  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 $q > 2$.
