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Summary: On the Set of Realizations of
Edge-Weighted Graphs in Euclidean Spaces
Abdo Y. Alfakih
Department of Mathematics and Statistics
University of Windsor
Windsor, Ontario N9B 3P4
Canada
February 22, 2005
AMS classi cation: 51K05, 90C22, 52A20, 05C50, 15A57
Keywords: graph realizations, Euclidean distance matrices, semide nite program-
ming, low rank solutions, Gale transform, convex sets.
Abstract
Let G = (V; E; !) be an edge-weighted 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 =
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