The lattice of integral flows and the lattice of integral cuts on a finite graph

Bacher, Roland; Harpe, Pierre de la; Nagnibeda, Tatiana

doi:10.24033/bsmf.2303

Cited by 160 publications

(231 citation statements)

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“…Finally, we recall some facts from [BN] and [BdlHN97] about the graph-theoretic analogue of the Abel-Jacobi map from a Riemann surface to its Jacobian.…”

Section: 2mentioning

confidence: 99%

“…In [BN], the authors investigated some new analogies between graphs and Riemann surfaces, formulating the notion of a linear system on a graph and proving a graph-theoretic analogue of the classical Riemann-Roch theorem. The theory of linear systems on graphs has applications to understanding the Jacobian of a finite graph, a group which is analogous to the Jacobian of a Riemann surface, and which has appeared in many different guises throughout the literature (e.g., as the "Picard group" in [BdlHN97], the "critical group" in [Big97], the "sandpile group" in [Dha90], and the "group of components" in [Lor91]). …”

Section: 2mentioning

confidence: 99%

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Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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Abstract. 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 RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

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“…Finally, we recall some facts from [BN] and [BdlHN97] about the graph-theoretic analogue of the Abel-Jacobi map from a Riemann surface to its Jacobian.…”

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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“…The same group was studied from different viewpoints and with different names: Berman [Be86], Lorenzini [Lo89], Dahr [Da90] (sandpile group), Bacher-de la Harpe-Nagnibeda [BDN97] (Picard group), Biggs [Bi99] (critical group). (7.5) When is the r-torsion subgroup of Φ Γ isomorphic to (Z/rZ) ⊕b 1 ?…”

Section: Finite Néron D-models Of the Torsor Of Rth Rootsmentioning

confidence: 99%

Quantitative Néron theory for torsion bundles

Chiodo

2009

manuscripta math.

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Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth curve C K over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme Pic C K [r] of r-torsion line bundles on C K . We determine when there exists a finite R-group scheme, which is a model of Pic C K [r] over R; in other words, we establish when the Néron model of Pic C K [r] is finite. To this effect, one needs to analyse the points of the Néron model over R, which, in general, do not represent r-torsion line bundles on a semistable reduction of C K . Instead, we recast the notion of models on a stacktheoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of C K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it.

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“…Other important references are [Big99], [BdlHN97] and [CR02]. We addressed this problem in [BMS06], where in particular we constructed a family of graphs with cyclic complexity group.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, given a stable (or, more generally, nodal) curve C we call ∆ C the complexity group of the dual graph of the curve. This group has been extensively studied as an invariant of graphs, with applications to Physics, Chemistry, and many other areas, and it goes under many different names, such as critical group [Big99] [CR02], determinant group [BdlHN97], Picard group [BdlHN97], Jacobian group [BN07], abelian sandpiles group [CE02]. From the point of view of geometry, the complexity group was introduced in [Cap94], with the name of degree class group, in order to describe and handle the fibres of the compactification of the universal Picard variety P d,g over M g (also constructed in the same article).…”

Section: Introductionmentioning

confidence: 99%

On the complexity group of stable curves

Busonero

Melo

Stoppino

2011

Advg

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In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the Néron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus g of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for g ≫ 0. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction.

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The lattice of integral flows and the lattice of integral cuts on a finite graph

Cited by 160 publications

References 10 publications

Harmonic Morphisms and Hyperelliptic Graphs

Harmonic Morphisms and Hyperelliptic Graphs

Quantitative Néron theory for torsion bundles

On the complexity group of stable curves

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