  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.
