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On the Gomori-Hu inequality (1211.2389v1)
Published 11 Nov 2012 in math.MG
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.