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arXiv · 0908.1956

Cellular spanning trees and Laplacians of cubical complexes

Art M. Duval·Caroline J. Klivans·Jeremy L. Martin
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Abstract

We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin's enumeration of spanning trees of a complete colorful simplicial complex from the cellular Matrix-Tree Theorem together with a result of Kook, Reiner and Stanton.

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Art M. Duval, Caroline J. Klivans, Jeremy L. Martin. 2009-08-13. Cellular spanning trees and Laplacians of cubical complexes. https://arxiv.org/abs/0908.1956

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