Combinatorica
On the second eigenvalue of a graph
Discrete Mathematics
Spectra of regular graphs and hypergraphs and orthogonal polynomials
European Journal of Combinatorics
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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 0, there exist a positive constant c = c(ε, k) and a non-negative integer g = g(ε, 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 ≤ - (2 - ε) √k - 1 is at least c|X|. This implies a result of Winnie Li.