Suppose you have a finite group and you consider its Cayley graph with respect to some fixed generating set of nonidentity elements closed under inversion. Are there any results known to the effect that structural information about the group can be recovered from the spectrum of the adjacency matrix of the Cayley graph?
The eigenvalues of the adjacency matrix of the graph are images of the generating set $S$ under the left regular representation of the group $G$. One would think that this should give some information about the representations of $G$ and thus information about $G$. However, in any many cases, a graph which may be viewed as a Cayley graph can be viewed as a Cayley graph in many different ways, with different groups. For example, take the graph of the cube, with eigenvalues $\pm 1, \pm 3.$ I think it can be viewed as a Cayley graph in any of the five groups of order 8. So the spectrum of the adjacency matrix of the graph is a bit coarse?