Characterization of power digraphs modulo n

A power digraph modulo n, denoted by G(n,k), is a directed graph with Z(n) = {0,1,... ,n - 1} as the set of vertices and E = {(a,b) : a(k) = b (mod n)} as the edge set, where n and k are any positive integers. In this paper we find necessary and sufficient conditions on n and k such that the digraph G(n,k) has at least one isolated fixed point. We also establish necessary and sufficient conditions on n and k such that the digraph G(n,k) contains exactly two components. The primality of Fermat number is also discussed.