Structures of power digraphs over the congruence equation $ x^p\equiv y\; (\text{mod}\; m) $ and enumerations

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 $ p $th 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 $ \mathbb{Z}_{m} = \{0, 1, 2, 3, ..., m-1\} $ and there will be a directed edge from vertices $ u\in \mathbb{Z}_{m} $ to $ v\in \mathbb{Z}_{m} $ if and only if $ u^{p}\equiv v\; (\text{mod} \; m) $. We study the structures of $ G(p, m) $. For the classes of numbers $ 2^{r} $ and $ p^{r} $ where $ r\in \mathbb{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) $.