Ultrametric spaces generated by labeled star graphs

Abstract

For arbitrary star graph $S$ with a non-degenerate vertex labeling $l:V(S)\to {\mathbb{R}}^{+}$ we denote by $d_l$ the corresponding ultrametric on the vertex set $V(S)$ of $S$. We characterize the class $\mathbf{US}$ 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 $S(l)$.

23 pages, 4 figures