Riemann-Roch Theory for Graph Orientations

Backman, Spencer

doi:10.48550/arxiv.1401.3309

Cited by 7 publications

(20 citation statements)

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“…(4) Acyclic: There are no directed cycles. (5) Cut directed: For each potential cut, if the minimum edge of the cut is neutral then the cut contains an oriented edge directed in agreement with the reference orientation of this minimum edge. (6) Cut negative: The minimum edge in each potential cut is neutral or is oriented in agreement with its reference orientation.…”

Section: Specializationsmentioning

confidence: 99%

“…In [4] and [5] the first author investigated two different extensions of Gioan's cyclecocycle reversal systems for partial orientations. One extension, which we call the cycle/cocycle reversal systems for partial orientations describes the set of partial orientations modulo cycle and/or cocycle reversals.…”

Section: Remark 411mentioning

confidence: 99%

“…The (co)graphic Lawrence ideals and their connection to fourientations are discussed in §4.6. The other extension, which we call the generalized cycle/cocycle reversal systems for partial orientations was introduced in [5] for the study of chip-firing in the context of Baker and Norine's combinatorial Riemann-Roch theorem [6]. In the next section we explain how this extension allows for a direct and aesthetically pleasing interpretation of the graphical Riemann-Roch duality in terms of fourientations.…”

Section: Remark 411mentioning

confidence: 99%

“…The generalized cycle/cocycle reversal systems and Riemann-Roch theory for fourientations. In [5], an edge pivot for partial orientations was defined as follows: given an edge e oriented towards a vertex v and e ′ a neutral edge incident to v, we may unorient e and orient e ′ towards v. This name is motivated by the image of an oriented edge nailed down at its head which can pivot to other unoriented edges. The generalized cycle, cocycle and cycle-cocycle reversal systems for partial orientations are defined to be these systems extended to partial orientations by the inclusion of edge pivots.…”

Section: 3mentioning

confidence: 99%

“…In §4 we outline how several of our cut and cycle properties relate to geometric, combinatorial, and algebraic objects including bigraphical hyperplane arrangements, cycle-cocycle reversal systems, graphic Lawrence ideals, divisors on graphs, zonotopal algebras, and the reliability polynomial. Recent developments in the study of divisors on graphs, including the commutative algebra of the abelian sandpile model [49] [39] [21] [48] [47], Riemann-Roch theory for graphs [6] [5], and geometrizations of the Matrix-Tree theorem [1], highlight the algebraic significance of the relationship between graph orientations and their indegree sequences. The partial orientation classes we define, which we term min-edge classes, appear to arise in many situations where one is interested in indegree sequences.…”

Section: Introductionmentioning

confidence: 99%

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Fourientations and the Tutte polynomial

Backman

Hopkins

2017

Res Math Sci

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A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we investigate properties of cuts and cycles in fourientations which give trivariate generating functions that are generalized Tutte polynomial evaluations of the formfor α, γ ∈ {0, 1, 2} and β, δ ∈ {0, 1}. We introduce an intersection lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion-contraction argument and classify axiomatically the set of fourientation classes to which our deletion-contraction argument applies. This work unifies and extends earlier results for fourientations due to , . arXiv:1205. We conclude by describing how these classes of fourientations relate to geometric, combinatorial, and algebraic objects including bigraphical arrangements, cycle-cocycle reversal systems, graphic Lawrence ideals, Riemann-Roch theory for graphs, zonotopal algebra, and the reliability polynomial.

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Section: Specializationsmentioning

confidence: 99%

Section: Remark 411mentioning

confidence: 99%

Section: Remark 411mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fourientations and the Tutte polynomial

Backman

Hopkins

2017

Res Math Sci

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Partial graph orientations and the Tutte polynomial

Backman¹

2018

Advances in Applied Mathematics

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Gessel and Sagan [9] investigated the Tutte polynomial, T (x, y) using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2 g T (3, 1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2 n−1 T (1/2, 3). We then prove that the number of partial orientations modulo cycle reversals is 2 g T (3, 1) and the number of partial orientations modulo cut reversals is 2 n−1 T (1, 3). To prove these results, we introduce cut and cycle minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals. These extend classes of total orientations introduced by Gioan [10], and Greene and Zaslavksy [12], and we highlight a close connection with graphic and cographic Lawrence ideals. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities, p with 0 < p < 1/2.

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Canonical Representatives for Divisor Classes on Tropical Curves and the Matrix–tree Theorem

et al. 2014

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Let Γ be a compact tropical curve (or metric graph) of genus g. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree g on Γ . We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an 'integral' version of this result which is of independent interest. As an application, we provide a 'geometric proof' of (a dual version of) Kirchhoff's celebrated matrixtree theorem. Indeed, we show that each weighted graph model G for Γ gives rise to a canonical polyhedral decomposition of the g-dimensional real torus Pic g (Γ ) into parallelotopes C T , one for each spanning tree T of G, and the dual Kirchhoff theorem becomes the statement that the volume of Pic g (Γ ) is the sum of the volumes of the cells in the decomposition.

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Riemann-Roch Theory for Graph Orientations

Cited by 7 publications

References 0 publications

Fourientations and the Tutte polynomial

Fourientations and the Tutte polynomial

Partial graph orientations and the Tutte polynomial

Canonical Representatives for Divisor Classes on Tropical Curves and the Matrix–tree Theorem

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