Complete Graphs and Polyominoes

TL;DR

This paper investigates chip-firing diffusion on complete graphs, quantifies distribution configurations, and reveals connections to polyominoes, advancing understanding of combinatorial structures in diffusion models.

Contribution

It provides a detailed enumeration of chip distributions on complete graphs and establishes a novel link to polyominoes, enriching combinatorial theory.

Findings

Number of chip distributions on unlabelled complete graphs determined

Connections established between chip-firing configurations and polyominoes

Enhanced understanding of combinatorial structures in diffusion processes

Abstract

In the chip-firing variant, Diffusion, chips flow from places of high concentration to places of low concentration (or equivalently, from the rich to the poor). We explore this model on complete graphs, determining the number of different ways that chips can be distributed on an unlabelled complete graph and demonstrate connections to polyominoes.