Spherical two-distance sets and graph eigenvalues
Jiang Zhou
TL;DR
The paper develops a spectral-graph framework to study spherical two-distance sets, establishing precise correspondences between spherical {α,β}-codes and α-graphs/β-graphs via μ=(1−β)/(α−β) and Gram-rank representations. It then leverages graph eigenvalue properties and rank-perturbation lemmas to derive bounds on the maximal size N_{α,β}(d) of such codes, with both global (via G_{μ,r}) and local (via N(r,p,μ)) approaches. The contributions include explicit constructive converses between graphs and codes, as well as a suite of bounds that connect Turán-type results and hierarchical graph decompositions to the geometry of spherical two-distance sets. Overall, the work advances understanding of how spectral constraints limit spherical codes and provides tools for evaluating or bounding code sizes in arbitrary dimensions without fixing α,β a priori.
Abstract
A set of unit vectors in $\mathbb{R}^d$ is a called a spherical two-distance set if the inner products of distinct vectors only take two values. In this paper, we give explicit correspondence between spherical two-distance sets and graphs with specific spectral properties, and derive some bounds on the maximum size of spherical two-distance sets.