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

Zilin Jiang, Alexandr Polyanskii

Published 2017-08-07Version 1

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 for every $\lambda < \lambda^* := \sqrt{2+\sqrt{5}} \approx 2.058$, whereas $\mathcal{F}(\lambda)$ cannot for every $\lambda \ge \lambda^*$. 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 $\mathcal{R}^n$ --- 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 $\mathcal{R}^n$ with angle $\arccos\alpha$. We establish an approach to determine the constant $c$ such that $N_\alpha(n) = cn + O_\alpha(1)$ for every $\alpha > \frac{1}{1+2\lambda^*}$. We also show that $N_\alpha(n) \le 1.81n + O_\alpha(1)$ for every $\alpha \neq \tfrac{1}{3}$, which improves a recent result of Balla, Dr\"axler, Keevash and Sudakov.