Simplicial matrix-tree theorems

Details Simplicial matrix-tree theorems Simplicial matrix-tree theorems

2008-02-19

arxiv.org/abs/0802.2576v2

Trans. Amer. Math. Soc. 361 (2009), no. 11, 6073-6114

Mathematics

Combinatorics

36 pages, 2 figures. Final version, to appear in Trans. Amer. Math. Soc

Scientific paper

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.

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