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An Algebraic Analogue of a Formula of Knuth

Abstract : 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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Lionel Levine. An Algebraic Analogue of a Formula of Knuth. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.379-390. ⟨hal-01186296⟩

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