Even signings, signed switching classes, and (−1, 1)-matrices

Abstract

In this paper, algebraic and combinatorial techniques are used to establish results concerning even signings of graphs, switching classes of signed graphs, and (−1, 1)-matrices. These results primarily deal with enumeration of isomorphism types, and determining whether there are fixed elements under the action of automorphisms. A formula is given for the number of isomorphism types of even signings of any fixed simple graph. This is shown to be equal to the number of isomorphism types of switching classes of signings of the graph. A necessary and sufficient criterion is found for all switching classes fixed by a given graph automorphism to contain signings fixed by that automorphism. It is determined whether this criterion is met for all automorphisms of various graphs, including complete graphs, which yields a known result of Mallows and Sloane. As an application, a formula is developed for the number of H-equivalence classes of (−1, 1)-matrices of fixed size. Independently, using Molien's theorem and following a suggestion of Cameron's, generating series for these numbers are given. As a final application, a necessary and sufficient condition that a square (−1, 1)-matrix be switching equivalent to a symmetric matrix is given.