On the Characteristic Polynomial of $n$-Cayley Digraphs

Abstract

A digraph $\Gamma$ is called $n$-Cayley digraph over a group $G$‎, ‎if there exists a semiregular subgroup $R_G$ of Aut$(\Gamma)$ isomorphic to $G$ with $n$ orbits‎. ‎In this paper‎, ‎we represent the adjacency matrix of $\Gamma$ as a diagonal block‎ ‎matrix in terms of irreducible representations of $G$ and determine its characteristic polynomial‎. ‎As corollaries of this result we find‎:  ‎the spectrum of  semi-Cayley graphs over abelian groups‎, ‎a relation between the characteristic polynomial of an $n$-Cayley graph and its complement‎, ‎and   the spectrum of‎ ‎Calye graphs over groups with cyclic subgroups‎. ‎Finally we determine the eigenspace of $n$-Cayley digraphs and their main eigenvalues‎.