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

M. Haris Mateen; Muhammad Khalid Mahmmod; Dilshad Alghazzawi; Jia-Bao Liu; M. Haris Mateen; Muhammad Khalid Mahmmod; Dilshad Alghazzawi; Jia-Bao Liu

doi:10.3934/math.2021270

AIMS Mathematics

2021, Volume 6, Issue 5: 4581-4596. doi: 10.3934/math.2021270

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Research article

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

1.
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
2.
Department of Mathematics, King Abdulaziz University, Rabigh 21589, Saudi Arabia
3.
Department of Mathematics, School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Received: 05 November 2020 Accepted: 08 February 2021 Published: 22 February 2021
MSC : 05C25, 11E04, 20G15

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 $G(p, m)$ .

Keywords:

Citation: M. Haris Mateen, Muhammad Khalid Mahmmod, Dilshad Alghazzawi, Jia-Bao Liu. Structures of power digraphs over the congruence equation $x^p\equiv y\; (\text{mod}\; m)$ and enumerations[J]. AIMS Mathematics, 2021, 6(5): 4581-4596. doi: 10.3934/math.2021270

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Abstract

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