On the heights of power digraphs modulo n

Abstract

A power digraph, denoted by G(n, k), is a directed graph with ℤ n = {0, 1, &h., n − 1} as the set of vertices and E = {(a, b): a k ≡ b (mod n)} as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křžek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křžek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of G(n, k) for n ⩾ 1 and k ⩾ 2 are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on n such that each vertex of indegree 0 of a certain subdigraph of G(n, k) is at height q ⩾ 1.