Parallel chip firing games on graphs

We study the periodic behaviour of parallel dynamics associated with the chip firing game introduced by Spencer (1986). We prove that on trees the periods are of length one (fixed points) or two. For general graphs we give counter-examples where periods depend on the graph size. We also prove that the chip firing game may be used as an algorithm to evaluate logical functions, which shows that the dynamics have nontrivial computing capabilities. Finally, we study a related game on graphs.