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

• Main Authors: M. Haris Mateen, Muhammad Khalid Mahmmod, Dilshad Alghazzawi, Jia-Bao Liu
• Format: Article
• Language: English
• Published: AIMS Press 2021-02-01
• Series: AIMS Mathematics
• Subjects: cycles; components; power digraphs; congruence equation
• Online Access: http://awstest.aimspress.com/article/doi/10.3934/math.2021270?viewType=HTML

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 $G(p,m)$.