhang}						%ZHANG YP
\authorspace
\authref  See also \hyperlink{Xiaogang Liu}{X.G.~Liu}.
\authrefspace

\author{Yuan-ping Zhang \& Xiao-gang Liu}
\authorspace
\bsgitem{2012a}{}  Some eigenvalue relations of signed graphs.  \emph{J.\ Guangzhou Univ.\ (Nat.\ Sci.\ Ed.)}\ 11 (2012), no.\ 1, 83--86.
\Zbl{1265.05291}.  
\codes{\CSG:~\CAdj,~\CLap:~\CEig}%
\itemspace

\author{Yuanping Zhang, Xiaogang Liu, Bingyan Zhang, \& Xuerong Yong}
\authorspace
\bsgitem{2009a}{}  The lollipop graph is determined by its $Q$-spectrum.  \emph{Discrete Math.}\ 309 (2009), 3364--3369.
\MR{2526754} (2010g:05239).  \Zbl{1182.05084}.					
\annot{} 
	
[Correction by \xref{Hamidzade and Kiani}{2010a}{Hasti Hamidzade \& Dariush Kiani}.]
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Zhang Zhang}						%ZHANG Z
\authorspace
\authref  See \hyperlink{Matt DeVos}{M.~DeVos} and \hyperlink{Lingsheng Shi}{L.-S.~Shi}.
\itemspace

\author{Zhi-Li Zhang}						%ZHANG ZL
\authorspace
\authref  See \hyperlink{Yanhua Li}{Y.-H.~Li}.
\itemspace

\author{Guopeng Zhao}						%ZHAO GP
\authorspace
\authref  See \hyperlink{Ke Li}{K.~Li} and \hyperlink{Ligong Wang}{L.G.~Wang}.
\itemspace

\author{Qin Zhao}							%ZHAO Q
\authorspace
\authref  See \hyperlink{Qin Wen}{Q.~Wen}.
\itemspace

\author{Xuehua Zhao}						%ZHAO XH
\authorspace
\authref  See \hyperlink{Bo Yang}{B.~Yang}.
\itemspace

\author{Yanhua Zhao}						%ZHAO YH
\authorspace
\authref  See \hyperlink{Hangtian Guo}{H.T.~Guo}.
\itemspace

\author{Yufei Zhao}							%ZHAO YF
\authorspace
\authref  See \hyperlink{Zilin Jiang}{Z.-L.~Jiang}.
\itemspace

\authorx{Q.\ Zheng}							%ZHENG Q
\author{Q.\ Zheng, D.B.\ Skillicorn, \& O.\ Walther}
\authorspace
\bsgitem{2015a}{}  Signed directed social network analysis applied to group conflict.  In:  Peng Cui \etal, eds., \emph{IEEE International Conference on Data Mining Workshop} (ICDMW 2015, Atlantic City, N.J.), pp.\ 1007--1014.  IEEE, 2015.
\par
\codes{\CSD:~\CAdj,~\CPsS}%
\itemspace

\author{Qing Yu Zheng}						%ZHENG QY
\authorspace
\authref  See also \hyperlink{Hong Shan Du}{H.S.~Du}.
\authrefspace

\author{Qing Yu Zheng \& Qing Jun Ren}
\authorspace
\bsgitem{2001a}{}  Quasi-Laplacian characteristic polynomials of graphs [or, The quasi-Laplacian characteristic polynomial].  (In Chinese.)  \emph{Qufu Shifan Daxue Xuebao Ziran Kexue Ban} [\emph{J.\ Qufu Normal Univ., Nat.\ Sci.}] 27 (2001), no.\ 4, 40--43.
\MR{1873005} (no rev).  \Zbl{1013.05050}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Shanshan Zheng}						%ZHENG SS
\authorspace
\authref  See \hyperlink{Hongyan Cai}{H.-Y.~Cai}.
\itemspace

\author{Wei Xing Zhen}						%ZHENG WX
\authorspace
\authref  See \hyperlink{Shidong Zhai}{S.-D.~Zhai}.
\itemspace

\author{Zhehui Zhong}						%ZHONG ZH
\authorspace
\authref  See \hyperlink{Gbemisola Adejumo}{G.~Adejumo}.
\itemspace

\author{Bo Zhou}							%ZHOU B
\authorspace
\authref  See also \hyperlink{Modjtaba Ghorbani}{M.~Ghorbani}, \hyperlink{Ivan Gutman}{I.~Gutman}, \hyperlink{Gui-Xian Tian}{G.X.~Tian}, \hyperlink{Shilin Wang}{S.L.~ Wang}, and \hyperlink{Rundan Xing}{R.D.~Xing}.
\authrefspace

\bsgitem{2008a}{}  A connection between ordinary and Laplacian spectra of bipartite graphs.  \emph{Linear Multilinear Algebra} 56 (2008), no.\ 3, 305--310.
\MR{2384657} (2008k:05139).  \Zbl{1163.05043}.					
\annot{} $\Gamma$ is bipartite.  Subdivide every edge of $\Gamma$ once.  The eigenvalues are the square roots of the Laplacian eigenvalues of $\Gamma$, and $0$.  
[\Problem\ 1.  Generalize to all graphs and the ``signless Laplacian''.  \Problem\ 2.  Generalize to signed graphs via negative subdivision (every positive edge is subdivided into two negative edges).]
\dated{28 Aug 2011} 
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\bsgitem{2010a}{}  Signless Laplacian spectral radius and Hamiltonicity.  \emph{Linear Algebra Appl.}\ 432 (2010), no.\ 2-3, 566--570. 
\MR{2577702} (2011c:05219).  \Zbl{1188.05086}.					
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Bo Zhou \& Aleksandar Ili\'c}
\authorx{Bo Zhou \& Aleksandar Ilic}
\authorspace
\bsgitem{2010a}{}  On the sum of powers of Laplacian eigenvalues of bipartite graphs.  \emph{Czechoslovak Math.\ J.}\ 60(135) (2010), no.\ 4, 1161--1169.
\MR{2738977} (2011m:05192).  \Zbl{1224.05333}.  
\annot{} See \xref{Tian, Huang, and Zhou}{2009a}{Gui-Xian Tian, Ting-Zhu Huang, \& Bo Zhou}.  Lower and upper bounds in terms of sums of squared degrees.  Thus, bounds on incidence energy \etal
\dated{24 Jan 2012} 
\codes{\Cpar:~\Cbal:~\CLap:~\CEig}%
\itemspace

\author{Chuixiang Zhou}						%ZHOU CX
\authorspace
\authref  See \hyperlink{Mengmeng Xie}{M.-M.~Xie}.
\itemspace

\author{Guanglu Zhou}						%ZHOU GL
\authorspace
\authref  See \hyperlink{Dong Li}{D.~Li}.
\itemspace

\author{Haijun Zhou}							%ZHOU HJ
\authorspace
\bsgitem{2005a}{}  Long-range frustration in a spin-glass model of the vertex-cover problem.  \emph{Phys.\ Rev.\ Lett.}\ 94 (2005), no.\ 21, art.\ 217203, 4 pp.
\annot{} ``Long-range frustration'' means correlation between spins ($\pm1$) of vertices at considerable distance, within the same ``state'' (a configuration domain separated by energy barriers).  [This should be generalized to signed graphs.]
	
\dated{12 Sept 2010}
\codes{\Csg:~\CPhys:~\CState}%
\itemspace

\author{Harrison H.\ Zhou}						%ZHOU HH
\authorspace
\authref  See \hyperlink{Derek Feng}{D.~Feng}.
\itemspace

\author{Hou Chun Zhou}						%ZHOU HC
\authorspace
\authref  See \hyperlink{Hong Shan Du}{H.S.~Du}.
\itemspace

\author{Jiang Zhou}							%ZHOU J
\authorspace
\authref  See \hyperlink{Changjiang Bu}{C.J.~Bu}.
\itemspace

\authorx{Jun Zhou}							%ZHOU J
\author{Jun Zhou, Yi-Zheng Fan \& Yi Wang}
\authorspace
\authref  See also \hyperlink{Yi-Zheng Fan}{Y.Z.~Fan}.
\authrefspace

\bsgitem{2007a}{}  On the second largest eigenvalue of a mixed graph.  \emph{Discuss.\ Math.\ Graph Theory} 27 (2007), no.\ 2, 373--384.
\MR{2355728} (2008j:05223).  \Zbl{1134.05067}.					
\annot{} Sufficient condition for $\lambda_2(\Lap(\Sigma)) \geq d_2$, the second largest eigenvalue and degree, respectively.
Dictionary:  See \xref{X.D.~Zhang and Li}{2002a}{Xiao-Dong Zhang \& Jiong-Sheng Li}.
\dated{28 Oct 2011}
\codes{\Csg:~\CLap:~\CEig}%
\itemspace

\author{Min Zhou}							%ZHOU M
\authorspace
\authref  See \hyperlink{Chang-Xiang He}{C.-X.~He}.
\itemspace

\author{Qi Zhou}							%ZHOU Q
\authorspace
\authref  See \hyperlink{Feng Xu}{F.~Xu}.
\itemspace

\author{Qiannan Zhou}						%ZHOU QN
\authorspace
\authref  See \hyperlink{Yong Lu}{Y.~Lu}.
\itemspace

\author{Sanming Zhou}						%ZHOU SM
\authorspace
\authref  See \hyperlink{Fan Yang}{F.~Yang}.
\itemspace

\author{Xiangqian Zhou}						%ZHOU X
\authorspace
\authref  See \hyperlink{Yu Qing Chen}{Y.Q.~Chen}, \hyperlink{Deborah Chun}{D.~Chun}, \hyperlink{Hongxun Qin}{H.~Qin}, \hyperlink{Jakayla R.\ Robbins}{J.~Robbins}, and \hyperlink{Daniel Slilaty \& Xiangqian Zhou}{D.~Slilaty}.
\itemspace

\author{Yue Zhou}							%ZHOU Y
\authorspace
\authref  See \hyperlink{Francesco Belardo}{F.~Belardo}.
\itemspace

\authorx{Yulong Zhou}						%ZHOU YL
\author{Yulong Zhou \& Jiangping Hu}
\authorspace
\bsgitem{2013a}{}  Event-based bipartite consensus on signed networks.  \emph{Proceedings of the 2013 IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems} (Nanjing, 2013), pp.\ 326--330.  IEEE, 2013.
\arXiv{1310.8377}.  
\codes{\CSG:~\CLap:~\CEig}%
\itemspace

\author{Zhi-Hua Zhou}						%ZHOU ZH
\authorspace
\authref  See \hyperlink{Leting Wu}{L.T.~Wu}.
\itemspace

\author{Bao-Xuan Zhu}						%ZHU BX
\authorspace
\bsgitem{2010a}{}  On the signless Laplacian spectral radius of graphs with cut vertices.  \emph{Linear Algebra Appl.}\ 433 (2010), no.\ 5, 928--933.
\MR{2658643} (2011e:05163).  \Zbl{1215.05108}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\bsgitem{2011a}{}  Bounds on the eigenvalues of graphs with cut vertices or edges.  \emph{Linear Algebra Appl.}\ 434 (2011), 2030--2041.
\MR{2780398} (2012g:05150).  \Zbl{1216.05082}.  
\par
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Guangjun Zhu}						%ZHU GJ
\authorspace
\authref  See \hyperlink{Yi-Huang Shen}{Y.-H.~Shen}.
\itemspace

\author{Jianming Zhu}						%ZHU JM
\authorspace
\bsgitem{2020a}{}  Unicyclic signed graphs with the first $\lfloor(n+1)/2\rfloor$ largest energies.  \emph{Discrete Appl.\ Math.}\  285 (2020), 350--363.
\MR{4114942}.  \Zbl{1447.05100}.  
\codes{\CSG:~\CAdj:~\CEig}%
\itemspace

\author{Li Zhu}							%ZHU L
\authorspace
\authref  See \hyperlink{Rong Huang}{R.~Huang}.
\itemspace

\author{Min Zhu}							%ZHU M
\authorspace
\authref  See also \hyperlink{Fenglei Tian}{F.L.~Tian}.
\authrefspace

\author{Min Zhu, Yihao Guo, Fenglei Tian, \& Lingfei Lu}
\authorspace
\bsgitem{2013a}{}  The least eigenvalues of the signless Laplacian of non-bipartite graphs with fixed diameter.  \emph{Int.\ J.\ Math.\ Engineering Sci.}\ 2 (2013), no.\ 6, 12 pp.
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace


\author{Qunxiong Zhu}						%ZHU QX
\authorspace
\authref  See \hyperlink{Di Peng}{D.~Peng}.
\itemspace

\author{Shirley Zhu}							%ZHU S
\authorspace
\authref  See \hyperlink{Lucas J.\ Rusnak}{L.J.~Rusnak}.
\itemspace

\authorx{Xiao Xin Zhu}						%ZHU XX
\author{Xiao Xin Zhu, Zhi Ren Sun, \& Chun Zheng Cao}
\authorspace
\bsgitem{2008a}{}  A bound for the quasi-Laplacian spectral radius of connected graphs [or, A bound on quasi-Laplacian spectral radius of connected graphs].  (In Chinese.)  \emph{J.\ Nanjing Normal Univ.\ Nat.\ Sci.\ Ed.}\ 31 (2008), no.\ 2, 27--30.
\MR{2444764} (2009e:05196).  \Zbl{1174.05448}.  
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{Xuding Zhu}							%ZHU XD
\authorspace
\authref  See also \hyperlink{Yiting Jiang}{Y.-T.~Jiang}, \hyperlink{Ligang Jin}{L.-G.~Jin}, \hyperlink{Ringi Kim}{R.-G.~Kim}, \hyperlink{Zhishi Pan}{Z.-S.~Pan}, \hyperlink{Hao Qi}{H.~Qi}, and \hyperlink{Andre Raspaud}{A.~Raspaud}.
\authrefspace

\bsgitem{2015a}{}  Circular flow number of highly edge connected signed graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 112 (2015), 93--103.
\MR{3323036}.  \Zbl{1310.05112}.  \arXiv{1211.3179}.				
\codes{\CSG:~\CInvar}%
\itemspace

\bsgitem{2017a}{}  A note on two conjectures that strengthen the four colour theorem.  Manuscript, 2017.
\arXiv{1711.02848}.  
\annot{} Thm.:  If every signed planar graph is 4-colorable (\cf\ \xref{M\'a\v{c}ajov\'a, Raspaud, and \v{S}koviera}{2016a}{Edita Macajova, Andre Raspaud, \& Martin Skoviera}), then every planar graph is 2-list-bipartite-colorable.
\dated{10 Nov 2017} 
\codes{\CSG:~\CCol}%
\itemspace

\bsgitem{2020b}{}  A refinement of choosability of graphs.  \emph{J.\ Combin.\ Theory Ser.\ B} 141 (2020), 143--164.
\MR{4046290}.  \Zbl{1430.05046}.  \arXiv{1811.08587}.				
\annot{} See \xref{Jin, Wong, and Zhu}{20xxa}{Ligang Jin, Tsai-Lien Wong, \& Xuding Zhu}.
\codes{\CSG.~\CGG:~\CCol}%
\itemspace

\author{Zhongxun Zhu}						%ZHU ZX
\authorspace
\bsgitem{2011a}{}  The signless Laplacian spectral radius of bicyclic graphs with a given girth.  \emph{Electronic J.\ Linear Algebra} 22 (2011), 378--388.
\MR{2788652} (2012i:05178).  \Zbl{1227.05191}.					
\codes{\Cpar:~\CLap:~\CEig}%
\itemspace

\author{G\"unter M.\ Ziegler}						%ZIEGLER
\authorx{Gunter M.\ Ziegler}
\authorspace
\authref   See \hyperlink{Anders Bjorner}{A.~Bj\"orner}, \hyperlink{Laszlo Lovasz}{L.~Lov\'asz}, and \hyperlink{Tuan Tran}{T.~Tran}.
\itemspace

\author{Howard E.\ Zimmerman}					%ZIMMERMAN
\authorspace
\bsgitem{1966a}{}  On molecular orbital correlation diagrams, the occurrence of M\"obius systems in cyclization reactions, and factors controlling ground- and excited-state reactions.  I.  \emph{J.\ Amer.\ Chem.\ Soc.}\ 88 (1966), no.\ 7, 1564--1565.
\annot{} Energies of M\"obius cycles with one negative sign, inspired by \xref{Heilbronner}{1964a}{E.\ Heilbronner}.
\dated{23 Nov 2012} 
\codes{\CChem:~\Csg:~\Cbal}%
\itemspace

\author{Franti\v{s}ek Z{\'\i}tek}						%ZITEK
\authorx{Frantisek Zitek}
\authorspace
\bsgitem{1972a}{}  Polarisovan\'e grafy.  [Polarized graphs.]  Lecture at the Czechoslovak Conf.\ on Graph Theory, \v{S}ti\v{r}\'{\i}n, May, 1972.
\annot{} For definitions see \xref{Zelinka}{1976a}{Bohdan Zelinka}.  For work on these objects see many papers of Zelinka.
\codes{\Csg:~\COri,~\Csw}%
\itemspace

\bsgitem{1975a}{}  Some remarks on polar graphs.  In:  \emph{Recent Advances in Graph Theory} (Proc.\ Second Czechoslovak Symp., Prague, 1974), pp.\ 537--539.  Academia, Prague, 1975.
\MR{0398890} (53 \#2741).  \Zbl{324.05113}.					
\annot{} \Cf\ \xref{Zelinka}{1976a}{Bohdan Zelinka} for definitions.  The number of labelled polar trees; the same with given degrees.
\dated{27 Jul 2013} 
\codes{\Csg:~\COri}%
\itemspace

\author{Arjana \v{Z}itnik}						%ZITNIK
\authorx{Arjana Zitnik}
\authorspace
\authref  See \hyperlink{Tomaz Pisanski}{T~Pisanski}.
\itemspace

\author{J.\ Zittartz}							%ZITTARTZ
\authorspace
\authref  See \hyperlink{P.\ Hoever}{P.~Hoever}, \hyperlink{M.H.\ Waldor}{M.H.~Waldor}, and \hyperlink{W.F.\ Wolff}{W.F.~Wolff}.
\itemspace

\authorx{Etay Ziv}							%ZIV
\author{Etay Ziv, Robin Koytcheff, Manuel Middendorf, \& Chris Wiggins}
\authorspace
\bsgitem{2005a}{}  Systematic identification of statistically significant network measures.  \emph{Phys.\ Rev.\ E} 71 (2005), no.\ 1, art.\ 016110, 8 pp.
\arXiv{cond-mat/0306610}.  
\annot{} Statistical analysis on a space of graphs.  Mentions easy generalization to signed (and weighted) graphs.
\dated{8 Sept 2010} 
\codes{\CSGw:~\CAlgor}%
\itemspace

\author{Dejan \v{Z}ivkovi\'c}						%ZIVKOVIC
\authorx{Dejan Zivkovic}
\authorspace
\authref  See \hyperlink{Slobodan K.\ Simic}{S.K.~Simi\'c}.
\itemspace

\authorx{Dorota Zuchowska-Skiba}					%ZUCHOWSKA-SKIBA
\author{Dorota \.Zuchowska-Skiba}
\authorspace
\authref  See \hyperlink{Forough Hassanibesheli}{F.~Hassanibesheli}.
\itemspace

\author{Mark Zuckerberg}						%ZUCKERBERG
\authorspace
\bsgitem{2016a}{}  Geometric proofs for convex hull defining formulations.  \emph{Operations Res.\ Lett.}\  44 (2016), no.\ 5, 625--629.
\MR{3546882}.  \Zbl{1408.90193}.  
\annot{} Example 3:  The constraint matrix is the incidence matrix of a balanced signed graph.
\dated{16 July 2016} 
\codes{\CSG:~\CBal:~\CInc}%
\itemspace

\author{Alejandro Zu\~niga}						%ZUNIGA
\authorx{Alejandro Zuniga}
\authorspace
\authref  See \hyperlink{Julio Aracena}{J.~Aracena}.
\itemspace

\author{[Anke van Zuylen]}						%ZUYLEN
\authorspace
\authref  See \hyperlink{Anke van Zuylen}{A.~van~Zuylen} (under ``V'').
\itemspace

\author{Alexei Zverovich}						%ZVEROVICH
\authorspace
\authref  See \hyperlink{Gregory Gutin}{G.~Gutin}.
\itemspace

\author{Igor E.\ Zverovich}						%ZVEROVICH
\authorspace
\authref  See also \hyperlink{Endre Boros}{E.~Boros}.
\authrefspace

\bsgitem{2002a}{}  Arc signed graphs of oriented graphs.  \emph{Ars Combin.}\ 62 (2002), 289--297.
\MR{1881967} (2002k:05104).  \Zbl{1073.05539}.
\annot{} The arc signed graph $\Lambda_Z(\vec\Gamma)$ of a digraph $\vec\Gamma$ (simple $\Gamma$) is the line graph $\Lambda(\Gamma)$ with $\sigma_Z(\vec{uv}\vec{wv}), \sigma_Z(\vec{vu}\vec{vw}) := +$ and $\sigma_Z(\vec{uv}\vec{vw}) := -$.  [Thus, it is $-\Lambda(+\Gamma)$ where $+\Gamma$ has orientation $\vec\Gamma$; \cf\ \xref{Zaslavsky}{2012c}{Thomas Zaslavsky}, \xrefd{20xxa}{Thomas Zaslavsky}.]  Thm.\ 1: A Krausz-type characterization of $\Lambda_Z$.  Cor.\ 1:  $\Lambda_Z$ determines $\vec\Gamma$ up to isolated vertices and reversing the orientation.  Thm.\ 2:  Characterization by induced subgraphs: a finite list plus antibalanced circles of length $\geq4$.  Cor.\ 2:  $\Lambda_Z(\vec\Gamma)$ graphs can be recognized and $\vec\Gamma$ reconstructed in polynomial time.
Dictionary:  ``$(+)$-complete'' means $+K_n$;  ``bicomplete'' means complete and balanced.  
[Antibalanced circles are forbidden due to having all-positive base graphs.] 
\codes{\CSG:~\CLG}%
\itemspace

\author{A.\ Zverovitch}						%ZVEROVITCH
\authorspace
\authref  See \hyperlink{N.\ Gulpinar}{N.~G\"ulpinar}.
\itemspace

\author{Stefan H.M.\ van Zwam}					%ZWAM
\authorspace
\authref  See also \hyperlink{Kevin Grace}{K.~Grace}, \hyperlink{Dillon Mayhew}{D.~Mayhew} and \hyperlink{R.A.\ Pendavingh}{R.A.~Pendavingh}.
\authrefspace

\bsgitem{2009a}{}  \emph{Partial Fields in Matroid Theory}.  Doctoral dissertation, Technische Universiteit Eindhoven, 2009. 
\annot{} \S3.2, ``The Dowling lift of a partial field''.  [Expanded in \xref{Pendavingh \& van Zwam}{2013a}{R.A.\ Pendavingh \& S.H.M.\ van Zwam}.]
\codes{\CMatrd:~\Cgg}%
\itemspace

\author{Uri Zwick}							%ZWICK
\authorspace
\authref   See \hyperlink{Raphael Yuster}{R.~Yuster}.
\itemspace

\author{Krzysztof Zwierzy\'nski}						%ZWIERZYNSKI
\authorx{Krzysztof Zwierzynski}
\authorspace
\authref  See \xref{D.~Stevanovi\'c}{2007a}{Dragan Stevanovic}.
\itemspace

\author{Lisa Zyga}							%ZYGA
\authorspace
\bsgitem{2009a}{}  Physics model determines dynamics of friends and enemies.  \emph{PhysOrg.com}, December 2, 2009.  URL \url{http://www.physorg.com/news178954961.html}
\annot{} Popular account of \xref{Marvel, Strogatz, and Kleinberg}{2009a}{Seth A.\ Marvel, Steven H.\ Strogatz, \& Jon M.\ Kleinberg}.
\dated{26 Jan 2011}
\codes{\CSG:~\CPhys,~\CFr,~\CState:~\CDyn:~\CExp}%
\itemspace

\author{Ond\v{r}ej Z\'yka}						%ZYKA
\authorx{Ondrej Zyka}
\authorspace
\authref   See also \hyperlink{Jan Kratochvil}{J.~Kratochv{\'\i}l}.
\authrefspace

\bsgitem{1987a}{}  Nowhere-zero 30-flow on bidirected graphs.  KAM-DIMATIA Series 87-26, Charles University, Praha, 1987.
\annot{} \Cf\ \xref{Bouchet}{1983a}{Andre Bouchet}.  There is a nowhere-zero 30-flow [presumably, if the signed-graphic matroid has no coloop; there might be other restrictions].
\dated{15 Jan 2015} 
\codes{\CSG:~\CFlows}%
\itemspace


\end{document}

