Characteristic polynomials of graphs having a semifree action

Abstract

J.H. Kwak and J. Lee (Linear and Multilinear Algebra 32 (1992) 61–73) computed the characteristic polynomial of a finite graph G having an abelian automorphism group which acts freely on G. For a finite weighted symmetric pseudograph G having an abelian automorphism group which acts semifreely on G, K. Wang (Linear Algebra Appl. 51 (1983) 121–125) showed that the characteristic polynomial of G is factorized into a product of a polynomial associated to the orbit graph and a polynomial associated to the free part of the action. But he did not explicitly compute the characteristic polynomial of such a graph G. In this paper, we introduce a new method to construct a finite pseudograph G having an automorphism group which acts semifreely on G, and obtain an explicit formula to compute the characteristic polynomial of such a graph by using the construction method.