On the extreme eigenvalues of regular graphs

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 Serreʹs 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.