# Linear algebraic techniques for weighted spanning tree enumeration

## TL;DR

This paper introduces linear algebra techniques, specifically the Matrix Determinant Lemma and Schur complements, to efficiently compute weighted spanning tree enumerators for various graph families, simplifying calculations beyond the standard Matrix-Tree Theorem.

## Contribution

It demonstrates how classical linear algebra tools can be applied to improve the computation of weighted spanning tree enumerators for specific graph classes.

## Key findings

- Efficient computation of weighted spanning tree enumerators using linear algebra methods.
- Application of the Matrix Determinant Lemma to graph Laplacians.
- Use of Schur complements to simplify determinant calculations.

## Abstract

The weighted spanning tree enumerator of a graph $G$ with weighted edges is the sum of the products of edge weights over all the spanning trees in $G$. In the special case that all of the edge weights equal $1$, the weighted spanning tree enumerator counts the number of spanning trees in $G$. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of $G$. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Matrix Determinant Lemma and the method of Schur complements, can be used to elegantly compute the weighted spanning tree enumerator for several families of graphs.