Diameter and diametrical pairs of points in ultrametric spaces
Metric Geometry
2011-11-01 v1
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.
Keywords
Cite
@article{arxiv.1110.0781,
title = {Diameter and diametrical pairs of points in ultrametric spaces},
author = {D. Dordovskyi and O. Dovgoshey and E. Petrov},
journal= {arXiv preprint arXiv:1110.0781},
year = {2011}
}