Enhanced Power Graph of Certain Non-abelian Groups

TL;DR

This paper investigates the structural and spectral properties of the enhanced power graph of semidihedral, dihedral, and quaternion groups, revealing new insights into their graph-theoretic and spectral characteristics.

Contribution

It introduces the study of various graph properties and spectra specifically for the enhanced power graph of certain non-abelian groups, including semidihedral, dihedral, and quaternion groups.

Findings

Determined the metric dimension and resolving polynomial of the enhanced power graph.

Computed the Laplacian spectrum for the enhanced power graphs of the groups studied.

Analyzed properties like closure, interior, and eccentric subgraphs of these graphs.

Abstract

The enhanced power graph of a group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if they belong to same cyclic subgroup. In this paper, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraph of the enhanced power graph of semidihedral group. Consequently, we obtained the metric dimension and resolving polynomial of the enhanced power graph of semidihedral group. At the final part of this paper, we obtained the Laplacian spectrum of the enhanced power graph of semidihedral, dihedral and generalized quaternion groups.