Sandpile groups and spanning trees of directed line graphs

TL;DR

This paper extends a theorem relating spanning trees and sandpile groups of directed graphs and their line graphs, providing new insights and explicit calculations for well-known graph families.

Contribution

It generalizes Knuth's theorem to directed line graphs and computes sandpile groups for de Bruijn and Kautz graphs.

Findings

Sandpile group of G is isomorphic to a quotient of that of LG by its k-torsion subgroup.

Explicit computation of sandpile groups for de Bruijn and Kautz graphs.

Generalization of the relationship between spanning trees and sandpile groups to 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.