On the extreme eigenvalues of regular graphs

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Abstract

In this paper, we present an elementary proof of a theorem of Serre concerning the greatest eigenvalues of k-regular graphs. We also prove an analogue of Serre's theorem regarding the least eigenvalues of k-regular graphs: given ε{lunate} > 0, there exist a positive constant c = c ( ε{lunate}, k ) and a non-negative integer g = g ( ε{lunate}, k ) such that for any k-regular graph X with no odd cycles of length less than g, the number of eigenvalues μ of X such that μ {less-than or slanted equal to} - ( 2 - ε{lunate} ) sqrt(k - 1) is at least c | X |. This implies a result of Winnie Li. © 2005 Elsevier Inc. All rights reserved.

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Cioabǎ, S. M. (2006). On the extreme eigenvalues of regular graphs. Journal of Combinatorial Theory. Series B, 96(3), 367–373. https://doi.org/10.1016/j.jctb.2005.09.002

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