Collapse of the Metric Hierarchy for Bipartite Graphs

We show that among connected bipartite graphs the following classes are identical: graphs isometrically embeddable in a hypercube, graphs isometrically embeddable in l 1 , hypermetric graphs, graphs of negative type, and graphs whose distance matrices have just one positive eigenvalue. Here we regard any connected graph as a metric space, the distance between two vertices being the number of edges in a shortest path between them.

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