Graph polynomials and their applications I: The Tutte polynomial

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.

This is a preliminary version of one of two book chapters for inclusion in a volume on graph structures. Minor changes

References in corpus (6)

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• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
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Cited by in corpus (7)

• Feynman graph polynomials
• The multivariate signed Bollobas-Riordan polynomial
• Generalization of the Bollobás-Riordan polynomial for tensor graphs
• Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
• Tutte and Jones Polynomial of Link Families
• Irreducibility of the Tutte polynomial of an embedded graph
• Tutte polynomials and a stronger version of the Akiyama-Harary problem