A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb Z_{n}=\{0,1,\dots ,n-1\}$ as the set of vertices and $E=\{(a,b)\colon a^{k}\equiv b\pmod n\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: 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 $n\geq 1$ and $k\geq 2$ 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 $q\geq 1$.
A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = xkfor all x \in G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed., Uzma Ahmad, Muqadas Moeen., and Obsahuje seznam literatury