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...

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Veröffentlicht in:arXiv.org 2017-09
Hauptverfasser: Richey, Joseph, Shutty, Noah, Stover, Matthew
Format: Artikel
Sprache:eng
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Zusammenfassung: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 \(\mu_1(X) < (2 - \epsilon) \sqrt{k - 1}\).
ISSN:2331-8422
DOI:10.48550/arxiv.1306.6548