Compact ultrametric spaces generated by labeled star graphs

TL;DR

This work provides a complete metric description of compact ultrametric spaces generated by labeled star graphs. It proves that compact ${\bf US}$-spaces are exactly the completions of totally bounded spaces generated by decreasingly labeled rays, and it establishes four-point weak-similarity obstructions that characterize finite ${\bf US}$-spaces. The authors confirm two conjectures of Dov-Rov: (i) a finite ${\bf US}$-space lacks four-point subspaces weakly similar to $(X_4,d_4)$ or $(Y_4,\rho_4)$, and (ii) compact ${\bf US}$-spaces arise as completions from labeled rays and correspond to star-graph generation. They also connect the theory to the ${\mathbb{R}^+}$ ultrametric example $(\mathbb{R}^+,d^+)$ and pose two further conjectures linking weak similarity, completions, and starlike trees, enriching the structural understanding of ultrametric spaces generated by trees.

Abstract

Let US be the class of all ultrametric spaces generated by labeled star graphs. We prove that compact US-spaces are the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. We characterize the ultrametric spaces which are weakly similar to finite US-spaces and describe these spaces by certain four-point conditions.

Figures (4)

• Figure 1: $(X_4,d_4)$ and $(Y_4,\rho_4)$ are not ${\bf US}$-spaces by Theorem \ref{['xz']}.
• Figure 2: The unique point $x_0$ satisfying \ref{['zg']} for all $x,y\in Y$ which satisfy \ref{['cd']}, is the center of the labeled star graph $S(l)$.
• Figure 3: The ultrametric space $(V(T_1), d_{l_1})$ contains four-point subspace which is isometric to $(X_4, d_{4})$. Similarly, $(V(T_2), d_{l_2})$ contains a subspace isometric to $(Y_4, \rho_{4})$.
• Figure 4: Trees $T_1$ and $T_2$ are starlike and rayless.

Theorems & Definitions (42)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• Definition 2.6
• Proposition 2.7
• Definition 2.8
• Remark 2.9
• Proposition 2.10