On the Spectrum of Unitary Finite–Euclidean Graphs

Let Γ be an additive group. For S ⊆ Γ, 0 6∈ S and S = {−s : s ∈ S} = S, the Cayley graph G = C(Γ, S) is the undirected graph having vertex set V (G) = Γ and edge set E(G) = {(a, b) : a − b ∈ S}. The Cayley graph G = C(Γ, S) is regular of degree |S|. For a positive integer n > 1, the unitary Cayley graph Xn = C(Zn,Z∗n) is defined by the additive group of the ring Zn of integers modulo n and the multiplicative group Z∗n of its units. So Xn has vertex set V (Xn) = Zn = {0, 1, . . . , n− 1} and edge set E(Xn) = {(a, b) : a, b ∈ Zn, gcd(a− b, n) = 1}.