Abelian sandpile model and Biggs-Merino polynomial for directed graphs

Swee Hong Chan

Published 2014-12-15Version 1

This paper is motivated by the result of Merino L{\'o}pez that for an undirected graph G and a specified sink s, the Biggs-Merino polynomial, which is defined as a generating function of recurrent configurations of abelian sandpile model with sink, is equal to the Tutte polynomial of G. Perrot and Pham extended the definition of Biggs-Merino polynomial to directed graphs and conjectured that this polynomial is independent of the choice of sink. In this paper, we give a proof of the conjecture of Perrot and Pham, and answer the conjecture with an affirmative answer. We also observe that the Biggs-Merino polynomial is equal to the greedoid Tutte polynomial when G is an Eulerian digraph, generalizing Merino's Theorem to the setting of Eulerian digraphs.