Signed Graphs and Geometry

TL;DR

This paper provides an introductory survey of signed graphs, emphasizing their geometric origins from root systems, and discusses properties that extend unsigned graph theory with some novel results.

Contribution

It offers a coherent, mostly expository overview of signed graphs with some original results, focusing on their geometric connections and generalizations from unsigned graphs.

Findings

Signed graphs relate naturally to classical root systems.

Many properties of signed graphs generalize those of unsigned graphs.

Some results presented are new and previously unpublished.

Abstract

These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The emphasis is on the way signed graphs arise naturally from geometry, especially from the geometry of the classical root systems. Most of the properties I discuss generalize those of unsigned graphs, but the constructions and proofs are often more complicated. My aim is a coherent presentation of the subject, with a few illustrative proofs and adequate references. Hence the arrangement of the notes is topical with only occasional remarks about the historical course of development. Though this is mainly an expository survey, some of the results have not hitherto been published.