On Picard groups and Jacobians of directed graphs

This repository serves as an archive for the code used for the research in this paper.

We focus on the study of Chip-Firing games and how different combinations of directed and undirected edges affect its winning strategies. The paper itself can be found with this arXiv link.

Abstract

The Picard group of an undirected graph is a finitely generated abelian group, and the Jacobian is the torsion subgroup of the Picard group. These groups can be computed by using the Smith normal form of the Laplacian matrix of the graph or by using chip-firing games associated with the graph. One may consider its generalization to directed graphs based on the Laplacian matrix. We compute Picard groups and Jacobians for several classes of directed trees, cycles, wheel, and multipartite graphs.

Submissions and Conferences

This paper was accepted into the Joint Mathematics Meetings 2023 and presented no January 6th 2023. We also plan to submit this paper to the Journal of Experimental Mathematics and the Journal of Combinatorial Theory.

Documentation

The documentation for this project of course comes within the paper itself, but also within several presentations that I had prepared over the course of working on this project. These can be found in the Docs/ folder.

Code

The code is split into several files with each having a specific role.

• The game.py file contains the utility classes for Graphs and Divisors that were instrumental in representing the abstract concepts associated with our research.

• The algorithms.py file contains the algorithms that we used to either computationally prove or guide our rigorous proofs with experimental results.

• The utils.py file centers around the computations for the smith normal form. These can be quite heavy, so I made several optimizations to the code in the places that I could.