Let $G$ be a finite group. Do eigenvalues of its Cayley graph say anything about the algebraic properties of $G$? The spectrum of Cayley graph may depend on the presentation, so it's not a good invariant, but maybe something interesting can still be said here?

In the case of an infinite group, can Cayley graph be replaced by some suitable infinite-dimensional object (say, linear operator, a generalization of the graph's adjacency matrix) so that the object's spectral properties may carry some algebraic data about the group?