On The Commuting Graph of Semidihedral Group

TL;DR

This paper explores the properties of the commuting graph of the semidihedral group, analyzing various graph invariants, spectrum, and metric properties to deepen understanding of its structure.

Contribution

It provides a detailed analysis of the commuting graph of semidihedral groups, including new results on invariants, spectrum, and metric properties.

Findings

Determined minimum degree, vertex connectivity, independence number, and matching number.

Computed Laplacian spectrum and metric dimension.

Analyzed detour properties of the commuting graph.

Abstract

The commuting graph $\Delta(G)$ of a finite non-abelian group $G$ is a simple graph with vertex set $G$ and two distinct vertices $x, y$ are adjacent if $xy = yx$. In this paper, among some properties of $\Delta(G)$, we investigate $\Delta(SD_{8n})$ the commuting graph of the semidihedral group $SD_{8n}$. In this connection, we discuss various graph invariants of $\Delta(SD_{8n})$ including minimum degree, vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $\Delta(SD_{8n})$.