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

TL;DR

This paper precisely determines the vertex connectivity of enhanced power graphs for all finite nilpotent groups, advancing understanding of their structural properties.

Contribution

It provides an exact enumeration of vertex connectivity for enhanced power graphs of finite nilpotent groups, a novel result in group and graph theory.

Findings

Vertex connectivity is explicitly calculated for all finite nilpotent groups.

The results reveal structural insights into enhanced power graphs.

The work extends previous studies on power graphs and connectivity.

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