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A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
Aalipour, Ghodratollah Duval, Art M. Kook, Woong Lee, Kang-Ju Martin, Jeremy L.
Aalipour, Ghodratollah
Duval, Art M.
Kook, Woong
Lee, Kang-Ju
Martin, Jeremy L.
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Abstract
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube.
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Date
2018-03-26
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Elsevier
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Keywords
Matrix-tree theorem, Laplacian, Complete colorful complex, Hypercube, Euler characteristic
Citation
Ghodratollah Aalipour, Art M. Duval, Woong Kook, Kang-Ju Lee, Jeremy L. Martin, "A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes", Journal of Combinatorial Theory, Series A, Volume 158, 2018, Pages 362-386, ISSN 0097-3165,
https://doi.org/10.1016/j.jcta.2018.03.009.