Matrix Invariants of Finite Metric Spaces

Abstract

Finite metric spaces are characterized by a polyhedral cone defined in terms of the positivity of the distance functions and the triangle inequalities. Their classification is based on the decomposition of an associated polyhedral cone, called the "metric fan". The complete classification of $n$-point metric spaces is available only for $n\le 6$. As the number of classes increases rapidly with the number of elements, it is desirable to have coarser equivalence class decompositions based on certain invariants of finite metric spaces. If $(X,d)$ is a finite metric space with elements $P_i$ and with distance functions $d_{ij}$, the Gromov product at $P_i$ is defined as $\Delta_{ijk}=1/2(d_{ij}+d_{ik}-d_{jk})$. Assuming that the set of Gromov product at $P_i$ has a unique smallest element $\Delta_{ijk}$, the association of the edge $P_jP_k$ to $P_i$ defines the "Gromov product structure". The "pendant-free" reduction of the finite metric space is the graph obtained by removing the edges $P_jP_k$ corresponding to the minimal Gromov products $\Delta_{ijk}$ at $P_i$. In the present work, we define a matrix representation for a Gromov product structure $S$ on an $n$-point metric space, by $n\times n$ matrix $G_S$. We prove that if two metric spaces have Gromov product structures that can be mapped to each other by a permutation of the indices, then their matrices are similar via the corresponding permutation matrix. Matrix invariants of $G_S$ are used to define subclasses of Gromov product structures and their application to $n=5$ and $n=6$-point spaces are given.