Main functions and the spectrum of super graphs

TL;DR

This paper introduces a new class of supergraphs based on group equivalence relations, computes their spectra for specific groups, and shows these graphs are generally not integral, expanding understanding of graph spectra in algebraic structures.

Contribution

It defines superA graphs based on group equivalence relations and computes their spectra for dihedral and dicyclic groups, revealing they are not integral.

Findings

Spectra of equality/conjugacy supercommuting graphs for dihedral groups

Spectra of these graphs for dicyclic groups

These graphs are not integral

Abstract

Let A be a graph type and B an equivalence relation on a group $G$. Let $[g]$ be the equivalence class of $g$ with respect to the equivalence relation B. The B superA graph of $G$ is an undirected graph whose vertex set is $G$ and two distinct vertices $g, h \in G$ are adjacent if $[g] = [h]$ or there exist $x \in [g]$ and $y \in [h]$ such that $x$ and $y$ are adjacent in the A graph of $G$. In this paper, we compute spectrum of equality/conjugacy supercommuting graphs of dihedral/dicyclic groups and show that these graphs are not integral.