Integral quadratic forms and graphs

Abstract

The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define an integral quadratic form. The main purpose of this paper is to raise this question: “What can quadratic forms tell us about graphs?” As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. The essential definitions and facts from the theory of quadratic forms will be sketched without proof.