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 $O(sqrt{D})$.