On Representations of Graphs as Two-Distance Sets

TL;DR

This paper provides explicit formulas for representing graphs as two-distance sets in Euclidean space, spherical space, and J-spherical space, based on eigenvalues of adjacency matrices, simplifying previous complex results.

Contribution

The paper derives simple, exact formulas for the minimal dimensions of Euclidean, spherical, and J-spherical representations of graphs, extending and simplifying prior research.

Findings

Formulas for dim_E(G) and dim_S(G) in terms of eigenvalues of V^TAV.

Explicit criteria for J-spherical representations based on the largest eigenvalue of the complement graph.

Answers to previously posed questions about graph representations.

Abstract

Let a \neq b be two positive scalars. A Euclidean representation of a simple graph G in R^r is a mapping of the nodes of G into points in R^r such that the squared Euclidean distance between any two points is a if the corresponding nodes are adjacent and b otherwise. A Euclidean representation is spherical if the points lie on an (r-1)-sphere, and is J-spherical if this sphere has radius 1 and a=2 < b. Let dim_E(G), dim_S(G) and dim_J(G) denote, respectively, the smallest dimension r for which G admits a Euclidean, spherical and J-spherical representation. In this paper, we extend and simplify the results of Roy[18] and Nozaki and shinohara[17] by deriving exact simple formulas for dim_E(G) and dim_S(G) in terms of the eigenvalues of V^TAV, where A is the adjacency matrix of G and V is the matrix whose columns form an orthonormal basis for the orthogonal complement of the vector of all 1's. We also extend and simplify the results of Musin [16] by deriving explicit formulas for determining the J-spherical representation of G and for determining dim_J(G)in terms of the largest eigenvalue of \bar{A}, the adjacency matrix of the complement graph \bar{G}. As a byproduct, we obtain several related results and in particular we answer a question raised by Musin in [16].