  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.
