Abelian sandpile model and Biggs-Merino polynomial for directed graphs

TL;DR

This paper extends the theory of the sandpile model to directed graphs, proving a sink-independence conjecture, establishing a bijection with parking functions, and linking the polynomial to the greedoid polynomial in Eulerian graphs.

Contribution

It proves sink-independence of the Biggs-Merino polynomial for directed graphs and connects it to arborescences, parking functions, and greedoid polynomials, extending known results from undirected graphs.

Findings

Proved the sink-independence conjecture for the polynomial.

Established a bijection between arborescences and reverse G-parking functions.

Extended Merino's Theorem to Eulerian directed graphs.

Abstract

We prove several results concerning a polynomial that arises from the sandpile model on directed graphs; these results are previously only known for undirected graphs. Implicit in the sandpile model is the choice of a sink vertex, and it is conjectured by Perrot and Pham that the polynomial $c_0+c_1y+\ldots c_n y^n$, where $c_i$ is the number of recurrent classes of the sandpile model with level $i$, is independent of the choice of the sink. We prove their conjecture by expressing the polynomial as an invariant of the sinkless sandpile model. We then present a bijection between arborescences of directed graphs and reverse $G$-parking functions that preserves external activity by generalizing Cori-Le Borgne bijection for undirected graphs. As an application of this bijection, we extend Merino's Theorem by showing that for Eulerian directed graphs the polynomial $c_0+c_1y+\ldots c_n y^n$ is equal to the greedoid polynomial of the graph.