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Ultrametric spaces generated by labeled star graphs

Oleksiy Dovgoshey, Olga Rovenska

TL;DR

The paper characterizes ultrametric spaces that can be generated by labeled star graphs and links these spaces to graph isomorphisms, providing a metric criterion for US-spaces, a uniqueness condition for the generating star, and a precise isometry–isomorphism correspondence between the space and its generating labeled trees. It shows that for US-spaces with a fixed generator, self-isometries correspond to isomorphisms of labeled trees, and it gives conditions under which the full isometry group equals the automorphism group of the generator. The authors present two conjectures on finite four-point configurations and on infinite compact US-spaces, and discuss completions via labeled rays, thereby connecting ultrametric geometry, labeled trees, and graph symmetry in a coherent framework.

Abstract

For arbitrary star graph $S$ with a non-degenerate vertex labeling $l\colon V(S) \to \mathbb{R}^+$ we denote by $d_l$ the corresponding ultrametric on the vertex set $V(S)$ of $S$. We characterize the class $\bf US$ of all ultrametric spaces $(V(S), d_l)$ up to isometry. We also find the necessary and sufficient conditions under which the group of all self-isometries of ultrametric space $(V(S), d_l)$ coincides with the group of all self-isomorphisms of the labeled star graph $S(l)$.

Ultrametric spaces generated by labeled star graphs

TL;DR

The paper characterizes ultrametric spaces that can be generated by labeled star graphs and links these spaces to graph isomorphisms, providing a metric criterion for US-spaces, a uniqueness condition for the generating star, and a precise isometry–isomorphism correspondence between the space and its generating labeled trees. It shows that for US-spaces with a fixed generator, self-isometries correspond to isomorphisms of labeled trees, and it gives conditions under which the full isometry group equals the automorphism group of the generator. The authors present two conjectures on finite four-point configurations and on infinite compact US-spaces, and discuss completions via labeled rays, thereby connecting ultrametric geometry, labeled trees, and graph symmetry in a coherent framework.

Abstract

For arbitrary star graph with a non-degenerate vertex labeling we denote by the corresponding ultrametric on the vertex set of . We characterize the class of all ultrametric spaces up to isometry. We also find the necessary and sufficient conditions under which the group of all self-isometries of ultrametric space coincides with the group of all self-isomorphisms of the labeled star graph .

Paper Structure

This paper contains 4 sections, 10 theorems, 123 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Characterization of US-spaces and unique- ness of generating star graphs
  3. From isometries to isomorphism and back
  4. Two conjectures

Key Result

Theorem 1.1

Let $T = T(l)$ be a labeled tree. Then the function $d_l$ is an ultrametric on $V(T)$ if and only if the inequality holds for every edge $\{u, v\}$ of $T$.

Figures (4)

  • Figure 1: The labeled star graph $T(l)$ generates the ultrametric triangle $(X, d)$.
  • Figure 2: The labeled path $T_1(l_1)$ and the labeled star graph $T_2(l_2)$ generate isometric spaces $(V(T_1), d_{l_1})$ and $(V(T_2), d_{l_2})$ if $0 <a_1 \leq a_2 \leq a_3 \leq a_4$.
  • Figure 3: $(X_4,d_4)$ and $(Y_4,\rho_4)$ are not ${\bf US}$-spaces.
  • Figure 4: $(V(S),d_{l^*}) \in {\bf US}$ is a completion of $(V(R),d_l)\notin {\bf US}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Definition 1.4
  • Definition 1.5
  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • ...and 23 more