Divisors and Sandpiles: An Introduction to Chip-Firing

The book has three parts that are quite independent from each other. The first part, “Divisors”, consists of five chapters focusing on the dollar game, in which vertices of a graph lend money to each other, with the goal of all vertices becoming debt-free by the end of the game. In this context, a divisor is just an element of the free abelian group on the vertices of the graph. The subsequent chapters of this part deepen the analysis of the game, with the use of techniques known from a diverse collection of areas, such as the Laplacian operator, the Smith normal form, and Riemann surfaces.

The second part discusses the sandpile model. Each vertex of a graph is given an amount of sand, and if a vertex has too much sand, its sandpile will topple, or fire, sending sand to its neighbor. One vertex is designated as the sink, and sand that is sent to the sink will disappear there. Markov chains are introduced. The part, that consists of three chapters, can mostly be read without reading Part 1 first.

Part 3 is almost as long as the preceding two parts combined, but is not as focused. It consists of seven independent chapters on various subjects that are connected to the preceding topics in some way, such as the matrix-tree theorem, parking functions, and matroids. These can be read in any order.

There are many exercises, but they do not come with solutions. This reviewer has only one concern, the audience. The authors say that the book is for advanced undergraduates. They should be really advanced. In the first chapter, the book uses the notion of a free abelian group, in the second chapter, we see commutative diagrams and group homomorphisms, later in the first part, we hear about the genus of a graph and Riemann surfaces, and the list goes on and on. Sure, there could be a few undergraduates here and there who are ready for this, but do not reserve an auditorium for the course.

Buy Now

Miklós Bóna is Professor of Mathematics at the University of Florida.