Algebraic and combinatorial aspects of sandpile monoids on directed graphs

Details Algebraic and combinatorial aspects of sandpile monoids on directed graphs Algebraic and combinatorial aspects of sandpile monoids on directed graphs

2011-05-12

arxiv.org/abs/1105.2357v1

Mathematics

Combinatorics

19 pages

Scientific paper

We study the abelian sandpile model on a finite directed graph. We begin by reviewing the necessary background material starting with the identification, by Babai and Toumpakari, of the sandpile group on a directed graph as the minimal ideal of its sandpile monoid, and continuing through some of their recent results concerning the connections between the idempotent structure of a sandpile monoid and the cycle structure of its graph. We then build on these results to give our first main result, which is a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph X in terms of the sandpile groups of certain easily-identifiable subgraphs of X. We then return to undirected graphs and give our second main result, which is a combinatorial classification of the sandpile group identity of every undirected distance regular graph. Along the way we give several new algebraic results for sandpiles based on directed graphs, and we point out parallels to previously known results for undirected graphs.

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