  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.
