Metric graphs of negative type
Rutger Campbell, Kevin Hendrey, Ben Lund, Casey Tompkins

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 does not contain a theta submetric as an embedding, then has negative type. We show the converse: if a metric graph contains a theta, then it does not have negative type.
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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
