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arXiv · 1306.6548

Explicit bounds from the Alon-Boppana theorem

Joseph Richey·Noah Shutty·Matthew Stover
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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 $μ_1(X)$ 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 $μ_1(X) < (2 - ε) \sqrt{k - 1}$, 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 $μ_1(X) < (2 - ε) \sqrt{k - 1}$.

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BibTeXRIS

Joseph Richey, Noah Shutty, Matthew Stover. 2017-09-15. Explicit bounds from the Alon-Boppana theorem. https://doi.org/10.1080/10586458.2017.1311813

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