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E. Petrov

Publications and source records attributed to E. Petrov.

10 recordsLinked to original sources
arXiv2017

Extremal properties and morphisms of finite ultrametric spaces and their representing trees

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 $Y$.

math.MG↗
arXiv2015

How rigid the finite ultrametric spaces can be?

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.

math.MG↗
arXiv2015

Minimal universal metric spaces

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $\mathfrak{M}$ of metric spaces such that $X$ is minimal $\mathfrak{M}$-universal. We generalize the notion of minimal $\mathfrak{M}$-universal metric space to notion of minimal $\mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $\mathfrak{M}$ is minimal $\mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.

math.MG↗
arXiv2014

On spaces extremal for the Gomory-Hu inequality

Let $(X,d)$ be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality $|Sp(X)|\leqslant |X|$ where $Sp(X)=\{d(x,y)\colon x,y \in X\}$. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric $(X,d)$ satisfying the equality $|Sp(X)|=|X|$ for given $Sp(X)$. Moreover it is shown that every finite semimetric space $Z$ is an image under a composition of mappings $f\colon X\to Y$ and $g\colon Y\to Z$ such that $X$ and $Y$ are finite ultrametric space, $X$ satisfies the above equality, $f$ is an $\varepsilon$-isometry with an arbitrary $\varepsilon>0$, and $g$ is a ball-preserving map.

math.MG↗
arXiv2012

On the Gomori-Hu inequality

It was proved by Gomori and Hu in 1961 that for every finite nonempty ultrametric space $(X,d)$ the following inequality $|\Sp(X)|\leqslant |X|-1$ holds with $\Sp(X)=\{d(x,y):x,y \in X, x\neq y\}$. We characterize the spaces $X$, for which the equality in this inequality is attained by the structural properties of some graphs and show that the set of isometric types of such $X$ is dense in the Gromov-Hausdorff space of the compact ultrametric spaces.

math.MG↗
arXiv2012

Metric products and continuation of isotone functions

Let $\mathbb{R}_+=[0,\infty)$ and let $A\subseteq\mathbb{R}^n_+$. We have found the necessary and sufficient conditions under which a function $Φ:A\to\mathbb{R}_+$ has an isotone subadditive continuation on $\mathbb{R}^n_+$. It allows us to describe the metrics, defined on the Cartesian product $X_1\times...\times X_n$ of given metric spaces $(X_1,d_{X_1}),...,(X_n,...,d_{X_n})$, generated by the isotone metric preserving functions on $\mathbb{R}^n_+$. It also shows that the isotone metric preserving functions $Φ:\mathbb{R}^n_+\to\mathbb{R}_+$ coincide with the first moduli of continuity of the nonconstant bornologous functions $g:\mathbb{R}^n_+\to\mathbb{R}_+$. We discuss some algebraic properties of sets $X\subseteq \mathbb{R}$ providing the existence of isometric embeddings $f:B\to X$ for every three-point $B\subseteq \mathbb{R}$. In particular, we prove that every finite subset of $\mathbb{R}$ is isometric to some subset of transcendental real numbers.

math.MG↗
arXiv2011

Subdominant pseudoultrametric on graphs

Let (G,w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G)-->R^+ can be extended to a pseudoultrametric on V(G) are found. A criterion of the uniqueness of this extension is also obtained. It is proved that G is complete k-partite with k >= 2 if and only if, for every pseudoultrametrizable weight w, there exists the smallest pseudoultrametric agreed with w. We characterize the structure of graphs for which the subdominant pseudoultrametric is an ultrametric for every strictly positive pseudoultrametrizable weight.

math.MG↗
arXiv2011

Diameter and diametrical pairs of points in ultrametric spaces

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function f:F(X)-->R there is an ultrametric on X such that f(A)=diam A for every A\in F(X). For finite nondegenerate ultrametric spaces (X,d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k>= 2, and, conversely, every finite complete k-partite graph with k>=2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X,d) having the minimal card{(x,y):d(x,y)=diam X, x,y \in X} for given card X.

math.MG↗