Lionel Levine - An Algebraic Analogue of a Formula of Knuth

dmtcs:2867 - Discrete Mathematics & Theoretical Computer Science, January 1, 2010, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) - https://doi.org/10.46298/dmtcs.2867
An Algebraic Analogue of a Formula of KnuthArticle

Authors: Lionel Levine ORCID1

  • 1 Department of Mathematics [MIT]

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.


Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
Section: Proceedings
Published on: January 1, 2010
Imported on: January 31, 2017
Keywords: weighted Laplacian,critical group,de Bruijn graph,iterated line digraph,Kautz graph,matrix-tree theorem,oriented spanning tree,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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