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