Signed graphs

Abstract

A signed graph is a graph with a sign attached to each arc. This article introduces the matroids of signed graphs, which generalize both the polygon matroids and the even-circle (or unoriented cycle) matroids of ordinary graphs. The concepts of balance, switching, restriction and contraction, double covering graphs, and linear representation of signed graphs are treated in terms of the matroid, and a matrix-tree theorem for signed graphs is proved. The examples treated include the all-positive and all-negative graphs (whose matroids are the polygon and even-circle matroids), sign-symmetric graphs (related to the classical root systems), and signed complete graphs (equivalent to two-graphs).

Replacing the sign group by an arbitrary group leads to voltage graphs. Most of our results on signed graphs extend to all voltage graphs.