Simplicial matrix-tree theorems

TL;DR

This paper extends the classical matrix-tree theorem to higher-dimensional simplicial complexes, providing a way to count and analyze their spanning trees using Laplacian matrices and eigenvalues.

Contribution

It introduces a generalized simplicial matrix-tree theorem, including weighted counts and eigenvalue interpretations for shifted complexes, expanding combinatorial and algebraic understanding.

Findings

Counts simplicial spanning trees via Laplacian matrices

Provides eigenvalue-based face characterization for shifted complexes

Generalizes classical graph results to higher dimensions

Abstract

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.