Non-Isomorphic Cayley Graphs of Metacyclic Groups of Order 8p with the Same Spectrum

The spectrum of a graph Γ, denoted by Spec(Γ), is the multiset of eigenvalues of its adjacency matrix. A Cayley graph Cay(G,S) of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph Cay(G,T), Spec(Cay(G,S))=Spec(Cay(G,T)) implies Cay(G,S)≅Cay(G,T). A group G is said to be Cay-DS if all Cayley graphs of G are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups M8p of order 8p (with center of order 4), where p is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of M8p (p≥5) with the same spectrum are found. As a conclusion, this paper concludes that M8p is Cay-DS if and only if p=3.