Gromov Product Decomposition of 7-point Metric Spaces

Abstract

Let $X$ be a finite metric space with elements $P_i$, $i=1,\dots ,n$ and with distance functions $d_{ij}$. The Gromov product of the triangle with vertices $P_i$, $P_j$ and $P_k$ at the vertex $P_i$ is defined by ${\Delta }_{ijk}=\frac{1}{2}(d_{ij}+d_{ik}-d_{jk})$. A metric space is called $\Delta$-generic, if the set of Gromov products at each $P_i$ has a unique smallest element ${\Delta }_{ijk}$. For a $\Delta$-generic metric space, the map $P_i\to (P_j{P}_k)$, where $(P_j{P}_k)$ is the edge joining $P_j$ to $P_k$ is a well defined map called the "Gromov product structure" [Bilge, Celik and Kocak, "An equivalence class decomposition of finite metric spaces", Discrete Mathmetics, Vol 340, (2017) 1928-1932]. For n=5, the 3 $\Delta$-equivalence classes coincide with the classification of $5$-point metrics. For $n=6$, there are 26 $\Delta$-equivalence classes refined by 339 metric classes. In this paper, we present the first systematic treatment of 7-point spaces and we obtain the $\Delta$-equivalence decomposition of 7-point metric spaces that consist of 431 equivalence classes.