Bilge, Ayşe HümeyraBilge, Ayşe HümeyraÇelik, DeryaKoçak, Şahin2020-12-222020-12-22201720012-365X1872-681X0012-365X1872-681Xhttps://hdl.handle.net/20.500.12469/3625https://doi.org/10.1016/j.disc.2017.03.023Let (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.eninfo:eu-repo/semantics/closedAccessFinite metric spacesGromov productWeighted graphsArticle192819328340WOS:00040221110001610.1016/j.disc.2017.03.0232-s2.0-85018902568Q3Q1