اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDistance matrices of subsets of the Hamming cube
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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 $det (D) ot= 0$ 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 Murugans classification of the subsets of $H_{n}$ that have strict $1$-negative type.
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