Spherical two-distance sets and graph eigenvalues

Let Nα,β(d) denote the maximum size of a spherical two-distance set in Rᵈ such that the inner products of distinct vectors only take α and β. By considering the correspondence between spherical two-distance sets and graphs with specific spectral properties, we determine Nα,β(d) for fixed -1≤β<0≤α<1 and sufficiently large d, which extends the work in [Equiangular lines with a fixed angle, Ann. Math. 194 (2021) 729-743] and [Spherical two-distance sets and eigenvalues of signed graphs, Combinatorica 43 (2023) 203-232]. The limit limd→∞Nα,β(d)/d is also derived from our work, that is, the problem of determining limd→∞Nα,β(d)/d for any fixed -1≤β<0≤α<1 is completely solved in the spherical \α,β\-code setting.