Husnine, S. M.;Ahmad, Uzma;Somer, Lawrence

Util. Math.85, 257-271 (2011).

Let us assign to each pair of positive integers \(n\) and \(k \geq 2\) a digraph \(G(n,k)\) whose set of vertices is \(H = \{0,1,\dots ,n-1\}\) and for which there is a directed edge from \(a \in H\) to \(b \in H\) if \(a^k \equiv b \pmod{n}\). \(G(n,k)\) is then called an iteration directed graph. 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. The authors give necessary and sufficient conditions for \(G(n,k)\) to be symmetric of order \(p\) when \(n = p^a q_1,\dots,q_m\) and \(p, q_i\) are odd prime divisors of \(n\) and \(a > 1\).

Keywords:

iteration digraph;symmetric digraphs;permutation