Generalized roundness of vertex transitive graphs

Details Generalized roundness of vertex transitive graphs Generalized roundness of vertex transitive graphs

2011-12-23

arxiv.org/abs/1112.5657v1

Mathematics

Functional Analysis

Scientific paper

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 $\mathbb{R}^n$.

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