Gonality of random graphs

Deveau, Andrew; Jensen, David; Kainic, Jenna; Mitropolsky, Dan

doi:10.2140/involve.2016.9.715

Cited by 14 publications

(13 citation statements)

References 22 publications

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“…Finally, we like to mention two independent simultaneous works [20] and [21]. In [20] the authors prove the above mentioned conjecture of [19], and they extend it to metric graphs.…”

Section: Introductionmentioning

confidence: 93%

A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

Amini¹,

Kool

2015

Int Math Res Notices

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Abstract. Let Γ be a compact metric graph, and denote by ∆ the Laplace operator on Γ with the first non-trivial eigenvalue λ1. We prove the following Yang-Li-Yau type inequality on divisorial gonality γ div of Γ. There is a universal (explicit) constant C such thatwhere the volume µ(Γ) is the total length of the edges in Γ, ℓ geo min is the non-zero minimum length of all the geodesic paths between points of Γ of valence different from two, and dmax is the largest valence of points of Γ. Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of Γ and their spectral gaps.

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“…Finally, we like to mention two independent simultaneous works [20] and [21]. In [20] the authors prove the above mentioned conjecture of [19], and they extend it to metric graphs.…”

Section: Introductionmentioning

confidence: 93%

A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

Amini¹,

Kool

2015

Int Math Res Notices

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“…For example, de Bruyn and Gijswijt connect the gonality of a graph to the notion of treewidth, an important concept in graph theory [31]. The authors of [32] study the gonality of Erdős-Rényi random graphs and prove the following theorem.…”

Section: Ranks Of Divisors and Gonality Of Graphsmentioning

confidence: 99%

Chip-Firing Games and Critical Groups

Glass

Kaplan

2020

Foundations for Undergraduate Research in Mathematics

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In this note we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also attempt to highlight some of the many contributions to this area made by undergraduate students.Suggested prerequisites. The basic definitions and themes of this note should be accessible to any student with some knowledge of linear algebra and group theory. As we go along, deeper understanding of graph theory, abstract algebra, and algebraic geometry will be of use in some sections.

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“…The spectral bound from [CFK15] was subsequently refined by Amini and Kool in [AK14] to a spectral lower bound for the divisorial gonality (i.e., the minimal degree of a rank 1 divisor) of a metric graph Γ. In [AK14], as well as in the related paper [DJKM14], this circle of ideas is applied to show that the expected gonality of a random graph is asymptotic to the number of vertices.…”

Section: Further Readingmentioning

confidence: 99%