Divisors and Sandpiles

Corry, Scott; Perkinson, David

doi:10.1090/mbk/114

Cited by 53 publications

(48 citation statements)

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“…Since |D| is finite, it follows that the polyhedron P D is bounded, and hence is a polytope. (For a direct proof of boundedness, see [8,Proposition 2.20].) If D ∼ D ′ with D ′ = D + Lf , then the polyhedra associated with these divisors differ by a translation: Q D = f + Q D ′ , and as discussed above, we may assume f n = 0 to write P D = f + P D ′ .…”

Section: 2mentioning

confidence: 99%

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Enumerating linear systems on graphs

2020

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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. The final section generalizes our results to a model based on integral M -matrices.2010 Mathematics Subject Classification. primary 05C30, secondary 05C25. For the combinatorial structure of |D| in the case of a metric graph (tropical curve), see [13]. 1 Divisor theory preliminariesLet G = (V, E) be a connected, undirected multigraph with finite vertex set V and finite edge multiset E. Many of our constructions will depend on fixing a vertex q ∈ V , which we do now, once and for all. Loops are allowed but our results are not affected if they are removed. We let N := Z ≥0 denote the natural numbers.We recall some of the theory of divisors on graphs, referring readers unfamiliar with this theory to [3] or to the textbooks [8] and [15]. A divisor on G is an element of the free abelian group on the vertices of G,The degree of a divisor D is the sum of its coefficients: deg(D) := v∈V D(v). For instance, if we consider v ∈ V as a divisor, then deg(v) = 1. We use the notation deg G (v) to refer to the ordinary degree of a vertex-the number of edges incident on v. The set of divisors of degree k is denoted by Div k (G). The (discrete) Laplacian operator of G is the function L : Z V → Z V given by

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Section: 2mentioning

confidence: 99%

“…In all of the above, if m k = 1 for some k, then the k-th factor of R and the k-th row of U may be dropped. we have ULW = diag (1,1,8) and the corresponding isomorphism…”

Section: Linear Systemsmentioning

confidence: 99%

Enumerating linear systems on graphs

2020

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“…If D ∈ Div 0 (G), we denote the equivalence class containing D by [D]. Jac(G) is always finite for a connected graph, and its order is equal to the number of spanning trees on G by the matrixtree theorem (see [1], or [6]).…”

Section: 1mentioning

confidence: 99%

“…The Jacobian of a graph is equal to the torsion subgroup of the cokernel of the graph's Laplacian, which is an n × n integer matrix denoted by L and defined as follows. Let ∆ be the diagonal matrix with (i, i)-entry equal to val(v i ), and let A be the adjacency matrix of G. Then L = ∆ − A. L gives a map from the the space of firing scripts to the space of principal divisors: if σ is a firing script, then Lσ is the principal divisor obtained by applying σ to the zero divisor (again, see [1] or [6]). For a given choice of i ∈ {1, ..., n} the reduced LaplacianL is L with the ith row and ith column removed.L is invertible if G is connected, regardless of the choice of i, and furthermore we have det(L) = |Jac(G)|.…”

Section: 1mentioning

confidence: 99%

Two-Vertex Generators of Jacobians of Graphs

Brandfonbrener

Devlin

Netanel

et al. 2018

Electron. J. Combin.

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We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.

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“…As it turns out, this chip-firing process is essentially the same as the Abelian Sandpile Model, originally introduced by the physicists Bak, Tang, and Wiesenfeld [BTW87] and subsequently developed by Dhar [Dha90,Dha99]. For more on chip-firing and sandpiles, we refer the reader to [LP10,CP18].…”

Section: Introductionmentioning

confidence: 99%

Root System Chip-Firing II: Central-Firing

Galashin

Hopkins

McConville

et al. 2019

International Mathematics Research Notices

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Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-McConville-Propp. We reinterpret Propp's labeled chip-firing moves in terms of root systems: a "central-firing" move consists of replacing a weight λ by λ + α for any positive root α that is orthogonal to λ. We show that centralfiring is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root systems we describe this unlabeled chip-firing as a number game on the Dynkin diagram. We also offer a conjectural classification of when central-firing is confluent from the origin or a fundamental weight.

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Divisors and Sandpiles

Cited by 53 publications

References 0 publications

Enumerating linear systems on graphs

Enumerating linear systems on graphs

Two-Vertex Generators of Jacobians of Graphs

Root System Chip-Firing II: Central-Firing

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