超度量与完全多部图
Ultrametrics and complete multipartite graphs
摘要：
我们描述了一类图，对于这类图，所有具有直径图的度量空间都是超度量的。证明了度量空间$(X, d)$是超度量的当且仅当对于每个$\varepsilon > 0$，度量空间$d_{\varepsilon}(x, y) = \max\{d(x, y), \varepsilon\}$的直径图为空或完全多部。对完全有界空间得到了上一个结果的一个改进。此外，利用完全多部图刻画了紧超度量化拓扑空间。有界超度量空间与无界超度量空间弱相似，并利用完全多部图刻画了有界超度量空间。
英文摘要：
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.
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