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    Explicit Bounds from the Alon–Boppana Theorem

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    Genre
    Pre-print
    Date
    2018-10-02
    Author
    Richey, J
    Shutty, N
    Stover, M
    Subject
    Graph spectrum
    graph Laplacian
    Alon-Boppana theorem
    spectra of regular graphs
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/4908
    
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    DOI
    10.1080/10586458.2017.1311813
    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).
    Citation to related work
    Informa UK Limited
    Has part
    Experimental Mathematics
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    ae974a485f413a2113503eed53cd6c53
    http://dx.doi.org/10.34944/dspace/4890
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