On connectivity, domination number and spectral radius of the proper enhanced power graphs of finite nilpotent groups

TL;DR

This paper investigates the structural properties of proper enhanced power graphs of finite nilpotent groups, including connectivity, domination number, and spectral radius, providing a comprehensive classification and spectral analysis.

Contribution

It characterizes dominating vertices, classifies connected cases, computes domination numbers, and analyzes spectral radius multiplicity for these graphs.

Findings

Characterized dominating vertices of enhanced power graphs.

Classified nilpotent groups with connected proper enhanced power graphs.

Determined the spectral radius multiplicity of the graphs.

Abstract

For a group $G,$ the enhanced power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x, y$ are adjacent if and only if there exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set $G \setminus S,$ where $S$ is the set of dominating vertices of the enhanced power graph. In this paper, we first characterize the dominating vertices of enhanced power graph of any finite nilpotent group. Thereafter, we classify all nilpotent groups $G$ such that the proper enhanced power graphs are connected and find out their diameter. We also explicitly find out the domination number of proper enhanced power graphs of finite nilpotent groups. Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.