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 non-isomorphic Cayley graphs of the finite simple groups $\operatorname{PSL}_d({\mathbb F}_q)$ for every $d \geq 5$ ($d \neq 6$) and prime power $q > 2$.