Generating Graphs of Finite Dihedral Groups

TL;DR

This paper investigates the generating graph of dihedral groups, analyzing its spectral properties and topological indices to deepen understanding of their algebraic and graph-theoretic structure.

Contribution

It provides a comprehensive spectral analysis and computes topological indices for the generating graphs of dihedral groups, a novel contribution in algebraic graph theory.

Findings

Complete spectrum of adjacency matrix of $(D_n)$ determined

Laplacian spectrum of $(D_n)$ computed

Distance and degree-based indices calculated

Abstract

For a group $G$, the generating graph $\Gamma(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $\Gamma(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where $D_n$ is a Dihedral group of order $2n$. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of $\Gamma(D_n)$. Moreover, we compute some distance and degree based topological indices of $\Gamma(D_n)$.