The Tutte polynomial and the automorphism group of a graph

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.