On the Gomori-Hu inequality

TL;DR

This paper explores the conditions under which the number of distinct distances in finite ultrametric spaces reaches the maximum possible, characterizing such spaces and analyzing their density in the space of all compact ultrametric spaces.

Contribution

It characterizes ultrametric spaces that attain equality in Gomori-Hu inequality and shows their density in the Gromov-Hausdorff space.

Findings

Spaces with maximum distinct distances are characterized by graph structures.

Such spaces are dense in the Gromov-Hausdorff space of compact ultrametric spaces.

The paper provides structural insights into ultrametric spaces with extremal properties.

Abstract

It was proved by Gomori and Hu in 1961 that for every finite nonempty ultrametric space $(X,d)$ the following inequality $|\Sp(X)|\leqslant |X|-1$ holds with $\Sp(X)=\{d(x,y):x,y \in X, x\neq y\}$. We characterize the spaces $X$, for which the equality in this inequality is attained by the structural properties of some graphs and show that the set of isometric types of such $X$ is dense in the Gromov-Hausdorff space of the compact ultrametric spaces.