An integral family of quasi-strongly regular Cayley graphs

TL;DR

This paper investigates a family of quasi-strongly regular Cayley graphs derived from finite groups and subgroups, providing conditions for integrality and explicit spectra when the subgroup is normal.

Contribution

It offers a sufficient condition for the graphs to be integral and explicitly computes their spectra in the normal subgroup case, advancing understanding of their spectral properties.

Findings

Provided a criterion for integrality of the graphs.

Explicit spectrum calculation when H is normal.

Spectrum depends only on group order and subgroup index.

Abstract

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include a sufficient condition for $\Gamma_H(G)$ 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 $[G:H]$.