On super integral groups

TL;DR

This paper investigates super integral groups, characterized by integer spectra of their commuting graphs, and identifies conditions under which various finite groups exhibit this property, including specific families and graph topologies.

Contribution

It computes spectra for several group families, establishes super integrality for n-centralizer and certain commutativity degrees, and links graph topology to super integrality.

Findings

Certain finite groups are proven to be super integral.

Super integrality is linked to group properties like centralizers and commutativity degree.

Planar and toroidal commuting graphs imply super integrality.

Abstract

A finite non-abelian group $G$ is called super integral if the spectrum, Laplacian spectrum and signless Laplacian spectrum of its commuting graph contain only integers. In this paper, we first compute various spectra of several families of finite non-abelian groups and conclude that those groups are super integral. As an application of our results we obtain some positive integers $n$ such that $n$-centralizer groups are super integral. We also obtain some positive rational numbers $r$ such that $G$ is super integral if it has commutativity degree $r$. In the last section, we show that $G$ is super integral if $G$ is not isomorphic to $S_4$ and its commuting graph is planar. We conclude the paper showing that $G$ is super integral if its commuting graph is toroidal.