# 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…

## 6 Citations

The Bernardi process and torsor structures on spanning trees

- Mathematics
- 2014

Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to O. Bernardi, we associate to any pair…

Geometric bijections between spanning trees and break divisors

- MathematicsJ. Comb. Theory, Ser. A
- 2017

GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

- MathematicsForum of Mathematics, Sigma
- 2019

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of…

Recognizing hyperelliptic graphs in polynomial time

- Mathematics, Computer ScienceWG
- 2018

Hyperelliptic graphs are considered and a safe and complete set of reduction rules for such multigraphs are provided, showing that they can be recognized in time.

The sandpile group of a trinity and a canonical definition for the planar Bernardi action

- Mathematics
- 2019

Baker and Wang define the so called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but they prove…

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