Explicit bounds from the Alon-Boppana theorem

Abstract

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 $\hat{I}{¼}_1(X)$ the second largest eigenvalue of its adjacency matrix. It follows from the well-known Alon-Boppana Theorem, that for any $\hat{I}\mu >0$ there are only finitely many such $X$ with $\hat{I}{¼}_1(X)<(2-\hat{I}\mu )\sqrt{k-1}$, and we effectively implement Serre's quantitative version of this result. For any $k$ and $\hat{I}\mu$, this gives an explicit upper bound on the number of vertices in a $k$-regular graph with $\hat{I}{¼}_1(X)<(2-\hat{I}\mu )\sqrt{k-1}$.

To appear in Exp. Math