Forbidden subgraphs for graphs of bounded spectral radius, with applications to equiangular lines

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}(\lambda )$ be the family of connected graphs of spectral radius $\le \lambda$. We show that $\mathcal{F}(\lambda )$ can be defined by a finite set of forbidden subgraphs if and only if $\lambda < \lambda^* := \sqrt{2+\sqrt{5}} \approx 2.058$and$\lambda \not\in {\alpha_2, \alpha_3, \dots}$,where$\alpha_m = \beta_m^{1/2} + \beta_m^{-1/2}$and$\beta_m$isthelargestrootof$x^{m+1}=1+x+\dots+x^{m-1}$. The study of forbidden subgraphs characterization for $\mathcal{F}(\lambda )$ is motivated by the problem of estimating the maximum cardinality of equiangular lines in the $n$-dimensional Euclidean space ${\mathbb{R}}^n$ — a family of lines through the origin such that the angle between any pair of them is the same. Denote by $N_{\alpha }(n)$ the maximum number of equiangular lines in ${\mathbb{R}}^n$ with angle $\arccos \alpha$. We establish the asymptotic formula $N_\alpha(n) = c_\alpha n + O_\alpha(1)$forevery$\alpha \ge \frac{1}{1+2\lambda^*}$. In particular, $N_{1\mathrm{/}3}(n)=2n+O(1)$ and $N_{1/5}(n), N_{1/(1+2\sqrt{2})}(n) = \frac{3}{2}n+O(1)$\mathrm{.}Besidesweshowthat$N_\alpha(n) \le 1.49n + O_\alpha(1)$forevery$\alpha \neq \tfrac{1}{3}, \tfrac{1}{5}, \tfrac{1}{1+2\sqrt{2}}$, which improves a recent result of Balla, Dr"axler, Keevash and Sudakov.