标题： 广义的芯片发放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： 芯片发放提供了一种研究图的沙堆群（也称为雅可比群）的方法。 我们使用芯片发放的一种推广版本来限制图上算术结构的临界群的不变因子的数量。 我们还表明，在适当的假设下，具有算术结构的图的楔和下的临界群是可加的。 这些结果使我们能够将任何给定树相关的临界群的不变因子数量与树分解为更简单树的分解联系起来。 我们利用这一点对所有算术结构都有循环临界群的树进行分类。 最后，我们展示了如何在树上构造具有预定临界群的算术结构。 特别是，每个有限阿贝尔群都可以作为树上某个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 suitable hypotheses, critical groups are additive under wedge sums of graphs with arithmetical structures. These results allow us 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.