2021/11/01 by Zilin Jiang, Jonathan Tidor, Yuan Yao +2 · 1 citation
Engineering · Mathematics · #graph theory and CDMA systems #Point processes and geometric inequalities #Graph theory and applications #Combinatorics #Mathematics #Adjacency matrix #Spectral radius #Equiangular polygon #Eigenvalues and eigenvectors #Sublinear function #Bounded function #Multiplicity (mathematics) #Vertex (graph theory) #Graph #Discrete mathematics #Geometry #Physics #Regular polygon #Mathematical analysis
paper · doi:10.4007/annals.2021.194.3.3
openalex publication_date 2021/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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\lt α \lt 1. Let α(d) denote the maximum number of lines through the origin in ℝd with pairwise common angle arccos α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1-α)/(2α). If k \lt ∞, then Nα(d) = \lfloor k(d-1)/(k-1)\rfloor for all sufficiently large d, and otherwise Nα(d) = d+ o(d). In particular, N1/(2k-1)(d) = \lfloor k(d-1)/(k-1)\rfloor for every integer k≥ 2 and all sufficiently large 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.