# Chip-Firing Graph Configurations

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This Demonstration shows a chip-firing game, which is also referred to as an Abelian sandpile model. Starting with a simple weighted directed graph and a number of chips to be distributed among the vertices of , a new graph is constructed with a vertex for each distribution of chips and an edge corresponding to each valid chip firing.

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Contributed by: Ken Levasseur (July 2018)

(UMass Lowell)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

If has vertices and there is a supply of chips to be distributed within the graph, there are

ways to distribute the chips. The chip-firing graph for with chips consists of vertices, one for each distribution of the chips. Directed edges of are drawn from one weighted graph to another if all valid chip-firing moves from the first weighted graph produces the second.

To run this Demonstration, select the number of vertices in and the number of chips. Then toggle the presence/absence of each edge. Mousing over any edge of the chip-firing graph displays the change in distribution of chips.

References

[1] N. L. Biggs. "Chip-Firing and the Critical Group of a Graph," *Journal of Algebraic Combinatorics*, 9(1), 1999 pp. 25–45. doi:10.1023/A:1018611014097.

[2] A. E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp and D. B. Wilson, "Chip-Firing and Rotor-Routing on Directed Graphs," *In and Out of Equilibrium 2. Progress in Probability* (V. Sidoravicius and M. E. Vares, eds.), Vol. 60, Basel: Birkhäuser Verlag, 2008 pp. 331–364. doi:10.1007/978-3-7643-8786-0_ 17.

[3] J. van den Heuvel, "Algorithmic Aspects of a Chip-Firing Game," *Combinatorics, Probability and Computing*, 10(6), 2001 505–529. doi:10.1017/S0963548301004886.

## Permanent Citation