Laplacian Spectrum of Super Graphs defined on Certain Non-abelian Groups

TL;DR

This paper investigates the Laplacian spectra of various super graphs constructed on non-abelian groups, extending previous work and proving these graphs are L-integral.

Contribution

It extends the spectral analysis to conjugacy superenhanced power graphs and supercommuting graphs of specific non-abelian groups, including semidihedral groups.

Findings

Laplacian spectra of conjugacy superenhanced power graphs are obtained.

Laplacian spectra of conjugacy supercommuting graphs are determined.

The graphs studied are proven to be L-integral.

Abstract

Given a graph $A$ on a group $G$ and an equivalence relation $B$ on $G$, the $B$ super$A$ graph, whose vertex set is $G$ and two vertices $g$, $h$ are adjacent if and only if there exist $g^{\prime} \in[g]$ and $h^{\prime} \in[h]$ such that $g^{\prime}$ and $h^{\prime}$ are adjacent in $A$. Recently, Dalal \emph{et al.} (Spectrum of super commuting graphs of some finite groups, \textit{Computational and Applied Mathematics}, 43(6):348, 2024) obtain the Laplacian spectrum of supercommuting graphs of certain non-abelian groups including the dihedral group and the generalized quaternion group. In this paper, we continue the study of Laplacian spectrum of certian $B$ super$A$ graphs. We obtain the Laplacian spectrum of conjugacy superenhanced power graphs of certain non-abelian groups, namely: dihedral group, generalized quaternion group and semidihedral group. Moreover to enhance the work of Dalal \emph{et al}, we obtain the Laplacian spectrum of conjugacy supercommuting graph of semidihedral group. We prove that graphs considered in this paper are $L$-integral.