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 $N$ 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

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

DISCRETE INVERSE CONDUCTIVITY PROBLEMS ON NETWORKS

The purpose of this dissertation is to present a mathematical model of network tomography through spectral graph theory analysis. In this regard, we explore the properties of harmonic functions and eigensystems of Laplacians for weighted graphs (networks) with and without boundary. We prove the solvability of the Dirichlet and Neumann boundary value problems. We also prove the global uniqueness of the inverse conductivity problem on a network under a suitable monotonicity condition. As a physical interpretation to the discrete inverse conductivity problem, we define a variant of the chip-firing game (a discrete balancing process) in which chips are added to the game from the boundary nodes and removed from the game if they are fired into the boundary of the graph. We find a bound on the length of the game, and examine the relations between set of spanning weighted forest rooted in the boundary of the graph and the set of critical configurations of the chips

Identity Configurations of the Sandpile Group

The abelian sandpile model on a connected graph yields a finite abelian group Q of recurrent configurations which is closely related to the combinatorial Laplacian. We consider the identity configuration of the sandpile group on graphs with large edge multiplicities, called “thick” graphs. We explicitly compute the identity configuration for all thick paths using a recursion formula. We then analyze the thick cycle and explicitly compute the identity configuration for the three-cycle, the four-cycle, and certain types of symmetric cycles. The latter is a special case of a more general symmetry theorem we prove that applies to an arbitrary graph.https://digitalcommons.imsa.edu/sci_dsw/1001/thumbnail.jp