The Tutte Polynomial of Some Matroids

Abstract

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engineering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work, we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were used to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.

Algebraic and Geometric Methods in Enumerative Combinatorics

“…[Wel93] However, the results of Section 7.6 allow us to compute the Tutte polynomial of some matroids of interest. We now survey some of the most interesting examples; see [MRIS12] for others. Some of these formulas are best expressed in terms of the coboundary polynomial This formula is straightforward in terms of coboundary polynomials: χ M (k) (X, Y ) = χ M (X, Y K ).…”

Tutte polynomials of hyperplane arrangements and the finite field method

“…However, the computation is possible in some cases. We now survey some of the most interesting examples; see [32] for others. Some of these formulas are best expressed in terms of the coboundary polynomial χ(A; X, Y ), which is equivalent to the Tutte polynomial T (A; x, y) by (5).…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…We refer the reader to [98] for a definition and a list of basic properties of the Tutte polynomial of a graph. 36 If r = 1, the complete unipartite graph K (n) consists of n distinct points, and…”

“…Finally we describe explicitly the exponential generating function for the Tutte polynomials of the weighted complete multipartite graphs. We refer the reader to [98] for a definition and a list of basic properties of the Tutte polynomial of a graph. 36 If r = 1, the complete unipartite graph K (n) consists of n distinct points, and…”