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In the abelian sandpile model, recurrent chip configurations are of interestas they are a natural choice of coset representatives under the quotient of thereduced Laplacian. We investigate graphs whose recurrent identities withrespect to different sinks are compatible with each other. The maximal stableconfiguration is the simplest recurrent chip configuration, and graphs whoserecurrent identities equal the maximal stable configuration are of particularinterest, and are said to have the complete maximal identity property. We provethat given any graph $G$ one can attach trees to the vertices of $G$ to yield agraph with the complete maximal identity property. We conclude with severalintriguing conjectures about the complete maximal identity property of variousgraph products.

Compatible Recurrent Identities of the Sandpile Group and Maximal Stable Configurations