# Linear algebraic techniques for spanning tree enumeration

## TL;DR

This paper introduces linear algebraic methods, specifically the Matrix Determinant Lemma and Schur complement, to efficiently count spanning trees in various graph families, simplifying computations beyond traditional determinant approaches.

## Contribution

It demonstrates novel applications of linear algebra techniques to improve spanning tree enumeration in specific graph classes.

## Key findings

- Efficient counting of spanning trees using linear algebra methods.
- Application of Matrix Determinant Lemma and Schur complement to graph theory.
- Simplification of complex determinant calculations for certain graph families.

## Abstract

Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the Matrix Determinant Lemma and the Schur complement, can be used to elegantly count the spanning trees in several significant families of graphs.