Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set $S_2$).
If we assume $\Gamma_1,\Gamma_2$ are isospectral, what restrictions does this place on the underlying groups $G_1$ and $G_2$?
Conversely, given two groups $G_1$ and $G_2$ is there a choice of $S_1$ and $S_2$ such that $\Gamma_1$ and $\Gamma_2$ are isospectral? If the answer is in general no, are there known conditions on $G_1$ and $G_2$ (other than $G_1=G_2$) that guarantee an affirmative answer?
(As noted in the comments below, the converse has an obvious answer if the groups are finite, so for these questions we can assume that $G_1$ and $G_2$ are infinite, while keeping the restriction that $S_1$ and $S_2$ are finite).
Finally,
- What can be said about the expansion properties of $\Gamma_1$ and $\Gamma_2$ if they are isospectral?