Chip-Firing and Riemann-Roch Theory for Directed Graphs

TL;DR

This paper extends Riemann-Roch theory to directed graphs, exploring chip-firing games, parking functions, and sandpile models, and introduces new criteria and algorithms for analyzing these structures.

Contribution

It generalizes Riemann-Roch criteria to all integer lattices orthogonal to positive vectors and links chip-firing games to directed graph properties.

Findings

Column chip-firing relates to directed parking functions.

Row chip-firing relates to the sandpile model.

Arithmetical graphs can satisfy Riemann-Roch conditions in specific cases.

Abstract

We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a $v_0$-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the "column" chip-firing game is related to directed $\vec{G}$-parking functions and the "row" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, which after a simple transformation may be viewed as a special class of directed graphs which will always have the Riemann-Roch property for the column chip-firing game. Examples of arithmetical graphs are provided which demonstrate that either, both, or neither of the two Riemann-Roch conditions may be satisfied for the row chip-firing game.