Generalized roundness of vertex transitive graphs

TL;DR

This paper characterizes the generalized roundness of finite metric spaces with permutation-based distance matrices, specifically analyzing subsets of the Hamming cube with strict 1-negative type through linear dependence of difference vectors.

Contribution

It provides a novel characterization of the generalized roundness and strict 1-negative type of subsets of the Hamming cube using linear algebraic conditions.

Findings

Subset has generalized roundness one iff difference vectors are linearly dependent.

Subset has strict 1-negative type iff difference vectors are linearly independent.

Characterizes geometric properties of Hamming cube subsets in terms of linear algebra.

Abstract

We study the generalized roundness of finite metric spaces whose distance matrix $D$ has the property that every row of $D$ is a permutation of the first row. The analysis provides a way to characterize subsets of the Hamming cube $\{ 0, 1 \}^{n} \subset \ell_{1}^{(n)}$ ($n \geq 1$) that have strict $1$-negative type. The result can be stated in two ways: a subset $S = \{ \vc{x}_0,\vc{x}_1,\ldots,\vc{x}_k \}$ of the Hamming cube $\{ 0, 1 \}^{n} \subset \ell_{1}^{(n)}$ has generalized roundness one if and only if the vectors $\{ \vc{x}_1 - \vc{x}_0,\vc{x}_2 - \vc{x}_0,\ldots,\vc{x}_k - \vc{x}_0 \}$ are linearly dependent in $\mathbb{R}^n$. Equivalently, $S$ has strict $1$-negative type if and only if the vectors $\{ \vc{x}_1 - \vc{x}_0,\vc{x}_2 - \vc{x}_0,\ldots,\vc{x}_k - \vc{x}_0 \}$ are linearly independent in $\mathbb{R}^n$.