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 ℱ(λ) be the family of connected graphs of spectral radius ≤ λ. We show that ℱ(λ) can be defined by a finite set of forbidden subgraphs if and only if λ<λ*:=2+5≈2.058 and λ ∉ {α₂, α₃, …}, where αm=βm1/2+βm−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 ℱ(λ) 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_α(n) = c_αn+ O_α(1). In particular, α≥11+2λ*. Besides we show that N1/3(n)=2n+O(1)andN1/5(n),N1/(1+22)(n)=32n+O(1)., which improves a recent result of Balla, Dräxler, Keevash and Sudakov.