Chip-Firing on Trees of Loops

Kailasa, Sameer; Kuperberg, Vivian; Wawrykow, Nicholas

doi:10.37236/7244

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Ranks Of Divisors and Gonality Of Graphs1

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2020

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“…There are several additional interesting directions in the Brill-Noether theory of graphs and metric graphs that have been the subject of successful research projects with undergraduate coauthors. See for example, [27,44,50,51].…”

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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Section: Ranks Of Divisors and Gonality Of Graphsmentioning

confidence: 99%