Immanantal invariants of graphs

Abstract

Something between an expository note and an extended research problem, this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra. There are a variety of ways to assign a real n× n matrix K( G) to each n-vertex graph G, so that G and H are isomorphic if and only if K( G) and K( H) are permutation similar. It follows that G and H are isomorphic only if K( G) and K( H) are similar, i.e., that similarity invariants of K( G) are graph theoretic invariants of G, an observation that helps to explain the enormous literature on spectral graph theory. The focus of this article is the permutation part, i.e., on matrix functions that are preserved under permutation similarity if not under all similarity.