On the symmetric digraphs from powers modulo n

For any positive integers n and k, let G(n, k) denote the digraph whose set of vertices is H = {0, 1, 2, ... , n - 1} and 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(modn). The digraph G(n. k) is called symmetric of order M if its set of connected components can be partitioned into subsets of size M with each subset containing M isomorphic components. In this paper, we establish a necessary and sufficient condition for G(n. k) to be symmetric of order M, where M has an odd prime divisor. (C) 2011 Elsevier B.V. All rights reserved.