chipfiring: A Python Package for Efficient Mathematical Analysis of Chip-Firing Games on Multigraphs

TL;DR

The paper introduces `chipfiring`, a Python package that facilitates efficient mathematical analysis of chip-firing games on multigraphs, supporting operations like firing, winnability, and divisor analysis.

Contribution

It provides a novel, comprehensive Python toolkit with optimized algorithms and object-oriented design for studying chip-firing games on finite graphs.

Findings

Efficient implementation of Dhar's algorithm for dollar game solvability

Robust object-oriented framework for graph and divisor manipulation

Illustrative examples demonstrating package capabilities

Abstract

This paper presents `chipfiring`, a comprehensive Python package for the mathematical analysis of chip-firing games on finite graphs. The package provides a robust toolkit for defining graphs and chip configurations (divisors), performing chip-firing operations, and analyzing fundamental properties such as winnability, linear equivalence, and divisor rank. We detail the core components of the library, including its object-oriented graph and divisor implementations, integrated Laplacian matrix computations, and an efficient implementation of Dhar's algorithm for determining the solvability of the dollar game. The `chipfiring` package is designed for researchers and students in graph theory, combinatorics, and algebraic geometry, providing essential algorithms and data structures for exploring these rich mathematical models. We describe the library's architecture, illustrate its usage with comprehensive examples, and highlight its specialized contributions compared to general-purpose graph libraries.