Extreme Vertices of the Psi-Divisible Graph of the Group Z_{p^n}

Amit Kumar; Vinod Kumar; Amit Sehgal

doi:10.22342/jims.v32i2.2145

Amit Kumar ⁽¹⁾ , Vinod Kumar ⁽²⁾ , Amit Sehgal ⁽³⁾

(1) Department of Mathematics, Baba Mastnath University Rohtak, India,

(2) Department of Mathematics, Baba Mastnath University Rohtak, India,

(3) Department of Mathematics, Pandit Neki Ram Sharma Govt. College Rohtak, India

https://doi.org/10.22342/jims.v32i2.2145

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
July 19, 2025

Accepted
January 26, 2026

Published
April 23, 2026

Keywords:

Extreme vertices, $\Psi$-divisible graph of a group, finite cyclic group $\mathbb{Z}_{p^n}$

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Abstract

The $\Psi$-divisible graph of a finite group $G$, denoted by $\Psi_G$ is a special type of simple undirected graph, in which the set of vertices contains non-trivial subgroups of $G$ and two distinct vertices $u$ and $v$ are adjacent if and only if $u$ is a proper subgroup of $v$ such that $\Psi(u)|\Psi(v)$ or $v$ is a proper subgroup of $u$ such that $\Psi(v)|\Psi(u)$. The existence of extreme vertices in the $\Psi$-divisible graph of the finite cyclic group $\mathbb{Z}_{p^n}$ is described in this article.

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References

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Authors

Amit Kumar

Department of Mathematics, Baba Mastnath University Rohtak

amitckt0612@gmail.com (Primary Contact)

Vinod Kumar

Department of Mathematics, Baba Mastnath University Rohtak

Amit Sehgal

Department of Mathematics, Pandit Neki Ram Sharma Govt. College Rohtak

Kumar, A., Kumar, V., & Sehgal, A. (2026). Extreme Vertices of the Psi-Divisible Graph of the Group Z_{p^n}. Journal of the Indonesian Mathematical Society, 32(2), 2145. https://doi.org/10.22342/jims.v32i2.2145

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1. INTRODUCTION

Even though Cayley’s work dates back to the nineteenth century[1], the connection between group and graph theories became well known in the 1950s. Numerous graphs are defined, particularly on a finite group $G$. Without going too far, we recall the following:

Chakrabarty et al.[2]initially investigated the power graph of a finite group $G$, $\mathbb { P } ( G )$, as the graph whose vertex set consists of the elements of the finite group $G$ and two diferent vertices $x _ { 1 } \neq x _ { 2 }$ are adjacent if and only if $\langle x _ { 2 } \rangle \subseteq \langle x _ { 1 } \rangle$ or $\langle x _ { 1 } \rangle \subseteq \langle x _ { 2 } \rangle$. (See [3,4])
For a finite group $G$, the reduced power graph is a graph with a set of vertices $G$ such that two vertices $x , y \in G$ are adjacent if and only if $x \neq x$ and $\langle x \rangle \subsetneq \langle y \rangle$ or $\langle y \rangle \subsetneq \langle x \rangle$. (See[5],[6],[7],[8])
For a finite group $G$, an equal-square graph[9]is an undirected graph with group $G$ as its vertex set, having distinct vertices $x$ and $y$ adjacent if and only if $x ^ { 2 } = y ^ { 2 }$.
The undirected superpower graph of a finite group $G$, $\mathbb { S } ( G )$ is a simple graph in which the set of vertices $G$ and two distinct vertices $u , v$ are adjacent if $o ( u ) | o ( v )$ or $o ( v ) | o ( u )$. The concept of $\mathbb { S } ( G )$ is an extension of the power graph $\mathbb { P } ( G )$ of the finite group $G$. (See [10,11,12,13])
For a finite abelian group $( G , + )$, a square power graph[14]is an undirected graph with an abelian group $G$ as its vertex set, having distinct vertices $x$ and $y$ adjacent if and only if $x + y = 2 z$ for some $z \in G$ and $2 z \neq 0$.

Numerous graphs, including the co-maximal ideal graph, co-prime graph, coprime order graph, commuting graph, intersection graph, Gruenberg-Kegel graph, zero-divisor graphs, and many more, are associated with algebraic structures in the literature. In[15], the concept of a $Ψ$-divisible graph was introduced.

The $Ψ$-divisible graph of a finite group $G$, denoted by $\Psi _ { G }$ is a special type of simple undirected graph in which the set of vertices contains non-trivial subgroups of $G$ and two distinct vertices $u$ and $v$ are adjacent if and only if $u$ is a proper subgroup of $v$ such that $\Psi ( u ) | \Psi ( v )$ or $v$ is a proper subgroup of $u$ such that $\Psi ( v ) | \Psi ( u )$.

The $Ψ$-divisible graph of a group has emerged as a significant topic of re search, with vertices representing subgroups and edges linking elements that exhibit a particular algebraic relationship, generally concerning the divisible order sum properties of the group’s subgroups. The connection between $Ψ$-divisibility and the square-free order property of finite groups was further investigated in[16]. Some results $Ψ$-divisible graph for finite cyclic groups were investigated in[17].

In a graph $Γ$, if the neighborhood of a vertex $u$ induces a complete subgraph, then the vertex $u$ is said to be an extreme vertex[18]. Important research on the extreme vertices of the power graph for the finite abelian, dicyclic, and dihedral groups was carried out by AbuGhneim et al.[19]. Kumari et al.[18]extended this work to the power graph for the permutation group $S _ { n }$.

In this article, the extreme vertices of the $Ψ$-divisible graph of a finite cyclic group $\mathbb { Z } _ { p ^ { n } }$ are examined. The rest of the paper is organized as follows. The fundamental findings of group theory and graph theory are presented in Section2. Our study of the extreme vertices of the $Ψ$-divisible graph of a finite cyclic group $\mathbb { Z } _ { p ^ { n } }$ is presented in Section3.

2. Notations and Preliminaries

This section provides an overview of the notation and fundamental results used throughout the paper. The following notations are used:

$o(x)$ order of element $x$,

$\Psi ( H )$ sum of elements orders of $H$,

$\Psi _ { G }$$Ψ$-divisible graph of $G$,

$uv$ vertices $u$, $v$ are adjacent,

$H \leqslant K$$H$is subgroup of $K$.

In addition, we make use of the following key results from group theory.

Lemma 2.1.[20]“Every subgroup of a cyclic group is cyclic. Moreover, If$\left| \left. \langle a\rangle \right. \right| = n$, then the order of any subgroup of$\langle a\rangle$is a divisor of$n$; and, for each positive divisor$k$of$n$, the group$\langle a \rangle$has exactly one subgroup of order$k$".

Lemma 2.2.[20]“If$H$and$K$are two subgroups of a finite cyclic group G and\(| H

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