标题： 广义芯片火化与树上算术结构的临界群 显示英文标题

标题： Generalized chip firing and critical groups of arithmetical structures on trees

摘要： Chip firing提供了一种研究图的沙堆群（也称为Jacobi行列式）的方法。 我们使用Chip firing的一个广义版本来限制图上算术结构的临界群的不变因子数量。 我们还证明了，在适当的假设下，临界群在具有算术结构的图的楔形和下是可加的。 我们将这些结果应用于关联到任意给定树的临界群的不变因子数量与树分解为更简单树的关系。 我们用这个方法来分类那些每个算术结构都有循环临界群的树。 最后，我们展示了如何在具有预设临界群的树上构造算术结构。 特别是，每个有限阿贝尔群都可以作为某个树上的算术结构的临界群来实现。 显示英文摘要

摘要： Chip firing provides a way to study the sandpile group (also known as the Jacobian) of a graph. We use a generalized version of chip firing to bound the number of invariant factors of the critical group of an arithmetical structure on a graph. We also show that, under appropriate hypotheses, critical groups are additive under wedge sums of graphs with arithmetical structures. We apply these results to relate the number of invariant factors of critical groups associated to any given tree to decompositions of the tree into simpler trees. We use this to classify those trees for which every arithmetical structure has cyclic critical group. Finally, we show how to construct arithmetical structures on trees with prescribed critical groups. In particular, every finite abelian group is realized as the critical group of some arithmetical structure on a tree.