Explicit bounds from the Alon-Boppana theorem

The purpose of this paper is to give explicit methods for bounding the number of vertices of finite $k$-regular graphs with given second eigenvalue. Let $X$ be a finite $k$-regular graph and $\mu_1(X)$ the second largest eigenvalue of its adjacency matrix. It follows from the well-known Alon-Boppana Theorem, that for any $\epsilon > 0$ there are only finitely many such $X$ with $\mu_1(X) < (2 - \epsilon) \sqrt{k - 1}$, and we effectively implement Serre's quantitative version of this result. For any $k$ and $\epsilon$, this gives an explicit upper bound on the number of vertices in a $k$-regular graph with $\mu_1(X) < (2 - \epsilon) \sqrt{k - 1}$.