 
Summary: On the Set of Realizations of
EdgeWeighted Graphs in Euclidean Spaces
Abdo Y. Alfakih
Department of Mathematics and Statistics
University of Windsor
Windsor, Ontario N9B 3P4
Canada
February 22, 2005
AMS classication: 51K05, 90C22, 52A20, 05C50, 15A57
Keywords: graph realizations, Euclidean distance matrices, semidenite program
ming, low rank solutions, Gale transform, convex sets.
Abstract
Let G = (V; E; !) be an edgeweighted graph. A realization of G in < r is
a mapping of the vertices 1; 2; : : : ; n of G into points p 1 ; p 2 ; : : : ; p n in < r such
that jjp i p j jj 2 = ! ij for every edge (i; j) 2 E. In this paper, we study the
geometry of the set of all realizations of G, and we present a simple randomized
algorithm for obtaining realizations of G in low dimensional Euclidean spaces.
Some numerical results on randomly generated problems are also presented.
1 Introduction
Let G = (V; E; !) be a given simple connected undirected graph, where V =
