Graph Polynomials and Their Applications II: Interrelations and Interpretations

A graph polynomial is an algebraic object associated with a graph that is usually invariant at least under graph isomorphism. As such, it encodes information about the graph, and enables algebraic methods for extracting this information. This chapter surveys a comprehensive, although not exhaustive, sampling of graph polynomials. It concludes Graph Polynomials and their Applications I: The Tutte Polynomial by continuing the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an understanding of one polynomial can guide research into others. We also discuss multivariable generalizations of some of these polynomials and the theory facilitated by this. We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields.