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.

受平面图的对偶图上的切屑燃烧概念的启发，我们考虑任意图G上的“积分流切屑燃烧” 。该芯片烧成规则由管辖\（{\ mathcal {L}} ^ *（G）\） ，双重拉普拉斯ģ确定通过选择用于在积分流量的晶格的基础ģ。我们证明任何图都承认这样的基，使得\({\mathcal {L}}^*(G)\)是一个M矩阵，导致在这些基元素上的触发规则是雪崩有限的。这是基于可能具有独立兴趣的积分格的更一般结果得出的。我们的结果提供了z的概念- 与G的生成树集合双射的超稳定流配置。我们表明，对于平面图以及图\(K_5\)和\(K_{3,3}\)，可以找到这样一个由底层图的循环组成的流 M 基。我们考虑任意图的问题并解决一些悬而未决的问题。