The integrality of distance spectra of quasiabelian 2-Cayley graphs

Abstract

A graph Γ is called an n-Cayley graph over a group G if there exists a semiregular subgroup of Aut(Γ) isomorphic to G with n orbits. In this paper, we first establish a decomposition formula for the distance spectra of n-Cayley graphs over arbitrary finite groups by using the lifted graphs, the matrix theory and the group representation theory. Secondly, we completely determine the distance splitting fields and the distance algebraic degrees of quasiabelian 2-Cayley graphs. Finally, we present some criteria for the distance integrality of quasiabelian 2-Cayley graphs. Moreover, we present some sufficient conditions for the equivalence between the integrality and the distance integrality of 2-Cayley graphs over abelian groups. We also give some sufficient conditions for the integrality of distance powers of 2-Cayley graphs over abelian groups. Our results generalize several previously known results.