The complement of proper power graphs of finite groups

TL;DR

This paper studies the complement of proper power graphs of finite groups, classifying their structure under various graph-theoretic properties and determining key parameters like diameter and girth.

Contribution

It provides a comprehensive classification of finite groups based on the properties of their complement proper power graphs, including structural and topological characteristics.

Findings

Classified groups with complete, bipartite, path, cycle, star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective complement graphs.

Determined the diameter and girth of the complement graphs for finite groups.

Identified structural conditions for the complement graphs to have specific properties.

Abstract

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.