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 |
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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 \(\mu_1(X) < (2 - \epsilon) \sqrt{k - 1}\). |
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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1306.6548 |