Cellular spanning trees and Laplacians of cubical complexes

TL;DR

This paper generalizes the Matrix-Tree Theorem to cubical complexes, providing explicit formulas for spanning trees and Laplacian eigenvalues, and introduces new concepts like cubical shiftedness.

Contribution

It extends the Matrix-Tree Theorem to cubical complexes, introduces cubical shiftedness, and derives explicit formulas for eigenvalues and spanning tree enumeration.

Findings

Eigenvalues of cubical complexes are integers.

Explicit formulas for spanning tree enumeration of cubes.

Weighted eigenvalue formulas support a conjecture on cubical spanning trees.

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.