Longest paths in trees and isometricity of ultrametric spaces

TL;DR

The paper investigates which trees $T$ yield a family of ultrametric spaces ${U(T)}$ that are all isometric to spaces generated by vertex labelings of star graphs, i.e., ${U(T)\subseteq{\bf US}}$. It proves a set of equivalent conditions, notably that the longest path in $T$ has length at most $3$ and that at most two vertices have degree at least $2$, and shows these are equivalent to ${U(T)\subseteq{\bf US}}$. The culmination is that ${U(T)\subseteq{\bf US}}$ if and only if $T$ is isomorphic to a star-graph or a double-star graph, linking ultrametric generation to specific tree topologies. This provides a structural characterization of when ultrametric spaces induced by trees reduce to those generated by star graphs, with implications for understanding interconnections between star and double-star graph constructions.

Abstract

Let $T$ be a tree of arbitrary finite or infinite order and let $U(T)$ be the set of all ultrametric spaces generated by vertex labelings of $T$. Let ${\bf US}$ denote the class of all ultrametric spaces generated by vertex labelings of star graphs. We prove that the inclusion $U(T)\subseteq {\bf US}$ holds if and only if the longest path in $T$ has a length not exceeding three.

Figures (4)

• Figure 1: The vertex labeling $l_2$ of the path $P_2$.
• Figure 2: The vertices $w$ and $z$ are adjacent with the vertex $u$.
• Figure 3: The vertex $u$ is adjacent with $w$ but not adjacent with $z$.
• Figure 4: The vertices $w$ and $z$ are adjacent with vertex $v$.

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• proof
• Lemma 3.2
• proof