An integral family of quasi-strongly regular Cayley graphs

TL;DR

This work investigates the spectral properties of a family of quasi-strongly regular Cayley graphs $\Gamma_H(G)$ built from a finite group $G$ and a subgroup $H$. It proves that $\Gamma_H(G)$ is integral when $H$ is normal in $G$ and, in the normal case, determines the complete spectrum, showing the eigenvalues depend only on $|G|$ and the index $[G:H]$, with isospectrality across groups of the same order and index. The results illuminate the interplay between group representation theory and graph spectra for higher-grade QSRGs, and they raise open questions about the necessity of normality for integrality and broader spectral structure. The methods combine atoms, Eulerian subsets, and irreducible character techniques to obtain explicit eigenvalue formulas and multiplicities, contributing new understanding to the spectrum of non-strongly-regular Cayley graphs.

Abstract

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $Γ_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include a sufficient condition for $Γ_H(G)$ to be integral and an explicit computation of its entire spectrum when $H$ is normal, revealing that the spectrum in this case depends only on $|G|$ and the index $[G:H]$.