Graphs and spherical two-distance sets

TL;DR

This paper explores the embedding of graphs as two-distance sets in Euclidean and spherical spaces, providing exact formulas for their representation numbers using polynomial multiplicities and extending results to graph joins.

Contribution

It introduces explicit formulas for Euclidean and spherical representation numbers of graphs, including joins, based on polynomial multiplicities and Caley-Menger determinants.

Findings

Exact formulas for Euclidean and spherical representation numbers.

Extension of Kuperberg's theorem to graph joins.

Characterization of graph embeddings via polynomial multiplicities.

Abstract

Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical representation numbers of G and give exact formulas for these numbers using multiplicities of polynomials that are defined by the Caley-Menger determinant. One of the main results of the paper are explicit formulas for the representation numbers of the join of graphs which are obtained from W. Kuperberg's type theorem for two-distance sets.