Geodesic
The sandpile groups of chain-cyclic graphs
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 94-112 
I. A. Krepkiy. The sandpile groups of chain-cyclic graphs. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIII, Tome 421 (2014), pp. 94-112. http://geodesic.mathdoc.fr/item/ZNSL_2014_421_a7/
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     title = {The sandpile groups of chain-cyclic graphs},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Firstly, we consider the graphs obtained by gluing a family of arbitrary finite graphs to the edges of a cyclic graph and prove that the sandpile group of the resulting graph does not depend on a specific way of doing that. Then, we define the class of chain-cyclic graphs, which are the graphs obtained by connecting a finite family of cyclic graphs along a line. Two kinds of formulas for calculating the sandpile groups of chain-cyclic graphs are proved.

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

[2] I. A. Krepkiy, Sandpile groups of triangular binary trees, http://www.pdmi.ras.ru/preprint/2012/12-21.html

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

[4] R. Cori, D. Rossin, “On the Sandpile group of dual graphs”, European Journal of Combinatorics, 21:4 (2000), 447–459 | DOI | MR | Zbl