An equivalence class decomposition of finite metric spaces via Gromov products

BİLGE A. H., ÇELİK D., Kocak S.

DISCRETE MATHEMATICS, vol.340, no.8, pp.1928-1932, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 340 Issue: 8
  • Publication Date: 2017
  • Doi Number: 10.1016/j.disc.2017.03.023
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1928-1932
  • Keywords: Finite metric spaces, Gromov product, Weighted graphs
  • Anadolu University Affiliated: Yes


Let (X, d) be a finite metric space with elements P-i, i = 1,..., n and with the distance functions d(ij) The Gromov Product of the "triangle" (P-i, P-j, P-k) with vertices P-t, P-j and P-k at the vertex Pi is defined by Delta(ijk) = 1/2(d(ij) + d(ik) - d(jk)). We show that the collection of Gromov products determines the metric. We call a metric space Delta-generic, if the set of all Gromov products at a fixed vertex P-i has a unique smallest element (for i = 1,., n). We consider the function assigning to each vertex P-i the edge {P-i, P-k} of the triangle (P-i, P-j, P-k) realizing the minimal Gromov product at P-i and we call this function the Gromov product structure of the metric space (X, d). We say two Delta-generic metric spaces (X, d) and (X, d') to be Gromov product equivalent, if the corresponding Gromov product structures are the same up to a permutation of X. For n = 3, 4 there is one (Delta-generic) Gromov equivalence class and for n = 5 there are three (Delta-generic) Gromov equivalence classes. For n = 6 we show by computer that there are 26 distinct (Delta-generic) Gromov equivalence classes. (C) 2017 Elsevier B.V. All rights reserved.