论文标题
锤子立方体子集的距离矩阵
Distance matrices of subsets of the Hamming cube
论文作者
论文摘要
格雷厄姆（Graham）和温克勒（Winkler）得出了一个公式，用于$ n + 1 $ $ \ \ {x_ {0}，x_ {1}，\ ldots，x_ {n} \} $ in Hamming cube $ h_ { \ ell_ {1}）$。在本文中，我们得出了$ m + 1 $ $ \ \ {x_ {0}，x_ {1}，\ ldots，x_ {m} \} $ in $ h_ {n} $中的距离矩阵$ d $的公式。从这个更一般的公式中遵循$ \ det（d）\ not = 0 $，并且仅当vectors $ x_ {0}，x_ {1}，\ ldots，x_ {m} $是亲密独立的。专门研究$ M = n $的情况，可以为Graham和Winkler的原始公式提供新的见解。注意到$ m <n $和$ m = n $之间会产生的显着差异。我们还表明，如果$ d $是$ n + 1 $顶点的未加权树的距离矩阵，则是$ \ langle d^{ - 1} \ Mathbf {1}，\ Mathbf {1} \ rangle = 2/n $ where $ \ mathbf {1} $是compart vector us us coildates us coildates us coildates us coildates us coildates us coildates。最后，我们得出了Murugan对具有严格$ 1 $ 1 $阴性类型的$ H_ {N} $子集的分类的新证明。
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.
