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 Zₙ=0, 1, 2, ⋯, n-1, and a directed edge from a vertex x to a vertex y exists if and only if xᵏ≡y (mod n), where x, y∈Zₙ. 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 p² |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).