A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
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 and the Crapo -invariant of uniform matroids.
Keywords
Cite
@article{arxiv.1510.00033,
title = {A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes},
author = {Ghodratollah Aalipour and Art M. Duval and Woong Kook and Kang-Ju Lee and Jeremy L. Martin},
journal= {arXiv preprint arXiv:1510.00033},
year = {2018}
}
Comments
22 pages, 2 figures. Sections 6 and 7 of previous version simplified and condensed. Final version to appear in J. Combin. Theory Ser. A