Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any hints or references would be appreciated?
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$\begingroup$ The spectrum of any Cayley graph is determined by Babai's paper (sciencedirect.com/science/article/pii/0095895679900790). $\endgroup$– Gordon RoyleCommented Apr 30, 2016 at 13:50
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I don't think there any deep relations. The eigenvalues are integers, whence the number of distinct eigenvalues is at most $2|C|+1$ (and you can apply trhis to the complement). If the diameter is $d$, then the number of distinct eigenvalues is at least $d+1$. (Since a shortest path between two vertices must use each element of $C$ at most once, $d\le|C|$.)
answered Apr 30, 2016 at 13:56