Metric graphs of negative type

TL;DR

This paper establishes a precise characterization of metric graphs with negative type, showing that the presence of a theta submetric prevents negative type, thus linking embeddability properties to subgraph structure.

Contribution

It proves the converse of a known result, demonstrating that containing a theta submetric implies a metric graph does not have negative type.

Findings

Graphs without theta submetrics have negative type

Presence of a theta submetric prevents negative type

Provides a complete characterization of negative type in metric graphs

Abstract

The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph $G$ does not contain a theta submetric as an embedding, then $G$ has negative type. We show the converse: if a metric graph $G$ contains a theta, then it does not have negative type.