On the Characteristic Polynomial of n-Cayley Digraphs

A digraph Γ is called n-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with n orbits. In this paper, we represent the adjacency matrix of Γ 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 Cayley graphs over groups having cyclic subgroups. Finally we determine the eigenspace of n-Cayley digraphs and their main eigenvalues.