Gonality of expander graphs

Dutta, Neelav; Jensen, David

doi:10.1016/j.disc.2018.06.012

Cited by 5 publications

(4 citation statements)

References 18 publications

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“…In [23], a lower bound for the divisorial gonality of a graph is given in terms of its expansion. The gonality of a graph G (in any sense) is larger than or equal to its treewidth tw(G) [22].…”

Section: Known Resultsmentioning

confidence: 99%

Recognizing Hyperelliptic Graphs in Polynomial Time

Bodewes

Bodlaender

Cornelissen

et al. 2018

Graph-Theoretic Concepts in Computer Science

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Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. Various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs, called stable gonality and stable divisorial gonality. We consider so-called hyperelliptic graphs (multigraphs of gonality 2, in any meaning of graph gonality) and provide a safe and complete set of reduction rules for such multigraphs. This results in an algorithm to recognize hyperelliptic graphs in time O (m + n log n), where n is the number of vertices and m the number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph G admits an involution σ such that the quotient G/ σ is a tree.

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Section: Known Resultsmentioning

confidence: 99%

Recognizing Hyperelliptic Graphs in Polynomial Time

Bodewes

Bodlaender

Cornelissen

et al. 2018

Graph-Theoretic Concepts in Computer Science

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“…Note that one can ask the same question about a graph: what is the expected gonality of a random graph? (The answer depends on what one means by a random graph; see [DJKM16,DJ18]. )…”

Section: Going Furthermentioning

confidence: 99%

Chip Firing and Algebraic Curves

Jensen¹

2021

Notices Amer. Math. Soc.

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“…Amini and Kool show in [2] that random d-regular graphs on n vertices have gonality bounded above and below by constant multiples of n. Connections to tropical geometry led the authors of [32] to ask about the gonality of random 3-regular graphs. Dutta and Jensen prove a lower bound for the gonality of a regular graph G in terms of the Cheeger constant of G, one of the most studied measures of graph expansion [34]. They also give a lower bound for gonality of a general graph G in terms of its algebraic connectivity, the second smallest eigenvalue of L(G).…”

Section: Ranks Of Divisors and Gonality Of Graphsmentioning

confidence: 99%

“…[34, Theorem 1.3] Let G be a random 3-regular graph on n vertices. Then gon(G) ≥ 0.0072n asymptotically almost surely.Research Project 10.…”

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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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Gonality of expander graphs

Abstract: We provide lower bounds on the gonality of a graph in terms of its spectral and edge expansion. As a consequence, we see that the gonality of a random 3-regular graph is asymptotically almost surely greater than one seventh its genus.

Cited by 5 publications

References 18 publications

Recognizing Hyperelliptic Graphs in Polynomial Time

Recognizing Hyperelliptic Graphs in Polynomial Time

Chip Firing and Algebraic Curves

Chip-Firing Games and Critical Groups

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