Isospectral Cayley graphs of some finite simple groups

Abstract

We apply spectral analysis of quotients of the Bruhat-Tits buildings of type ${\tilde{A}}_{d-1}$ to construct isospectral nonisomorphic Cayley graphs of the finite simple groups ${\mathrm{PSL}}_d({\mathbb{F}}_q)$ for every $d\ge 5$ ($d\ne 6$) and prime power $q>2$