Subdominant pseudoultrametric on graphs

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.

From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

“…Remark 3.5. The characterization of trees obtained in Proposition 3.4 is similar to the characterizations of trees which were given in Corollary 3.6 of [7] and Corollary 5 of [8].…”

Properties and Morphisms of Finite Ultrametric Spaces and Their Representing Trees

“…where l(u) and l(v) are defined by (1.1). Then ρ is the subdominant pseudoultrametric for the weight w (see [8] and Theorem 10.40 in [39] for details).…”

On ultrametric-preserving functions

“…The theory of ultrametric spaces is closely connected with various directions of investigations in mathematics, physics, linguistics, psychology and computer science. Different properties of ultrametric spaces have been studied in [3,9,[18][19][20]22,30,31,39,[43][44][45][46][47][48][49][54][55][56]61,62]. Note that the use of trees and tree-like structures gives a natural language for description of ultrametric spaces [2, 6, 10, 17, 23, 24, 26-28, 33, 36-38, 49-51].…”