Concrete and abstract structure of the sandpile group for thick trees with loops

Details Concrete and abstract structure of the sandpile group for thick trees with loops Concrete and abstract structure of the sandpile group for thick trees with loops

2007-01-13

arxiv.org/abs/math/0701381v1

Mathematics

Combinatorics

8 pages

Scientific paper

We answer a question of Laszlo Babai concerning the abelian sandpile model. Given a graph, the model yields a finite abelian group of recurrent configurations which is closely related to the combinatorial Laplacian of the graph. We explicitly describe the group elements and operations in the case of thick trees with loops--that is, graphs which are obtained from trees by setting arbitrary edge multiplicities and adding loops at vertices. We do this both concretely (by describing the so-called recurrent and identity configurations) and abstractly (by computing the group's abstract structure), and define maps identifying the two.

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