Chip-firing games, potential theory on graphs, and spanning trees
Abstract
We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors (-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an "efficient bijective" proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the literature. We also extend some of these considerations to metric graphs.
Cite
@article{arxiv.1107.1313,
title = {Chip-firing games, potential theory on graphs, and spanning trees},
author = {Matthew Baker and Farbod Shokrieh},
journal= {arXiv preprint arXiv:1107.1313},
year = {2012}
}
Comments
To appear in Journal of Combinatorial Theory, Series A -- Revised and updated. The discussion on metric graphs (now in Appendix A) will not appear in the journal version. Proofs of the Dhar theorem and the Cori-Le Borgne theorem are in v1 but not in v2