The Tutte polynomial of some matroids

TL;DR

This paper reviews known formulas for the Tutte polynomial of specific graph and matroid families, highlighting techniques used for their derivation, aiding researchers in combinatorics and related fields.

Contribution

It compiles existing formulas for the Tutte polynomial of various graph and matroid families and explains the methods used to derive these formulas.

Findings

Compilation of known Tutte polynomial formulas

Explanation of techniques for deriving formulas

Useful resource for combinatorics researchers

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 use to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.