汉明立方体子集的距离矩阵
Distance matrices of subsets of the Hamming cube
摘要：
Graham和Winkler导出了汉明立方体$H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$中$n + 1$点的全维集合$\{ x_{0}, x_{1}, \ldots , x_{n} \}$的距离矩阵的行列式公式。本文导出了$H_{n}$中任意一组$m + 1$点$\{ x_{0}, x_{1}, \ldots , x_{m} \}$的距离矩阵$D$的行列式公式。从这个更一般的公式可以得出，$\det (D) \not= 0$当且仅当向量$x_{0}, x_{1}, \ldots , x_{m}$是仿射无关的。$m = n$提供了对Graham和Winkler原始公式的新见解。注意到$m < n$和$m = n$病例之间的显著差异。我们还表明，如果$D$是$n + 1$顶点上的未加权树的距离矩阵，那么$\langle D^{-1} \mathbf{1}, \mathbf{1} \rangle = 2/n$，其中$\mathbf{1}$是列向量，其坐标都是$1$。最后，我们给出了严格$1$ -负型的$H_{n}$子集的Murugan分类的一个新的证明.
英文摘要：
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.
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