Spectral conditions for spherical two-distance sets
Iliyas Noman, Yuan Yao
TL;DR
This paper provides a clean spectral characterization of spherical 2-distance sets represented by graphs on $d+2$ vertices. By analyzing the adjacency matrix $A_G$ and the projection $P A_G P$, the authors establish that a spherical representation in $\mathbb{R}^d$ exists exactly when the largest eigenvalue of $P A_G P$ equals $\lambda_2$ and the multiplicities of $\lambda_2$ in $A_G$ and $P A_G P$ agree (excluding $\lambda_1$ if equal); the distance ratio is determined by $\lambda_2$ via $k = \sqrt{1/\lambda_2 + 1}$. They introduce a streamlined matrix formulation using $\overline{B_G} = \lambda_2 I - A_G$ and show that the positive-semidefinite and rank conditions reduce to eigenstructure constraints on $A_G$, enabling a minimal embedding dimension to be read off from the $\lambda_2$-multiplicity. The results unify and simplify prior bulky conditions, yield a practical spectral test for spherical representations, and imply that many small graphs admit spherical realizations, with implications for discrete geometry and coding-theory constructions. All math is presented with explicit spectral criteria, improving applicability for graph-based spherical representations.
Abstract
A set of points $S$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a 2-distance set if the set of pairwise distances between the points has cardinality two. The 2-distance set is called spherical if its points lie on the unit sphere in $\mathbb{R}^{d}$. We characterize the spherical 2-distance sets using the spectrum of the adjacency matrix of an associated graph and the spectrum of the projection of the adjacency matrix onto the orthogonal complement of the all-ones vector. We also determine the lowest dimensional space in which a given spherical 2-distance set could be represented using the graph spectrum.