A combinatorial Li-Yau inequality and rational points on curves

Cornelissen, Gunther; Kato, Fumiharu; Kool, Janne

doi:10.48550/arxiv.1211.2681

Cited by 3 publications

(4 citation statements)

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“…Other notions of gonality of a graph G have been proposed by Caporaso [5] and by Cornelissen, Kato, and Kool in [6]. These notions are based on harmonic morphisms from G to a tree.…”

Section: Other Notions Of Gonalitymentioning

confidence: 99%

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Treewidth is a lower bound on graph gonality

Bruyn¹,

Gijswijt²

2020

Algebraic Combinatorics

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We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include the grid graphs and the complete multipartite graphs. We prove that the treewidth lower bound also holds for metric graphs (tropical curves) by constructing for any positive rank divisor on a metric graph a positive rank divisor of the same degree on a subdivision of the underlying combinatorial graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al.

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“…Other notions of gonality of a graph G have been proposed by Caporaso [5] and by Cornelissen, Kato, and Kool in [6]. These notions are based on harmonic morphisms from G to a tree.…”

Section: Other Notions Of Gonalitymentioning

confidence: 99%

“…In [6], Cornelissen, Kato and Kool define the stable gonality sgon(G) of G to be the minimum degree of an indexed harmonic homomorphism from a refinement of G to a tree T . Note that a harmonic homomorphism is automatically non-degenerate.…”

Section: Other Notions Of Gonalitymentioning

confidence: 99%

Treewidth is a lower bound on graph gonality

Bruyn¹,

Gijswijt²

2020

Algebraic Combinatorics

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“…At the time of writing, we became aware of simultaneous work by Amini and Kool, in which they use an improvement on the spectral methods of [CFK13] to show that the gonality of a random graph is bounded above and below by constant multiples of n [AK14]. Our results are essentially a tightening of these bounds, so that both upper and lower bounds are asymptotic to n, which indeed is conjectured in [AK14, Section 5.2].…”

Section: Introductionmentioning

confidence: 69%

Gonality of random graphs

Deveau

Jensen

Kainic

et al. 2016

Involve

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“…Through the tropical Riemann-Roch theorem [BN07, GK08, MZ08], Baker's specialization lemma and its generalizations [Bak08b, AB12, AC13], the nonarchimedean Poincaré-Lelong formula [Thu05,BPR11], and the theory of harmonic morphisms of metric graphs [BN09,ABBR13], this metric is a powerful tool in the study of linear series on algebraic curves. It has been used to characterize dual graphs of special fibers of regular semistable models of curves of a given gonality [Cap12], to compute the gonality of curves that are generic with respect to their Newton polygon [CC12], to characterize the Newton polygons of Brill-Noether general curves in toric surfaces [Smi14], to bound the gonality of Drinfeld modular curves [CKK12], and to give new proofs of the Brill-Noether and Gieseker-Petri theorems [CDPR12,JP14].…”

Section: Relations To Complex Algebraic Geometrymentioning

confidence: 99%