Geodesic
Sandpile groups and the join of graphs
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 119-124 
I. A. Krepkiy. Sandpile groups and the join of graphs. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXII, Tome 411 (2013), pp. 119-124. http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a6/
@article{ZNSL_2013_411_a6,
     author = {I. A. Krepkiy},
     title = {Sandpile groups and the join of graphs},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--124},
     year = {2013},
     volume = {411},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_411_a6/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We introduce the procedure of joining two graphs by identifying an arbitrary pair of their vertices. The main result is that the sandpile group of the join of several finite graphs is the direct product of the sandpile groups of the components. Some consequences are derived.

[1] A. E. Holroyd, L. Levine, K. Meszaros, Yu. Peres, J. Propp, D. B. Wilson, Chip-firing and rotor-routing on directed graphs, arXiv: 0801.3306 | MR

[2] I. A. Krepkii, Pesochnye gruppy treugolnykh binarnykh derevev, Preprint , POMI, 2012 http://www.pdmi.ras.ru/preprint/2012/12-21.html

[3] K. R. Matthews, Smith normal form MP274: Linear Algebra, , Lecture Notes, University of Queensland, 1991 http://www.numbertheory.org/courses/MP274/smith.pdf