  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.
