We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.
We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces and compute a dynamical invariant attached to their isometry groups.
A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
The necessary and sufficient conditions under which a given family $mathcal{F}$ of subsets of finite set $X$ coincides with the family $mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $(mathbf{B}_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtained from the representing tree $T_X$ of ultrametric space $(X, d)$ by adding a leaf to every internal vertex of $T_X$.
We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric spaces into Euclidean spaces. We also consider roundness properties additive metric spaces which are not ultrametric.
O. Dovgoshey
,E. Petrov
,H.-M. Teichert
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(2016)
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"Extremal properties and morphisms of finite ultrametric spaces and their representing trees"
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Oleksiy Dovgoshey
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