Chip-firing may be much faster than you think

TL;DR

This paper introduces a new bound for the chip-firing game duration on graphs, which significantly improves upon classic bounds by leveraging the pseudo-inverse of the graph Laplacian, especially in dense and regular graphs.

Contribution

The paper derives a novel bound for chip-firing duration based on the pseudo-inverse of the Laplacian, outperforming previous bounds in various graph classes.

Findings

New bound always better than classic Bj{"o}rner-Lov{á}sz-Shor bound

Dramatic improvements for strongly regular graphs

Significant reduction in bounds for dense regular graphs

Abstract

A new bound (Theorem \ref{thm:main}) 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 Bj{\"o}rner, Lov\'{a}sz 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 - $d=(\frac{1}{2}+\epsilon)n$ - 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.