The Home Page of Signed, Gain, and Biased Graphs

by Thomas Zaslavsky

Why is this picture appropriate?*

Illustration by Hugh Thomson, courtesy of Henry Churchyard

Mathematical Bibliography

A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas
Electronic Journal of Combinatorics,
Dynamic Surveys in Combinatorics, #DS8.

      A signed graph is a graph with signs labelling its edges. A gain graph has elements of any group as edge labels (called "gains"), with the understanding that reversing the sense in which you traverse the edge will invert the gain. A bidirected graph has both ends of each edge directed independently; it can be regarded as an oriented signed graph.
      The Bibliography is a classified and copiously annotated list of all the publications (and suitable unpublished manuscripts, theses, etc.) of mathematical interest related to signed graphs, vertex-signed graphs, gain graphs (invertibly group-labelled graphs), and bidirected graphs that I've been able to find and examine and enter into the list. (That is possibly about half of all publications.) It includes all or part of the literature of signed digraphs, Dowling lattices, combinatorics of root systems, parity of cycles and paths and max-cut problems as well as signless Laplacian matrices (these concern all-negative signatures), generalized networks (networks with gains), qualitative matrix theory, quadratic pseudo-Boolean functions, correlation clustering, dynamic labeled 2-structures, etc., etc., as well as selected publications on applications to social science (psychology, sociology, anthropology, economics) and natural science (physics, chemistry, biology—sorry, no geology or astronomy—yet).
      The Bibliography is published as Dynamic Survey 8 in the Dynamic Surveys in Combinatorics of the Electronic Journal of Combinatorics. The published editions are also available here (see below) in PDF. If you want PS or the Tex file, see the Dynamic Surveys Web page.

Topical subsets

These are (very outdated) specialized sub-bibliographies for selected subtopics. Each has its preparation date on the title page.


Glossary of Signed and Gain Graphs and Allied Areas.
Second Edition, 1998 September 16. 43 pp.
Copyright © 1998 Thomas Zaslavsky

The Glossary has terminology, definitions, and notation; it is a companion to the Bibliography. The second edition is published as Dynamic Survey 9 in the Dynamic Surveys in Combinatorics of the Electronic Journal of Combinatorics.

The working version, which is slightly more up to date, is available here. View the second edition in HTML (98 kilobytes), or download either
... a topically arranged version in dvi (124 kB) or PostScript (488 kB) or
... an incomplete alphabetical version in dvi (?? kB) or PostScript (?? kB).

Problem List

A miscellany of research problems in PostScript or (smaller) in HTML. At present, tiny (especially the HTML version). PS version is to be expanded.


The Matroids page of Steve Pagano: painlessly learn what a matroid is and what it has to do with signed graphs.

The hyperplane arrangement that represents
the frame matroid of
±K3o, the complete signed graph of order 3;
also known as the hyperplane arrangement corresponding to the root system

Courtesy of John Stembridge

* Answer to the caption at head of page: We see one family, in the same room but hardly together. The family of signed, gain, and biased graphs is a lot like that. The Bibliography and Glossary are supposed to lead the family members to talk to one another. May they do so! (If you want to know more about the picture, click on it.)

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