Spectral properties of Cayley graphs 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?
 A: This paper, by A. Valette, is a survey devoted to this question, although he's more interested in infinite groups. In the infinite case, the "adjacency matrix" is a bounded operator on $\ell^2(\Gamma)$, and its spectrum makes sense. Of course, it depends on the generating set.
One of the first results he mentions is a theorem of Kesten : it is possible to recover the fact that $G$ is amenable, or free, by looking at this spectrum.
A: Just to keep the references close to the question, here are two computations of the spectra of a Cayley graph:


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*Lovász, L. Spectra of graphs with transitive groups. Period. Math. Hungar.  6  (1975), no. 2, 191--195. MR0398886

*Babai, L. Spectra of Cayley graphs. J. Combin. Theory Ser. B  27  (1979), no. 2, 180--189. MR0546860
(The prevalence of people named László in this list is interesting. It reminds me of a little story I recently got from Wikipedia while hunting for a reference on the Higman-Sims group: the extraordinary fact that two people named 'Higman' discovered the same sporadic simple group!)
A: Just to keep the reference close to the question, I think that the manuscript of Petteri Kaski, Eigenvectors and Spectra of Cayley graphs,Helsinki university of technology, Spring Term 2002, is a very good reference.
A: I know at least one special case where your second question makes sense.  If $G$ is a compact group, it has a category $\text{Rep}(G)$ of finite-dimensional unitary representations which break up into direct sums of irreducible representations.  Fix a representation $V$ such that every irreducible representation appears in $V^{\otimes n}$ for some $n$.  One can construct a graph $\Gamma(V)$ whose vertices are the irreducible representations of $G$ and where the number of edges from $A$ to $B$ is the multiplicity by which $B$ appears in $A \otimes V$.  By the assumption, $\Gamma(V)$ is connected, and its combinatorial properties encode information about the behavior of the tensor powers of $V$, hence behavior about $G$.  
When $G$ is finite, this graph has the property that its eigenvalues are precisely the character values $\chi_V(g)$ as $g$ runs through all conjugacy classes.  But the great thing is that this statement still makes sense even when $G$ is infinite in a sense which is made precise in this blog post.
Finally, if $G$ is abelian, all of the finite-dimensional irreducible representations are one-dimensional.  They can be identified with the Pontryagin dual $G^{\vee}$, which is discrete, and $\Gamma(V)$ becomes precisely the Cayley graph of $G^{\vee}$ with respect to the generators that make up $V$!  So this is one sense in which the Cayley graph of an infinite group gives you algebraic data, but about its dual group.
A: There is the paper Property (T) and Kazhdan constants for discrete groups by Andrzej Żuk, which gives a sufficient criterion for property (T) in terms of some spectral properties of a graph depending on a group $G$ with a generating set $S$.  This graph is not the Cayley graph. But maybe it is still in the spirit of the question.
A: Just for other reference, we showed that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order $4$, $8$, $16$ or $9$. You can find more results in the below link:
https://www.worldscientific.com/doi/10.1142/S0219498816501759
In these cases, the DS property of Cayley graphs of a group can say something about its structure.
