  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 $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the maximum number of lines
through the origin in $\mathbb{R}^d$ with pairwise common angle $\arccos
\alpha$. 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 $k < \infty$, 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.
