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 $0\lt \alpha \lt 1$. Let $_\alpha(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\mathrm{arccos}\ \alpha$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-\alpha)/(2\alpha)$. If $k \lt \infty$, then $N_\alpha(d) = \lfloor k(d-1)/(k-1)\rfloor$ for all sufficiently large $d$, and otherwise $N_\alpha(d) = d+ o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1)\rfloor$ for every integer $k\ge 2$ 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.