# 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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