Some properties of various graphs associated with finite groups

TL;DR

This paper explores properties of power and commuting graphs of finite groups, revealing characterizations and formulas for their tree-numbers, including a unique identification of the group L_2(7).

Contribution

It introduces new characterizations of finite groups via tree-numbers of associated graphs and provides explicit formulas for Suzuki simple groups.

Findings

L_2(7) characterized by its power graph's tree-number

Classification of groups with power-free decomposition

Explicit formula for commuting graph's tree-number of Suzuki groups

Abstract

In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.