Spectral Properties of the Gramian of Finite Ultrametric Spaces

The concept of $p$-negative type is such that a metric space $(X,d_{X})$ has $p$-negative type if and only if $(X,d_{X}^{p/2})$ embeds isometrically into a Hilbert space. If $X=\{x_{0},x_{1},\dots,x_{n}\}$ then the $p$-negative type of $X$ is intimately related to the Gramian matrix $G_{p}=(g_{ij})_{i,j=1}^{n}$ where $g_{ij}=\frac{1}{2}(d_{X}(x_{i},x_{0})^{p}+d_{X}(x_{j},x_{0})^{p}-d_{X}(x_{i},x_{j})^{p})$. In particular, $X$ has strict $p$-negative type if and only if $G_{p}$ is strictly positive semidefinite. As such, a natural measure of the degree of strictness of $p$-negative type that $X$ possesses is the minimum eigenvalue of the Gramian $\lambda_{min}(G_{p})$. In this article we compute the minimum eigenvalue of the Gramian of a finite ultrametric space. Namely, if $X$ is a finite ultrametric space with minimum nonzero distance $\alpha_{1}$ then we show that $\lambda_{min}(G_{p})=\alpha_{1}^{p}/2$. We also provide a description of the corresponding eigenspace.