The Complexity of Power Graphs Associated With Finite Groups

Abstract

The power graph P(G) of a finite group G is the graph whose vertex set is G, with two elements in G being adjacent if one of them is a power of the other. The purpose of this paper is twofold: (1) to find the complexity of a clique-replaced graph and study some applications; (2) to derive some explicit formulas concerning the complexity \kappa(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.