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 $\epsilon>0$, 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.Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some comments on the constant c in Section

Equiangular Lines and Spherical Codes in Euclidean Space

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.Comment: 24 pages, 0 figure