Distance matrices of subsets of the Hamming cube

TL;DR

This paper generalizes the formula for the determinant of the distance matrix of point sets in the Hamming cube, revealing conditions for invertibility, and explores implications for trees and negative type classifications.

Contribution

It derives a new, general formula for the determinant of the distance matrix of any subset in the Hamming cube, extending previous results and providing new insights.

Findings

Determinant formula for arbitrary point sets in Hamming cube

Invertibility linked to affine independence of points

New proof of subsets with strict 1-negative type

Abstract

Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{m} \}$ in $H_{n}$. It follows from this more general formula that $\det (D) \not= 0$ if and only if the vectors $x_{0}, x_{1}, \ldots , x_{m}$ are affinely independent. Specializing to the case $m = n$ provides new insights into the original formula of Graham and Winkler. A significant difference that arises between the cases $m < n$ and $m = n$ is noted. We also show that if $D$ is the distance matrix of an unweighted tree on $n + 1$ vertices, then $\langle D^{-1} \mathbf{1}, \mathbf{1} \rangle = 2/n$ where $\mathbf{1}$ is the column vector all of whose coordinates are $1$. Finally, we derive a new proof of Murugan's classification of the subsets of $H_{n}$ that have strict $1$-negative type.