Sandpile groups and spanning trees of directed line graphs
Combinatorics
2010-10-11 v2
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.
Cite
@article{arxiv.0906.2809,
title = {Sandpile groups and spanning trees of directed line graphs},
author = {Lionel Levine},
journal= {arXiv preprint arXiv:0906.2809},
year = {2010}
}
Comments
v2 has an expanded section on deletion/contraction for directed graphs, and a more detailed proof of Theorem 2.3. To appear in Journal of Combinatorial Theory A.