边删除对图上非交换距离的影响
The effect of edge deletion on noncommutative distances on graphs
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中文总结 AI 辅助
本文研究有限加权图上删除边对非交换距离的影响，证明删除边可减小顶点间距离，给出4-环的最优上界2/√3，并刻画n-环情形及删除边或顶点的远距离效应。
中文摘要 AI 辅助
对于有限加权图上的Dirac算子$D$，设$d^D$表示相关的非交换（Connes）距离。我们证明删除一条边实际上可以减小图中两个顶点之间的非交换距离。这种现象的最小例子来自加权4-环，并且从加权4-环中删除一条边的情况被完全确定：如果$d^{D'}$表示边删除后图中的非交换距离，我们证明对于4-环，$\sup d^D/d^{D'} = 2/\sqrt{3}$，并且进一步证明这个上界对所有图都成立。另一方面，我们证明当且仅当$n$能被4整除时，删除一条边可以减小$n$-环中两个顶点之间的非交换距离。在删除的边减小两个顶点$x$和$y$之间的非交换距离的情况下，我们证明被删除的边不必与$x$或$y$中的任何一个顶点相关联，事实上，被删除的边可以在图论（边数）距离和加权（测地）距离上任意远离这两个顶点。此外，删除一个在图论或加权距离上任意远离$x$和$y$的单个顶点，可以以任意大的因子改变$d^D(x,y)$。
英文摘要
For a Dirac operator $D$ on a finite weighted graph, let $d^D$ denote the associated noncommutative (Connes) distance. We show that deleting an edge can actually decrease the noncommutative distance between two vertices in the graph. The smallest example of this phenomenon comes from a weighted 4-cycle, and the case of deleting an edge from a weighted 4-cycle is determined completely: if $d^{D'}$ denotes the noncommutative distance in the graph after edge deletion, we prove that $\sup d^D/d^{D'} = 2/\sqrt{3}$ for the $4$-cycle, and further that this upper bound holds for all graphs. On the other hand, we show that deleting an edge can decrease the noncommutative distance between two vertices of an $n$-cycle precisely when $n$ is divisible by $4$. In cases when a deleted edge decreases the noncommutative distance between two vertices $x$ and $y$, we show that the deleted edge need not be incident to either of the vertices $x$ or $y$, and in fact the deleted edge can be arbitrarily far from both in the graph-theoretic (number of edges) distance and in the weighted (geodesic) distance. Moreover, deletion of a single vertex arbitrarily far from both $x$ and $y$ in either the graph-theoretic or weighted distance can change $d^D(x,y)$ by an arbitrarily large factor.
发表机构
- Department of Mathematics, William & Mary（威廉与玛丽学院数学系）
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