Diameter and diametrical pairs of points in ultrametric spaces

TL;DR

This paper characterizes when a function on finite subsets of a set can be realized as the diameter function of an ultrametric space, and describes the structure of finite ultrametric spaces via complete k-partite graphs.

Contribution

It provides necessary and sufficient conditions for a function to be a diameter function of an ultrametric space and characterizes finite ultrametric spaces using complete k-partite graphs.

Findings

Finite ultrametric spaces correspond to complete k-partite graphs with k>=2.

Conditions are established for functions to be realized as diameter functions.

Finite ultrametric spaces with minimal diametrical pairs are characterized.

Abstract

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 f:F(X)-->R there is an ultrametric on X such that f(A)=diam A for every A\in 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 \in X} for given card X.