Ultrametrics and Complete Multipartite Graphs

Authors

Viktoriia Viktorivna Bilet, Institute of Applied Mathematic and Mechanics of NAS of UkraineFollow
Oleksiy Dovgoshey, Institute of Applied Mathematics and Mechanics of NASUFollow
Yuriy Nikitovich Kononov, Institute of Applied Mathematics and Mechanics of NAS of UkraineFollow

Abstract

Let (X, d) be a semimetric space and let G be a graph. We say that G is the diametrical graph of (X, d) if X is the vertex set of G and the adjacency of vertices x and y is equivalent to the equality diam X = d(x, y). It is shown that a semimetric space (X, d) with diameter d* is ultrametric if the diametrical graph of (X, d _ε) with d _ε (x, y) = min{d(x, y), ε} is complete multipartite for every ε ∈ (0, d*). A refinement of the last result is given for totally bounded ultrametric 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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Recommended Citation

Bilet, Viktoriia Viktorivna; Dovgoshey, Oleksiy; and Kononov, Yuriy Nikitovich (2022) "Ultrametrics and Complete Multipartite Graphs," Theory and Applications of Graphs: Vol. 9: Iss. 1, Article 8.
DOI: 10.20429/tag.2022.090108
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss1/8