On the extreme eigenvalues of regular graphs

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 $\epsilon >0$, 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 $\mu \le -(2-\epsilon )\sqrt{k-1}$ is at least $c|X|$. This implies a result of Winnie Li.

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