Explicit Bounds from the Alon–Boppana Theorem

Abstract

© 2018, © 2018 Taylor & Francis. The purpose of this article 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 μ1(X) the second largest eigenvalue of its adjacency matrix. It follows from the well-known Alon–Boppana theorem that for any ε > 0 there are only finitely many such X with μ1(X) < (2 − ϵ)√k − 1, and we effectively implement Serre's quantitative version of this result. For any k and ε, this gives an explicit upper bound on the number of vertices in a k-regular graph with μ1(X) < (2 − ϵ)√k − 1).