The Complexity of Power Graphs Associated With Finite Groups

TL;DR

This paper investigates the computational complexity of power graphs derived from finite groups, providing formulas for specific groups and exploring applications of clique-replaced graphs.

Contribution

It introduces explicit formulas for the complexity of power graphs for various finite groups and studies the complexity of clique-replaced graphs.

Findings

Derived formulas for the complexity of power graphs of cyclic groups, simple groups, and others.

Analyzed the complexity of clique-replaced graphs and their applications.

Provided explicit calculations for specific classes of finite groups.

Abstract

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity of a clique--replaced graph and study some applications. Second, we derive some explicit formulas concerning the complexity $\kappa(\mathcal{P}(G))$ for various groups $G$ such as the cyclic group of order $n$, the simple groups $L_2(q)$, the extra--special $p$--groups of order $p^3$, the Frobenius groups, etc.