Spherical Two-Distance Sets and Eigenvalues of Signed Graphs

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-alpha,N- beta (d) denote the maximum number of unit vectors in R-d where all pairwise inner products lie in {alpha, beta}. For fixed -1 <= beta < 0 <= alpha < 1, we propose a conjecture for the limit of N-alpha,N- ss (d)/d as d -> infinity in terms of eigenvalue multiplicities of signed graphs. We determine this limit when alpha + 2 beta < 0 or (1 - alpha)/(alpha - beta) is an element of {1, root 2, root 3}. Our work builds on our recent resolution of the problem in the case of alpha = - beta (corresponding to equiangular lines). It is the first determination of lim d ->infinity N-alpha,N- beta (d)/d for any nontrivial fixed values of alpha and beta outside of the equiangular lines setting.