Spherical two-distance sets and eigenvalues of signed graphs

2020/06/11 by Zilin Jiang, Jonathan Tidor, Jiang, Zilin +7
Materials Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Point processes and geometric inequalities #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2006.06633

openalex publication_date 2020/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let N_α,β(d) denote the maximum number of unit vectors in \mathbb R^d where all pairwise inner products lie in \α,β\. For fixed -1≤β<0≤α<1, we propose a conjecture for the limit of N_α,β(d)/d as d → ∞ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when α+2β<0 or (1-α)/(α-β) ∈ \1, √(2), √(3)\. Our work builds on our recent resolution of the problem in the case of α= -β (corresponding to equiangular lines). It is the first determination of lim_{d → ∞} N_α,β(d)/d for any nontrivial fixed values of α and β outside of the equiangular lines setting.

Related

• Spherical two-distance sets and graph eigenvalues
• Bounds on two-distance sets in Euclidean space and Unit Sphere
• Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below
• Upper bounds for s-distance sets and equiangular lines
• Spectral conditions for spherical two-distance sets