Graph Polynomials and Their Applications I: The Tutte Polynomial

Chapter

Abstract

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.

Keywords

Tutte polynomial Graph polynomial Chromatic polynomial Flow polynomial Reliability polynomial Shelling polynomial Abelian sandpile model Spanning tree Beta invariant 

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Notes

Acknowledgments

We thank all the friends and colleagues who offered many helpful comments and suggestions during the writing of this chapter.

The first author was supported by the National Security Agency and by the Vermont Genetics Network through Grant Number P20 RR16462 from the INBRE Program of the National Center for Research Resources (NCRR), a component of the National Institutes of Health (NIH).

The second author was supported by CONACYT of Mexico, Grant 83977.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsSaint Michael’s CollegeColchesterUSA
  2. 2.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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