The Sandpile Group of a Tree
arXiv:math/0703868 · doi:10.1016/j.ejc.2008.02.014
Abstract
A wired tree is a graph obtained from a tree by collapsing the leaves to a single vertex. We describe a pair of short exact sequences relating the sandpile group of a wired tree to the sandpile groups of its principal subtrees. In the case of a regular tree these sequences split, enabling us to compute the full decomposition of the sandpile group as a product of cyclic groups. This resolves in the affirmative a conjecture of E. Toumpakari concerning the ranks of the Sylow p-subgroups.
v2 incorporates referee comments, corrects references, improves notation
References in corpus (3)
Cited by in corpus (11)
- Chip-Firing and Rotor-Routing on Directed Graphs
- On the Sandpile group of the cone of a graph
- The Rotor-Router Model on Regular Trees
- On the critical ideals of graphs
- Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra
- Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups
- Critical ideals of trees
- Algebraic and combinatorial aspects of sandpile monoids on directed graphs
- The Abelian Sandpile Model on Fractal Graphs
- The rotor-router group of directed covers of graphs
- Compatible Recurrent Identities of the Sandpile Group and Maximal Stable Configurations