Some Properties of The Spectrum of The Power Digraph Γ(n,k)

For every positive integer n and k , a power digraph modulo n, denoted by Γ(n,k) is constructed with the vertex set Z_n={0,1,2,⋯,n-1}, and a directed edge from a vertex x to a vertex y exists if and only if x^k≡y(mod n), where x,y∈Z_n. In this work, we define the out-adjacency (A_Γ^+) and the in-adjacency (A_Γ^-) matrices of the digraph Γ(n,k) and some results on A_Γ^+ and A_Γ^- are discussed. It is proved that the matrices A_Γ^+ and A_Γ^- are singular if k|ϕ(n) or p^2 |n, for some prime p. Some spectral properties of Γ(n,k) are also presented. Moreover, it is proved that the algebraic multiplicity of 1 as an eigenvalue of A_Γ^+ is the number of components of the digraph Γ(n,k).