Spectral Properties of the Gramian of Finite Ultrametric Spaces
Gavin Robertson
TL;DR
The paper determines the spectral properties of the Gramian $G_p$ for finite ultrametric spaces, tying the minimum eigenvalue to the degree of strict $p$-negative type. It proves that for a nondegenerate finite ultrametric space with minimum nonzero distance $\alpha_1$, $\lambda_{\min}(G_p) = \alpha_1^p/2$ for all $p>0$ and fully describes the corresponding eigenspace. The results leverage the $p$-negative type framework, the $p$-Gramian construction, and the coterie structure of ultrametrics, providing an exact, $p$-dependent measure of strict $p$-negative type and an explicit eigenbasis. An illustrative example confirms the formula and demonstrates the construction in a concrete setting, highlighting the practical computability of the Gramian's spectral properties in 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 $λ_{min}(G_{p})$. 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 $α_{1}$ then we show that $λ_{min}(G_{p})=α_{1}^{p}/2$. We also provide a description of the corresponding eigenspace.