We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $mathbb{Z}_2^d$ and $mathbb{Z}_
4^d$. We also provide some infinite families of Cayley graphs of $mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
A connected, locally finite graph $Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $Gamma$. Define the minimal degree of $G$ as the minimal
degree of a Cayley--Abels graph of $G$. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_d$ denotes the $d$-regular tree, then the minimal degree of ${rm Aut}(T_d)$ is $d$ for all $dgeq 2$.
