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arXiv · 1608.01521

The sandpile model on K_{m,n} and the rank of its configurations

Michele D'Adderio·Yvan Le Borgne
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Abstract

We present an algorithm to compute the rank of a configuration of the sandpile model for the complete bipartite graph K_{m,n} of complexity O(m+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs K_{m,n} according to rank, degree, and the sizes m and n. The results in the present paper are similar to those found by Robert Cori and the second named author for the complete graph K_{n+1}, and they rely on the analysis of certain operators on the stable sorted configurations of K_{m,n} developed in a previous work by the authors together with Jean-Christophe Aval and Mark Dukes.

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Michele D'Adderio, Yvan Le Borgne. 2016-08-04. The sandpile model on K_{m,n} and the rank of its configurations. https://arxiv.org/abs/1608.01521

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