The Mathematics of Chip-firing is a solid introduction and overview of the growing field of
chip-firing. It offers an appreciation for the richness and diversity of the subject. Chip-firing …

Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on
finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar …

For a graph $ G $, we construct two algebras whose dimensions are both equal to the
number of spanning trees of $ G $. One of these algebras is the quotient of the polynomial …

We study the interplay between chip-firing games and potential theory on graphs,
characterizing reduced divisors (G-parking functions) on graphs as the solution to an energy …

The process called the chip-firing game has been around for no more than 20 years, but it
has rapidly become an important and interesting object of study in structural combinatorics …

We develop a new framework for investigating linear equivalence of divisors on graphs
using a generalization of Gioan's cycle–cocycle reversal system for partial orientations. An …

Given an undirected graph G=(V, E), and a designated vertex q∈ V, the notion of a G-
parking function (with respect to q) was independently developed and studied by various …

The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics
implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply …

We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize
graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building …

We give some new enumerations of indegree sequences of orientations of a graph using the
Tutte polynomial. Then we introduce some discrete dynamical systems in digraphs …