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 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 $\mu$ of $X$ such that $\mu \le -(2-ϵ)\sqrt{k-1}$ is at least $c\mathrm{\mid }X\mathrm{\mid }$. This implies a result of Winnie Li.