On symmetric digraphs of the congruence xk ≡ y (mod n)

We assign to each pair of positive integers n and k >= 2 a digraph G(n, k) whose set of vertices is H = {0, 1, . . . , n - 1} and for which there is a directed edge from a is an element of H to b is an element of H if a(k) equivalent to b (mod n). The digraph G(n, k) is symmetric of order M if its set of components can be partitioned into subsets of size M with each subset containing M isomorphic components. We generalize earlier theorems by Szalay, Carlip, and Mincheva on symmetric digraphs G(n, 2) of order 2 to symmetric digraphs G(n, k) of order M when k >= 2 is arbitrary. (C) 2008 Elsevier B.V. All rights reserved.