Equiangular lines with a fixed angle

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0< α < 1. Let _α(d) denote the maximum number of lines through the origin in Rᵈ with pairwise common angle arccos\ α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1-α)/(2α). If k < ∞, then N_α(d) = k(d-1)/(k-1) for all sufficiently large d, and otherwise N_α(d) = d+ o(d). In particular, N1/(2k-1)(d) = k(d-1)/(k-1) for every integer k≥ 2 and all sufficiently large d. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.