Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines

Abstract

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let ${\mathcal F}\left( \lambda \right)$ be the family of connected graphs of spectral radius ≤ λ. We show that ${\mathcal F}\left( \lambda \right)$ can be defined by a finite set of forbidden subgraphs if and only if $\lambda > \lambda *: = \sqrt {2 + \sqrt 5 } \approx 2.058$ and λ ∉ {α₂, α₃, …}, where ${\alpha _m} = \beta _m^{1/2} + \beta _m^{ - 1/2}$ and β_m is the largest root of x^m+1 = 1+ x + … + x^m−1. The study of forbidden subgraphs characterization for ${\mathcal F}\left( \lambda \right)$ is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n-dimensional Euclidean space ℝⁿ family of lines through the origin such that the angle between any pair of them is the same. Denote by N_α(n) the maximum number of equiangular lines in ℝⁿ with angle arccos α. We establish the asymptotic formula N_α(n) = c_αn + O_α(1) for every ${N_\alpha }\left( n \right) = {c_\alpha }n + {O_\alpha }\left( 1 \right)$. In particular, $\alpha \ge {1 \over {1 + 2\lambda *}}$.

Besides we show that

$${N_{1/3}}\left( n \right) = 2n + O\left( 1 \right)\quad {\rm{and}}\quad {N_{1/5}}\left( n \right),\,{N_{1/(1 + 2\sqrt 2 )}}(n) = {3 \over 2}n\, + O\left( 1 \right).$$

, which improves a recent result of Balla, Dräxler, Keevash and Sudakov.