کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6419700 | 1631647 | 2011 | 28 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Cellular spanning trees and Laplacians of cubical complexes Cellular spanning trees and Laplacians of cubical complexes](/preview/png/6419700.png)
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 Adinʼs 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.
Journal: Advances in Applied Mathematics - Volume 46, Issues 1â4, January 2011, Pages 247-274