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On the sandpile group of dual graphs. (English) Zbl 0969.05034
E. Goles studied the notion of sandpile group in [Sand piles, combinatorial games and cellular automata. Instabilities and nonequilibrium structures III. Math. Appl., 64, 101-121 (1991)].
Let \(G= (X,E)\) be a (nondirected) finite connected graph without loops. The elements of \({\mathcal Z}^n\) (i.e., the vectors of length \(n\) whose components are integers) form an abelian group. By the sandpile group \(\text{SP}(G)\) of \(G\), a factor group of \({\mathcal Z}^n\) is meant (where \(n=|X|\)) with respect to a subgroup defined by means of the adjacency matrix of \(G\).
It is shown that \(\text{SP}(G)\) and \(\text{SP}(G^*)\) are isomorphic if \(G\) is a planar graph and \(G^*\) is its dual, and that to every finite abelian group \({\mathfrak G}\) there is a planar graph \(G\) such that \(\text{SP}(G)\) is isomorphic to \({\mathfrak G}\). The sandpile groups \(\text{SP}(K_n)\) are explicitly determined (\(K_n\) is the complete graph on \(n\) vertices).
Let \(v\) be a cut vertex of \(G\). Denote by \(G_i\) the subgraph induced by \(X_i\cup \{v\}\) where \(X_i\) is one of the vertex classes (split by \(v\)). It is stated that \(\text{SP}(G)\) is the direct product of all the groups \(\text{SP}(G_i)\).
The authors consider models formerly also discussed by D. Dhar [Phys. Rev. Lett. 64, No. 14, 1613-1616 (1990)].

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
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