Certain properties of the enhanced power graph associated with a finite group

Abstract

The enhanced power graph of a finite group G, denoted by PE(G), is a simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if x,y∈⟨z⟩ for some z∈G. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of PE(G) are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain the upper and lower bounds of the Wiener index of PE(G), where G is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.