Matching in power graphs of finite groups

TL;DR

This paper studies the matching numbers of power graphs of finite groups, providing bounds, conditions for perfect matchings, and a formula for nilpotent groups, linking power and enhanced power graphs.

Contribution

It introduces new bounds and formulas for matching numbers in power graphs, and establishes their equality with enhanced power graphs for finite groups.

Findings

Derived bounds for matching numbers in power graphs.

Identified conditions for perfect matchings in these graphs.

Proved the matching number equality between power and enhanced power graphs.

Abstract

The power graph $P(G)$ of a finite group $G$ is the undirected simple graph with vertex set $G$, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are investigated. We give upper and lower bounds, and conditions for the power graph of a group to possess a perfect matching. We give a formula for the matching number for any finite nilpotent group. In addition, using some elementary number theory, we show that the matching number of the enhanced power graph $P_e(G)$ of $G$ (in which two elements are adjacent if both are powers of a common element) is equal to that of the power graph of $G$.