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

We study the supremal $p$-negative type of finite metric spaces. An explicit expression for the supremal $p$-negative type $\mathrm{\wp }(X,d)$ 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.