2011/10/04 by D. Dordovskyi, O. Dovgoshey, E. Petrov · 1 citation
Mathematics · #math.MG #msc:54E35
published as P-adic Numbers, Ultrametric Analysis and Applications, 3:4, 2011, 253-262
arxiv created 2011/10/04 · arxiv updated 2011/11/01
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∈ 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.