Abstract In the abelian sandpile model, recurrent chip configurations are of interest as they are a natural choice of coset representatives under the quotient of the reduced Laplacian. We investigate graphs whose recurrent identities with respect to different sinks are compatible with each other. The maximal stable configuration is the simplest recurrent chip configuration, and graphs whose recurrent identities equal the maximal stable configuration are of particular interest, and are said to have the complete maximal identity property. We prove that given any graph G one can attach trees to the vertices of G to yield a graph with the complete maximal identity property. We conclude with several intriguing conjectures about the complete maximal identity property of various graph products.

中文翻译：

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沙堆群的相容循环恒等式和最大稳定构型

摘要 在阿贝尔沙堆模型中，循环芯片配置是令人感兴趣的，因为它们是缩减拉普拉斯算子商下陪集代表的自然选择。我们研究了关于不同接收器的循环身份彼此兼容的图。最大稳定配置是最简单的循环芯片配置，并且循环标识等于最大稳定配置的图特别有趣，并且被称为具有完全最大标识属性。我们证明给定任何图 G 可以将树附加到 G 的顶点以产生具有完全最大恒等属性的图。我们以关于各种图产品的完全最大恒等性的几个有趣猜想作为结论。