An exact enumeration of vertex connectivity of the enhanced power graphs of finite nilpotent groups

Abstract

The enhanced power graph of a group \(G\) is a graph with vertex set \(G\), where two distinct vertices \(x\) and \(y\) are adjacent if and only if there exists an element \(w\) in \(G\) such that both \(x\) and \(y\) are powers of \(w\). Kumar, Ma, Parveen and Singh in [22] found the exact vertex connectivity of the enhanced power graph of finite nilpotent groups whose all except one Sylow subgroups are cyclic. In this paper, we determine the exact vertex connectivity of the enhanced power graph of any finite nilpotent group in full generality, by connecting it to the minimum number of roots of a prime order element in its Sylow subgroups.