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 $ε>0$, there exist a positive constant $c=c(ε,k)$ and a nonnegative 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 $μ\leq -(2-ε)\sqrt{k-1}$ is at least $c|X|$. This implies a result of Winnie Li.