The Generating graph of Dicyclic Groups

TL;DR

This paper investigates the structural and spectral properties of the generating graphs of dicyclic groups, providing explicit spectra for various associated matrices and comparing them with dihedral groups.

Contribution

It offers a comprehensive spectral analysis of the generating graphs of dicyclic groups and dihedral groups, including complete spectra of key matrices.

Findings

Spectra of adjacency, Laplacian, distance, and eccentricity matrices for $ ext{Gamma}(Q_n)$ determined.

Complete spectra of distance and eccentricity matrices for dihedral groups $D_n$ obtained.

Insights into graph characteristics and spectral properties of these groups' generating graphs.

Abstract

For a group $G,$ the generating graph of $G,$ denoted by $\Gamma(G).$ We define $Q_n=\langle x,y: x^{2n}=y^4=1, x^n=y^2,y^{-1}xy=x^{-1}\rangle,$ the dicyclic group of order $4n.$ This paper primarily delves into exploring the graph characteristics and spectral properties of various matrices associated with $\Gamma(Q_n)$. Specifically, we determine the complete spectrum of the adjacency, Laplacian, distance, and eccentricity matrices. Additionally, we completely determine the spectrum pertaining to the distance and eccentricity matrices of the dihedral group of order $2n$, denoted as $D_n$.