Sandpiles, Spanning Trees, and Plane Duality

@article{Chan2015SandpilesST,
  title={Sandpiles, Spanning Trees, and Plane Duality},
  author={Melody Chan and Darren B. Glass and Matthew Macauley and David Perkinson and Caryn Werner and Qiaoyu Yang},
  journal={SIAM J. Discret. Math.},
  year={2015},
  volume={29},
  pages={461-471}
}
Let $G$ be a connected, loopless multigraph. The sandpile group of $G$ is a finite abelian group associated to $G$ whose order is equal to the number of spanning trees in $G$. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of $G$ on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on $G$, and a choice of a root vertex. Chan, Church, and… 
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