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Majid Arezoomand
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Bijan Taeri
Keywords:
$n$-Cayley digraph, linear representations of groups, characteristic polynomial of graphs
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.