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 $0<\alpha <1$. Let $N_{\alpha }(d)$ denote the maximum number of lines through the origin in ${\mathbb{R}}^d$ with pairwise common angle $\arccos \alpha$\mathrm{.}Let$k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-\alpha )\mathrm{/}(2\alpha )$. If $k<\mathrm{\infty }$, then $N_\alpha(d) = \lfloor k(d-1)/(k-1) \rfloor$forallsufficientlylarge$d$,andotherwise$N_\alpha(d) = d + o(d)$\mathrm{.}Inparticular,$N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$foreveryinteger$k\ge 2$andallsufficientlylarge$d$. 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.