Sandpile groups and spanning trees of directed line graphs

Abstract. We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs.

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Lionel Levine
• Keywords: Directed Graph, JCT 2011, Oriented Spanning Trees, Sandpile Groups