On the extreme eigenvalues of regular graphs

Details On the extreme eigenvalues of regular graphs On the extreme eigenvalues of regular graphs

2004-07-15

arxiv.org/abs/math/0407274v2

Mathematics

Combinatorics

accepted to J.Combin.Theory, Series B. added 5 new references, some comments on the constant c in Section 2

Scientific paper

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 $\epsilon>0$, there exist a positive constant $c=c(\epsilon,k)$ and a nonnegative integer $g=g(\epsilon,k)$ such that for any $k$-regular graph $X$ with no odd cycles of length less than $g$, the number of eigenvalues $\mu$ of $X$ such that $\mu \leq -(2-\epsilon)\sqrt{k-1}$ is at least $c|X|$. This implies a result of Winnie Li.

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