Geodesic
The Sandpile Model on Km,n and the Rank of its Configurations
Séminaire lotharingien de combinatoire, Tome 77 (2017-2018) 
Michele D'Adderio; Yvan Le Borgne. The Sandpile Model on Km,n and the Rank of its Configurations. Séminaire lotharingien de combinatoire, Tome 77 (2017-2018). http://geodesic.mathdoc.fr/item/SLC_2017-2018_77_a7/
@article{SLC_2017-2018_77_a7,
     author = {Michele D'Adderio and Yvan Le Borgne},
     title = {The {Sandpile} {Model} on {Km,n} and the {Rank} of its {Configurations}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2017-2018},
     volume = {77},
     url = {http://geodesic.mathdoc.fr/item/SLC_2017-2018_77_a7/}
}
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Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

We present an algorithm to compute the rank of a configuration of the sandpile model for the complete bipartite graph Km,n of complexity O(m+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs Km,n according to rank, degree, and the sizes m and n. The results in the present paper are similar to those found in a previous paper by Cori and Le Borgne [Electron. J. Combin. 23(1) (2016), Paper 1.31, 47 pp.] for the complete graph Kn+1, and they rely on the analysis of certain operators on the stable sorted configurations of Km,n developed by Aval, D'Adderio, Dukes and Le Borgne in [Adv. Appl. Math. 73 (2016), 59-98].