On symmetric digraphs of the congruence xk≡y(modn)

We assign to each pair of positive integers nn and k≥2k≥2 a digraph G(n,k)G(n,k) whose set of vertices is H={0,1,…,n−1}H={0,1,…,n−1} and for which there is a directed edge from a∈Ha∈H to b∈Hb∈H if ak≡b(modn). The digraph G(n,k)G(n,k) is symmetric of order MM if its set of components can be partitioned into subsets of size MM with each subset containing MM isomorphic components. We generalize earlier theorems by Szalay, Carlip, and Mincheva on symmetric digraphs G(n,2)G(n,2) of order 2 to symmetric digraphs G(n,k)G(n,k) of order MM when k≥2k≥2 is arbitrary.