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Closed geodesics and eigenvalues in a non-regular graph Answer

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Closed geodesics and eigenvalues in a non-regular graph

Let $Gamma$ breathe a graph the diploma of whose $n$ vertices is $leq D$ with out essentially being fixed. Say we’ve got bounds of kind $leq gamma^{2 ok}$ for the variety of closed geodesics of size $2 ok$ for any sizable $ok$, for some $gamma$. Can we sure the non-trivial eigenvalues of the adjacency matrix $A$ of $Gamma$?

(If the diploma have been fixed, this may breathe simple, by way of the Ihara zeta duty and/or Hashimoto’s operator. When the diploma is non-constant, the relation between the Ihara zeta duty, on the one hand, and the eigenvalues of $A$, on the opposite, is much less antiseptic.)

If it helps, you possibly can occupy $gamma$ is of measurement $O(sqrt{D})$.

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