An integral family of quasi-strongly regular Cayley graphs

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