Graph polynomials and their applications I: The Tutte polynomial

TL;DR

This survey explores the Tutte polynomial's properties, applications, and computational aspects, illustrating how it encapsulates combinatorial and algebraic information of graphs and relates to physical models.

Contribution

It provides a comprehensive overview of the Tutte polynomial, highlighting its definitions, properties, applications, and computational complexity in graph theory.

Findings

The Tutte polynomial encodes diverse combinatorial invariants.

It can be specialized or generalized for various applications.

Computational complexity of evaluating the Tutte polynomial varies with parameters.

Abstract

In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and 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 conclude with a brief discussion of computational complexity considerations.