Spectral conditions for spherical two-distance sets

TL;DR

This paper characterizes spherical two-distance sets in Euclidean space using spectral graph theory, providing conditions and bounds for their existence and minimal embedding dimension.

Contribution

It introduces spectral characterizations of spherical 2-distance sets and determines the minimal dimension for their representation based on graph spectra.

Findings

Spectral conditions characterize spherical 2-distance sets.

The minimal embedding dimension is determined by the graph spectrum.

The approach links geometric configurations with spectral graph theory.

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.