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

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.