No abstract

It was proved by Gomori and Hu in 1961 that for every finite nonempty ultrametric space (X, d) the following inequality | Sp(X)| |X| − 1 holds with Sp(X) = {d(x, y) : x, y ∈ X, x = 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. MSC: 54E35, 37E25.Keywords: finite ultrametric space, strictly binary tree, complete bipartite graph, spectrum of the ultrametric space, map preserving the balls, Gromov-Hausdorff metric. О неравенстве Гомори -Ху Е. А. Петров и А. А. Довгошей Аннотация В 1961 году Гомори и Ху доказали, что для любого конечного непустого ультраметрического пространства (X, d) выполняется неравенство | Sp(X)| |X| − 1, где Sp(X) = {d(x, y) : x, y ∈ X, x = y}. Мы характеризуем пространства X, для которых достигается равенство, посредством структурных свойств некоторых графов и показываем, что множество типов изометрий таких X плотно в пространстве Громова-Хаусдорфа изометрических типов компактных ультраметрических пространств.2010 MSC: 54E35, 37E25.Ключевые слова: конечное ультраметрическое пространство, строго бинарное дерево, полный двудольный граф, спектр ультраметрического пространства, отображение сохраняющее шары, метрика Громова-Хаусдорфа.

No abstract

Let (G, w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G) → R + can be extended to a pseudoultrametric on V (G) are found. A criterion of the uniqueness of this extension is also obtained. It is proved that G is complete k-partite with k ≥ 2 if and only if, for every pseudoultrametrizable weight w, there exists the smallest pseudoultrametric agreed with w. We characterize the structure of graphs for which the subdominant pseudoultrametric is an ultrametric for every strictly positive pseudoultrametrizable weight.

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function τ : F(X) → R there is an ultrametric on X such that τ (A) = diam A for every A ∈ F(X). For finite nondegenerate ultrametric spaces (X, d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k 2, and, conversely, every finite complete k-partite graph with k 2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X, d) having the minimal card{(x, y) : d(x, y) = diam X, x, y ∈ X} for given card X. (2000): 54E35. Mathematics Subject Classification