The Tutte polynomial and the automorphism group of a graph

TL;DR

This paper explores how the Tutte polynomial relates to the symmetry properties of graphs, specifically their automorphism groups, and provides conditions linking polynomial coefficients to graph periodicity.

Contribution

It establishes a necessary condition on the Tutte polynomial coefficients for graphs with certain automorphism group elements, advancing understanding of graph symmetry detection.

Findings

Coefficients of the Tutte polynomial satisfy specific conditions for p-periodic graphs.

The result helps identify non-periodic graphs using polynomial invariants.

Application to the Frucht graph demonstrates the practical utility of the condition.

Abstract

A graph $G$ is said to be $p$-periodic, if the automorphism group $Aut(G)$ contains an element of order $p$ which preserves no edges. In this paper, we investigate the behavior of graph polynomials (Negmai and Tutte) with respect to graph periodicity. In particular, we prove that if $p$ is a prime, then the coefficients of the Tutte polynomial of such a graph satisfy a certain necessary condition. This result is illustrated by an example where the Tutte polynomial is used to rule out the periodicity of the Frucht graph.