k-Power Graphs of Finite Groups

TL;DR

This paper explores the properties of k-power graphs of finite groups, analyzing their structure and parameters, with specific focus on cyclic groups and an application of these graphs.

Contribution

It introduces the concept of k-power graphs for finite groups, investigates their graph parameters, and examines properties of cyclic groups' k-power graphs, including an application.

Findings

Determined the number of edges and clique number of k-power graphs.

Analyzed connectedness properties of k-power graphs.

Identified specific properties for cyclic groups' k-power graphs.

Abstract

For a finite group $G$ and for a fixed positive integer $k$, $k\geq 2$, the $k$-power graph of $G$ is an undirected simple graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $x^k=y$ or $y^k=x$. In this paper, we investigate some graph parameters such as number of edges, clique number, connectedness, etc. of $k$-power graphs of finite groups. Also find some properties of $k$-power graphs of finite cyclic groups, and finally we present an application