Sandpile groups and spanning trees of directed line graphs

Authors:
Lionel Levine
Affiliations:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

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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.