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Some Properties of Proper Power Graphs in Finite Abelian Groups

Dhawlath. G, Raja. V

Abstract

The power graph of a group $G$, denoted as $P(G)$, constitutes a simple undirected graph characterized by its vertex set $G$. Specifically, vertices $a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup generated by $b$ or vice versa. The corresponding proper power graph of $G$ is obtained by taking $P(G)$ and removing a vertex corresponding to the identity element, which is denoted as $P^*(G)$. In the context of finite abelian groups, this article establishes the sufficient and necessary conditions for the proper power graph's connectedness. Moreover, a precise upper bound for the diameter of $P^*(G)$ in finite abelian groups is provided with sharpness. This article also explores the study of vertex connectivity, center, and planarity.

Some Properties of Proper Power Graphs in Finite Abelian Groups

Abstract

The power graph of a group , denoted as , constitutes a simple undirected graph characterized by its vertex set . Specifically, vertices exhibit adjacency exclusively if belongs to the cyclic subgroup generated by or vice versa. The corresponding proper power graph of is obtained by taking and removing a vertex corresponding to the identity element, which is denoted as . In the context of finite abelian groups, this article establishes the sufficient and necessary conditions for the proper power graph's connectedness. Moreover, a precise upper bound for the diameter of in finite abelian groups is provided with sharpness. This article also explores the study of vertex connectivity, center, and planarity.
Paper Structure (4 sections, 20 theorems, 2 figures)

This paper contains 4 sections, 20 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Foundational Lemmas
  3. Connectivity
  4. Planarity

Key Result

Lemma 2.1

Let $G$ be finite group $a,b \in G$ have same order then either $\langle a\rangle$=$\langle b\rangle$ or $\langle a\rangle\cap\langle b\rangle= e$.

Figures (2)

  • Figure 1: Proper Power Graph of $C_3 \oplus C_2 \oplus C_2 \oplus...\oplus C_2$
  • Figure 2: Proper Power Graph of $C_2 \oplus C_3 \oplus C_3 \oplus...\oplus C_3$

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Lemma 3.4
  • ...and 26 more