An Interesting Property of a Class of Circulant Graphs

Suppose that Π=Cay(Zn,Ω) and Λ=Cay(Zn,Ψm) are two Cayley graphs on the cyclic additive group Zn, where n is an even integer, m=n/2+1, Ω=t∈Zn∣t is odd, and Ψm=Ω∪{n/2} are the inverse-closed subsets of Zn-0. In this paper, it is shown that Π is a distance-transitive graph, and, by this fact, we determine the adjacency matrix spectrum of Π. Finally, we show that if n≥8 and n/2 is an even integer, then the adjacency matrix spectrum of Λ is n/2+11, 1-n/21, 1n-4/2, -1n/2 (we write multiplicities as exponents)