A chip-firing game and Dirichlet eigenvalues

We consider a variation of the chip-firing game in a induced subgraph S of a graph G. Starting from a given chip configuration, if a vertex v has at least as many chips as its degree, we can fire v by sending one chip along each edge from v to its neighbors. The game continues until no vertex can be fired. We will give an upper bound, in terms of Dirichlet eigenvalues, for the number of firings needed before a game terminates. We also examine the relations among three equinumerous families, the set of spanning forests on S with roots in the boundary of S, a set of "critical" configurations of chips, and a coset group, called the sandpile group associated with S.

• Type: Journal
• Year: 2002
• Where: DM
• Authors: Fan R. K. Chung, Robert B. Ellis
• Keywords: Chips, DM 2002, Game, Graph G