Ultrametrics and complete multipartite graphs

TL;DR

This paper characterizes ultrametric spaces through the structure of their diametrical graphs, specifically using complete multipartite graphs, and explores their properties in various topological and bounded contexts.

Contribution

It provides a new characterization of ultrametric spaces via complete multipartite graphs and extends this to totally bounded, compact, and bounded ultrametric spaces.

Findings

Ultrametric spaces are characterized by diametrical graphs being complete multipartite or empty.

The characterization extends to totally bounded and compact ultrametrizable spaces.

Bounded ultrametric spaces are described via weak similarities to unbounded ones.

Abstract

We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space $(X, d)$ is ultrametric iff the diametrical graph of the metric $d_{\varepsilon}(x, y) = \max\{d(x, y), \varepsilon\}$ is either empty or complete multipartite for every $\varepsilon > 0$. A refinement of the last result is obtained for totally bounded spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs.