Chip-firing games on graphs.

*(English)*Zbl 0729.05048Summary: We analyse the following (solitaire) game: each node of a graph contains a pile of chips, and a move consists of selecting a node with at least as many chips on it as its degree, and letting it send one chip to each of its neighbors. The game terminates if there is no such node. We show that the finiteness of the game and the terminating configuration are independent of the moves made. If the number of chips is less than the number of edges, the game is always finite. If the number of chips is at least the number of edges, the game can be infinite for an appropriately chosen initial configuration. If the number of chips is more than twice the number of edges minus the number of nodes, then the game is always infinite.

The independence of the finiteness and the terminating position follows from simple but powerful ‘exchange properties’ of the sequences of legal moves, and from some general results on ‘antimatroids with repetition’, i.e. languages having these exchange properties. We relate the number of steps in a finite game to the least positive eigenvalue of the Laplace operator of the graph.

The independence of the finiteness and the terminating position follows from simple but powerful ‘exchange properties’ of the sequences of legal moves, and from some general results on ‘antimatroids with repetition’, i.e. languages having these exchange properties. We relate the number of steps in a finite game to the least positive eigenvalue of the Laplace operator of the graph.

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\textit{A. Björner} et al., Eur. J. Comb. 12, No. 4, 283--291 (1991; Zbl 0729.05048)

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