Spherical two-distance sets and eigenvalues of signed graphs

Abstract

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting.