Geodesic
An Algebraic Analogue of a Formula of Knuth
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. 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 $\mathcal{L} G$ 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. 
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     title = {An {Algebraic} {Analogue} of a {Formula} of {Knuth}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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     year = {2010},
     doi = {10.46298/dmtcs.2867},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2867/}
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Levine, Lionel. An Algebraic Analogue of a Formula of Knuth. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2867. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2867/

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