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

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 $w.$ In this paper, we determine the vertex connectivity of the enhanced power graph of any finite nilpotent group.