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 Rr is a mapping of the nodes of G into points in Rr such that the squared Euclidean distance between any two distinct 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 dimE(G), dimS(G) and dimJ(G) denote, respectively, the smallest dimension r for which graph G admits a Euclidean, spherical and J-spherical representation.In this paper, we extend and simplify the results of Roy (2010) and Nozaki and Shinohara (2012) by deriving exact simple formulas for dimE(G) and dimS(G) in terms of the eigenvalues of VTAV, 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 (2018) by deriving explicit formulas for determining the J-spherical representation of G and for determining dimJ(G) in terms of the largest eigenvalue of Ā, the adjacency matrix of the complement graph Ḡ. As a by-product, we obtain several other related results and in particular we answer a question raised by Musin in Musin (2018).