Ultrametric spaces generated by labeled star graphs

TL;DR

This paper characterizes ultrametric spaces derived from labeled star graphs and identifies conditions for their isometry groups to match graph automorphisms.

Contribution

It provides a complete characterization of ultrametric spaces generated by labeled star graphs and establishes criteria for their isometry groups to coincide with graph automorphism groups.

Findings

Characterization of the class of ultrametric spaces from labeled star graphs

Necessary and sufficient conditions for isometry group equivalence

Insights into the symmetry groups of these ultrametric spaces

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 $S(l)$.