The totally nonnegative Grassmannian Gr ( k , n ) ≥ 0 is the subset of the real Grassmannian Gr ( k , n ) consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of Postnikov's positroid cell decomposition of Gr ( k , n ) ≥ 0 [10] . We provide a closed formula for the sum of its weighted chains in the spirit of Stembridge [11] .

The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chipfiring moves. We use Duval, Klivans, and Martin???s theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus 0) are exactly the graphs on which every divisor of degree 0, interpreted as an instance of the dollar game, is winnable.

The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G , and linear equivalence of divisors is determined by the discrete Laplacian operator for G . As in the case of Riemann surfaces, we are interested in the complete linear system | D | of a divisor D —the collection of nonnegative divisors linearly equivalent to D . Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G . If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. Our results also apply to a model in which the Laplacian is replaced by an invertible, integral M -matrix.

Let G be a finite graph, and let G_n be the n-th iterated cone over G. We study the structure of the critical group of G_n arising in divisor and sandpile theory.

A depth-first search version of Dhar’s burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph solves a problem posed by R. Stanley.

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 model as an example of what they called self-organized criticality. The abelian sandpile model is a variation, due to the physicist Deepak Dhar in 1990 [24], in which the avalanching obeys a useful commutativity rule. He realized that the model provided an expression of the dynamics inherent in the discrete Laplacian of a graph. The long-term behavior of the abelian sandpile model on a graph is encoded by the critical configurations. These critical configurations have connections to parking functions [45], to the Tutte polynomial [46], and to the lattices of integral flows and cuts of a graph [53]. Among other properties, the critical configurations of the sandpile model have the structure of a group, and this group is our main object of study. It has been discovered in several different contexts and received many names: the sandpile group for graphs [24] and digraphs [49], the critical group [8], the group of bicycles [7], the group of components [51], and the jacobian of the graph [54]. The abelian sandpile model and its close relative the chip-firing game [44] have become a crossroads of a wide range of mathematics, physics, and computer science. Of particular interest in this workshop were connections with: commutative algebra, algebraic and tropical geometry, pattern formation, models of computation, generalizations of chip-firing, matroid theory, graph orientations, and tree-bijections, and random graphs.

We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley in 1996. We also give bounds on the number of regions of a bigraphical arrangement.

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.

We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a 2m x 2n rectangular checkerboard and a new way of counting the number of domino tilings of a 2m x 2n checkerboard on a Mobius strip.

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 algebraic geometry to the Laplacian matrix, drawing out connections with the ASM.An extended summary of the ASM and of the required algebraic geometry is provided.New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree.A characterization of all the Betti numbers is conjectured.

It is known that the Pak-Stanley labeling of the Shi hyperplane arrangement provides a bijection between the regions of the arrangement and parking functions. For any graph G, we define the G-semiorder arrangement and show that the Pak-Stanley labeling of its regions produces all G-parking functions.

A real representation of a finite group naturally determines a polytope, generalizing the well-known Birkhoff polytope. This paper determines the structure of the polytope corresponding to the natural permutation representation of a general Frobenius group.

This note answers a question posed by Levine. The main result shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.

In this thesis, we define the toppling Betti numbers and minimal free resolutions of toppling ideals. We give a proof of the Riemann-Roch Theorem first stated and proved in (1), and use the theorem to prove that the last Betti number of an undirected graph is the number of minimal recurrent configurations. We describe a minimal complex that we conjecture to be a free resolution for a toppling ideal, and use the conjecture to compute the Betti numbers of several graphs.

Each group G of n×n permutation matrices has a corresponding permutation polytope, P(G):=conv(G)⊂Rn×n. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t,⌊n/2⌋} is a sharp upper bound on the diameter of the graph of P(G). We also show that P(G) achieves its maximal dimension of (n−1)2 precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83–90] on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.

We describe a class of facets of the polytope of convex combinations of the collection of even n×n permutation matrices. As a consequence, we prove the conjecture of Brualdi and Liu [J. Combin. Theory Ser. A 57 (1991) 243] that the number of facets of the polytope is not bounded by a polynomial in n.