Chip-firing may be much faster than you think

A new bound (Theorem 4.4) for the duration of the chip-firing game with N chips on a n-vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Björner, Lovász and Shor. In some cases the improvement is dramatic. For instance: for strongly regular graphs the classic and the new bounds reduce to O(nN) and O(n + N), respectively. For dense regular graphs d = ( 2 + ǫ)n the classic and the new bounds reduce to O(N) and O(n), respectively. This is a snapshot of a work in progress, so further results in this vein are in the works.