Chip-Firing and the Critical Group of a Graph
Journal of Algebraic Combinatorics: An International Journal
A chip-firing game and Dirichelt eigenvalues
Discrete Mathematics - Kleitman and combinatorics: a celebration
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
CG '00 Revised Papers from the Second International Conference on Computers and Games
Classes of lattices induced by chip firing (and sandpile) dynamics
European Journal of Combinatorics
Algorithmic Aspects of a Chip-Firing Game
Combinatorics, Probability and Computing
Sandpile models and lattices: a comprehensive survey
Theoretical Computer Science - Discrete applied problems, florilegium for E. Goles
On the sandpile group of regular trees
European Journal of Combinatorics
Strict partitions and discrete dynamical systems
Theoretical Computer Science
On the Complexity of Sandpile Prediction Problems
Electronic Notes in Theoretical Computer Science (ENTCS)
Combinatorics, Probability and Computing
Algebraic and combinatorial aspects of sandpile monoids on directed graphs
Journal of Combinatorial Theory Series A
Lattices generated by Chip Firing Game models: Criteria and recognition algorithms
European Journal of Combinatorics
Control of limit states in absorbing resource networks
Automation and Remote Control
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We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the “probabilistic abacus” can be derived from our results.