Graph polynomials and symmetries

TL;DR

This paper explores how graph symmetries influence various graph polynomials, demonstrating that certain symmetries determine polynomial invariants like the characteristic polynomial, extending previous work on the Tutte polynomial.

Contribution

It generalizes the relationship between graph symmetries and polynomial invariants to other polynomials beyond the Tutte polynomial, including characteristic polynomials over finite fields.

Findings

Symmetries of prime order determine the characteristic polynomial over finite fields.

Results extend to some generalizations of the Tutte polynomial.

Graph quotient structures reflect in polynomial invariants.

Abstract

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph $G$ has a symmetry of prime order $p$, then its characteristic polynomial, with coefficients in the finite filed $\mathbb{F}_p$, is determined by the characteristic polynomial of its quotient graph $\overline G$. Similar results are also proved for some generalization of the Tutte polynomial.