author={David Perkinson and Jacob Perlman and John Wilmes},
  journal={arXiv: Combinatorics},
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 the theory of lattice ideals from al- gebraic geometry to the Laplacian matrix, drawing out connections with the ASM. A extended summary of the ASM and of the required algebraic ge- ometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a… 
Abelian Sandpile Model Of A Directed Multi-graph
In this project, we study the combinatorics, algebraic and algebraic geometry of an Abelian Sandpile Model (ASM). We aim to explore the Algebraic properties of the Abelian Sandpile Model (ASM). We
Sandpile Groups
The physicists Bak, Tang, and Wiesenfeld [5] created an idealized version of a sandpile in which sand is stacked on the vertices of a graph and is subjected to certain avalanching rules. They used
The Riemann-Roch theorem for graphs and the rank in complete graphs
The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavour. This result
On Computation of Baker and Norine's Rank on Complete Graphs
An algorithm for the determination of the rank of configurations for the complete graph K_n and an apparently new parameter which is called the prerank is presented which provides an alternative description to some well known $ q,t$-Catalan numbers.
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory
Simplicial and Cellular Trees Art
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory
Mixing time and eigenvalues of the abelian sandpile Markov chain
The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph G G . By viewing this chain as a (nonreversible)
Sandpiles and Dominos
A new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a $2m \times 2n$ rectangular checkerboard and a new way of counting the many-sided domino Tilings on a Mobius strip are provided.
Degree and Algebraic Properties of Lattice and Matrix Ideals
We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of $\mathbb{Z}^s$ and in terms of


Trees, parking functions, syzygies, and deformations of monomial ideals
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 ring modulo certain monomial
We offer a categorical presentation of the critical group of a graph. We take the lattice-theoretic approach to critical groups developed in [1], and consider the extent to which this construction is
The Critical Group of a Line Graph
AbstractThe critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of
Harmonic morphisms and hyperelliptic graphs
We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula,
Cayley-Bacharach schemes and their canonical modules
A set of s points in P d is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s − 1 points has the same Hilbert function. We investigate the consequences of this «weak uniformity.» The
On the Sandpile Group of Dual Graphs
It is proved that the sandpile group of planar graph is isomorphic to that of its dual, and a combinatorial point of view on the subject is developed.
Asymmetric Abelian Sandpile Models
In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system.
Gorenstein algebras and the Cayley-Bacharach theorem
This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in p2 and properties of graded Gorenstein algebras. Introduction. It is known,
Polynomial ideals for sandpiles and their Gröbner bases