Petrov Y. Continuation of pseudoultrametrics, some classes of extremal metric spaces and weak similarities

The problem of continuation of the weight defined on the edges of arbitrary graph to pseudoultrametric defined on the vertices of this graph is considered. Some classes of extremal metric spaces are investigated. In particular the class of finite ultrametric spaces for which the equality is attained in the Gomori-Hu inequality. The notion of weak similarity is introduced. The conditions under which weak similar metric spaces are homeomorphic are found as well as conditions under which weak similar ultrametric spaces are isometric. It is established that weak similar geodesic metric spaces are similar.