On the difference of the enhanced power graph and the power graph of a finite group

Abstract

The difference graph $D\left(G\right)$ of a finite group G is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of $D\left(G\right)$ with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.