53 research outputs found
Chip-firing may be much faster than you think
A new bound (Theorem \ref{thm:main}) for the duration of the chip-firing game
with chips on a -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 and , respectively. For dense regular graphs -
- the classic and the new bounds reduce to
and , respectively.
This is a snapshot of a work in progress, so further results in this vein are
in the works
A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
Hereditary chip-firing models generalize the Abelian sandpile model and the
cluster firing model to an exponential family of games induced by covers of the
vertex set. This generalization retains some desirable properties, e.g.
stabilization is independent of firings chosen and each chip-firing equivalence
class contains a unique recurrent configuration. In this paper we present an
explicit bijection between the recurrent configurations of a hereditary
chip-firing model on a graph and its spanning trees.Comment: 13 page
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