# Closed geodesics and eigenvalues in a non-regular graph

Let $$Gamma$$ be a graph the degree of whose $$n$$ vertices is $$leq D$$ without necessarily being constant. Say we have bounds of type $$leq gamma^{2 k}$$ for the number of closed geodesics of length $$2 k$$ for any large $$k$$, for some $$gamma$$. Can we bound the non-trivial eigenvalues of the adjacency matrix $$A$$ of $$Gamma$$?

(If the degree were constant, this would be easy, via the Ihara zeta function and/or Hashimoto’s operator. When the degree is non-constant, the relation between the Ihara zeta function, on the one hand, and the eigenvalues of $$A$$, on the other, is less clean.)

If it helps, you can assume $$gamma$$ is of size $$O(sqrt{D})$$.