A generalized characteristic polynomial of a graph having a semifree action

For an abelian group ΓΓ, a formula to compute the characteristic polynomial of a ΓΓ-graph has been obtained by Lee and Kim [Characteristic polynomials of graphs having a semi-free action, Linear algebra Appl. 307 (2005) 35–46]. As a continuation of this work, we give a computational formula for generalized characteristic polynomial of a ΓΓ-graph when ΓΓ is a finite group. Moreover, after showing that the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, we compute the reciprocals of the Bartholdi zeta functions of wheels and complete bipartite graphs as an application of our formula.