An ultrametric space comes from a labeled star graph exactly when some point is no farther from any point than that point is from every other point, and its self-isometries match the graph's symmetries exactly when the nonzero distance set has no least element.
Dovgoshey, Isomorphism of trees and isometry of ultrametric spaces, Theory and Applications of Graphs, 7 (2020), no
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Ultrametric spaces generated by labeled star graphs
An ultrametric space comes from a labeled star graph exactly when some point is no farther from any point than that point is from every other point, and its self-isometries match the graph's symmetries exactly when the nonzero distance set has no least element.