Chip-firing games, potential theory on graphs, and spanning trees

Details Chip-firing games, potential theory on graphs, and spanning trees Chip-firing games, potential theory on graphs, and spanning trees

2011-07-07

arxiv.org/abs/1107.1313v1

Mathematics

Combinatorics

Scientific paper

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-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 time 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.

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