On the supremal p-negative type of finite metric spaces

We study the supremal p-negative type of finite metric spaces. An explicit expression for the supremal p-negative type ℘(X,d) of a finite metric space (X,d) is given in terms of 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 Kn,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.