Diameter and diametrical pairs of points in ultrametric spaces

Details Diameter and diametrical pairs of points in ultrametric spaces Diameter and diametrical pairs of points in ultrametric spaces

2011-10-04

arxiv.org/abs/1110.0781v1

P-adic Numbers, Ultrametric Analysis and Applications, 3:4, 2011, 253-262

Mathematics

Metric Geometry

Scientific paper

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.

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