论文标题
图表上的积分流量和周期芯片冲击
Integral flow and cycle chip-firing on graphs
论文作者
论文摘要
由平面图的双图上的芯片射击概念的动机，我们考虑在任意图$ g $上考虑“积分流芯片”。筹码规则由$ {\ Mathcal l}^*（g）$，$ $ {\ mathcal l}^*（g）$，$ g $的双laplacian通过选择$ g $的整体流量的基础来确定。我们表明，任何图表都允许这样一个基础，因此$ {\ Mathcal l}^*（g）$是$ M $ -MATRIX，导致对这些基础元素的触发规则，这是雪崩有限的。这是从更一般的结果基础上的整体晶格基础上产生的，这可能具有独立的关注。我们的结果提供了$ z $ - 敏锐的流程配置的概念，这些配置与$ g $的一组跨越树相同。我们表明，对于平面图，以及图形$ k_5 $和$ k_ {3,3} $，人们可以找到由基础图的周期组成的流动m基础。我们考虑任意图的问题并解决了一些开放问题。
Motivated by the notion of chip-firing on the dual graph of a planar graph, we consider `integral flow chip-firing' on an arbitrary graph $G$. The chip-firing rule is governed by ${\mathcal L}^*(G)$, the dual Laplacian of $G$ determined by choosing a basis for the lattice of integral flows on $G$. We show that any graph admits such a basis so that ${\mathcal L}^*(G)$ is an $M$-matrix, leading to a firing rule on these basis elements that is avalanche finite. This follows from a more general result on bases of integral lattices that may be of independent interest. Our results provide a notion of $z$-superstable flow configurations that are in bijection with the set of spanning trees of $G$. We show that for planar graphs, as well as for the graphs $K_5$ and $K_{3,3}$, one can find such a flow M-basis that consists of cycles of the underlying graph. We consider the question for arbitrary graphs and address some open questions.
