On representations of graphs as two-distance sets

Abstract

Let α = β 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 α if the corresponding nodes are adjacent and β 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 α = 2 < β. 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 T AV , 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 ofĀ, the adjacency matrix of the complement graphḠ. As a by-product, we obtain several other related results and in particular we answer a question raised by Musin in [16].