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We study the supremal $p$-negative type of finite metric spaces. An explicitexpression for the supremal $p$-negative type $\wp (X,d)$ of a finite metricspace $(X,d)$ is given in terms its associated distance matrix, from which thesupremal $p$-negative type of the space may be calculated. The method is thenused to give a straightforward calculation of the supremal $p$-negative type ofthe complete bipartite graphs $K_{n,m}$ endowed with the usual path metric. Agap in the spectrum of possible supremal $p$-negative type values of pathmetric graphs is also proven.

On the supremal $p$-negative type of a finite metric space