Abstract
In this work, we incorporate modular arithmetic and discuss a special class of graphs based on power functions in a given modulus, called power digraphs. In power digraphs, the study of cyclic structures and enumeration of components is a difficult task. In this manuscript, we attempt to solve the problem for pth power congruences over different classes of residues, where p is an odd prime. For any positive integer m, we build a digraph G(p, m) whose vertex set is Z(m) = {0, 1, 2, 3, ..., m - 1} and there will be a directed edge from vertices u is an element of Z(m) to v is an element of Z(m) if and only if u(p) equivalent to v (mod m). We study the structures of G(p, m). For the classes of numbers 2(r) and p(r) where r is an element of Z(+), we classify cyclic vertices and enumerate components of G(p, m). Additionally, we investigate two induced subdigraphs of G(p, m) whose vertices are coprime to m and not coprime to m, respectively. Finally, we characterize regularity and semiregularity of G(p, m) and establish some necessary conditions for cyclicity of G(p, m).