Integral normal Cayley graphs

Source abstract

Abstract Given a finite group G , we say that a subset C of G is power-closed if, for every $$x\in C$$ x ∈ C and $$y\in \langle x\rangle$$ y ∈ ⟨ x ⟩ with $$\langle x\rangle =\langle y\rangle$$ ⟨ x ⟩ = ⟨ y ⟩ , we have $$y\in C$$ y ∈ C . In this paper, we are interested in finite Cayley digraphs $$\textrm{Cay}(G,C)$$ Cay ( G , C ) over G with connection set C , where C is a union of conjugacy classes of G . We show that each eigenvalue of $$\textrm{Cay}(G,C)$$ Cay ( G , C ) is integral if and only if C is power-closed. This result will follow from a more general result.