Equiangular lines with a fixed angle

Abstract

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 $0<\alpha <1$. Let $N_{\alpha }(d)$ denote the maximum number of lines through the origin in ${\mathbb{R}}^d$ with pairwise common angle $\arccos \alpha$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-\alpha )/(2\alpha )$. If $k<\infty$, then $N_{\alpha }(d)=\lfloor k(d-1)/(k-1)\rfloor$ for all sufficiently large $d$, and otherwise $N_{\alpha }(d)=d+o(d)$. In particular, $N_{1/(2k-1)}(d)=\lfloor k(d-1)/(k-1)\rfloor$ for every integer $k\ge 2$ 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.

11 pages. Fixed a minor issue at the end of the proof of Theorem 1.2

References in corpus (2)

• Eigenvalue multiplicity and volume growth
• Spherical two-distance sets and eigenvalues of signed graphs

Cited by in corpus (11)

• Discrete and Continuous Welch Bounds for Banach Spaces with Applications
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• Robust Eigenvectors of Symmetric Tensors
• The Lemmens-Seidel conjecture and forbidden subgraphs
• Equiangular lines and the Lemmens-Seidel conjecture
• Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below
• Bounds for the sum of distances of spherical sets of small size
• Support of Closed Walks and Second Eigenvalue Multiplicity of the Normalized Adjacency Matrix
• p-adic Welch Bounds and p-adic Zauner Conjecture