In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials such as the chromatic, flow, reliability, and shelling 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.

An algebraic theory of graphs

Graph Polynomials and Their Applications I: The Tutte Polynomial

This chapter treats error-correcting codes and their weight enumerator as the center of several closely related topics such as arrangements of hyperplanes, graph theory, matroids, posets and geometric lattices and their characteristic, chromatic, Tutte, Mobius and coboundary polynomial, respectively. Their interrelations and many examples and counterexamples are given. It is concluded with a section with references to the literature for further reading and open questions.

/pdf/codes-arrangements-and-matroids-4s9twyrofe.pdf

Codes, arrangements and matroids

Hopf algebras and Tutte polynomials

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.

/pdf/graph-polynomials-and-their-applications-i-the-tutte-z30iqv3dz4.pdf

Graph polynomials and their applications I: The Tutte polynomial

The Tutte polynomial of a graph or a matroid is a polynomial in two variables x, y, and
is of central importance in the study of combinatorial counting problems. Awards: Winner of the Mollie Holman Doctoral Medal for Excellence, Faculty of Information Technology, 2010.

An algebraic theory of graphs

Citations

Graph Polynomials and Their Applications I: The Tutte Polynomial

Codes, arrangements and matroids

Hopf algebras and Tutte polynomials

Graph polynomials and their applications I: The Tutte polynomial

Correlation inequalities for Tutte polynomials

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