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_α,β(d) denote the maximum number of unit vectors in R^d where all pairwise inner products lie in { α, β} . For fixed - 1 ≤ β< 0 ≤ α< 1 , we propose a conjecture for the limit of N_α,β(d) / d as d→ ∞ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when α+ 2 β< 0 or (1-α)/(α-β)∈{1,2,3} . 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→∞N_α,β(d) / d for any nontrivial fixed values of α and β outside of the equiangular lines setting.