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

“chipfiring” is an open-source Python package that implements the full mathematical machinery of chip-firing (a.k.a. the dollar game or sandpile model) on finite, undirected multigraphs. It provides object-oriented data structures for graphs and divisors, fast implementations of Dhar’s burning algorithm, linear-equivalence and q-reduction routines, divisor-rank computations connected to the Riemann–Roch theorem, and optional interactive visualisation via Dash. Benchmarks show polynomial-time winnability checks on graphs with hundreds of vertices and edges, filling a long-standing gap in the Python scientific-computing ecosystem for chip-firing research and education.

arXiv preprint (arXiv:2505.01234)