The power digraphs associated with generalized dihedral groups

Abstract

We study the digraphs based on dihedral group [Formula: see text] by using the power mapping, i.e., the set of vertices of these digraphs is [Formula: see text] and the set of edges is [Formula: see text]. These are called the power digraphs and denoted by [Formula: see text]. The cycle and in-degree structure of these digraphs are completely examined. This investigation leads to the derivation of various formulae regarding the number of cycle vertices, the length of the cycles, the number of cycles of certain lengths and the in-degrees of all vertices. We also establish necessary and sufficient conditions for a vertex to be a cycle vertex. The analysis of distance between vertices culminates at different expressions in terms of [Formula: see text] and [Formula: see text] to determine the heights of vertices, components and the power digraph itself. Moreover, all regular and semi-regular power digraphs [Formula: see text] are completely classified.