Spectral Properties of the Gramian of Finite Ultrametric Spaces

Abstract

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.