| Summary: | 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 $u^{p}\equiv v~ (\textmd{mod} ~m)$. 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
$G(p,m)$.
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