阿贝尔网络III。所述关键组
Abelian networks III. The critical group
摘要：
阿贝尔网络的临界群是一个有限的阿贝尔群，它控制着网络在大输入下的行为。它推广了图的沙堆群。我们证明了不可约阿贝尔网络的临界群对网络的常返状态具有自由传递性。我们展示了临界群作为一个商的自由阿贝尔群的一个子群包含图像的拉普拉斯，与平等的情况下，网络是矩形的。我们推广Dhar的燃烧算法的阿贝尔网络，并估计任意输入的阿贝尔网络的运行时间到一个常数的附加误差。
英文摘要：
The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We generalize Dhar's burning algorithm to abelian networks, and estimate the running time of an abelian network on an arbitrary input up to a constant additive error.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
