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Simplicial matrix-tree theorems
Duval, Art M. Klivans, Caroline J. Martin, Jeremy L.
Duval, Art M.
Klivans, Caroline J.
Martin, Jeremy L.
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Abstract
We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes Δ, 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 Δ. 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 Δ and replacing the entries of the Laplacian with Laurent monomials. When Δ 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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First published in Transactions of the American Mathematical Society in volume 361 (2009), no. 11, 6073--6114, published by the American Mathematical Society
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2009
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American Mathematical Society
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Simplicial matrix-tree theorems (with Art M. Duval and Caroline J. Klivans), Transactions of the American Mathematical Society 361 (2009), no. 11, 6073--6114.