Graph Polynomials and Their Applications I: The Tutte Polynomial

We begin our exploration of graph polynomials and their applications with the Tutte polynomial, a renown tool for analyzing properties of graphs and networks. This two-variable graph polynomial, due to W. T. Tutte [Tut47,Tut54, Tut67], has the important universal property that essentially any multiplicative graph invariant with a deletion/contraction reduction must be an evaluation of it. These deletion/contraction operations are natural reductions for many network models arising from a wide range of problems at the hearts of computer science, engineering, optimization, physics, and biology. In addition to surveying a selection of the Tutte polynomial’s many properties and applications, we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We begin with the Tutte polynomial because it has a rich and well-developed theory, and thus it serves as an ideal model for exploring other graph polynomials in the next chapter, Graph Polynomials and Their Applications II: Interrelations and Interpretations. Furthermore, because of the Tutte polynomial’s long history, extensive study, and its universality property, it is often a ‘point of contact’ for research into other graph polynomials in that their study frequently includes exploring their relations to the Tutte polynomial. These interrelationships will be a central theme of the following chapter. In this chapter we give both recursive and generating function formulations of the Tutte polynomial, and state its universality in the form of a recipe theorem.