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 τ : F(X) → R there is an ultrametric on X such that τ (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. (2000): 54E35. Mathematics Subject Classification