Chip-Firing Games and Critical Groups
In this note we introduce a finite abelian group that can be associated with 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 highlight some of the many contributions to this area made by undergraduate students.
Glass, Darren B. and Nathan Kaplan. "Chip Firing Games and Critical Groups." In A Project-Based Guide to Undergraduate Research in Mathematics, edited by Pamela Harris, Erik Insko, and Aaron Wootton, 107–152. Cham: Birkhäuser Basel, 2020.
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