# An elementary proof of a matrix tree theorem for directed graphs

## TL;DR

This paper provides an elementary proof of a generalized matrix tree theorem for directed, weighted graphs, using Laplacian matrix factorizations based on incidence matrices, with applications to eigenvector computation.

## Contribution

It introduces a simple, topologically based proof of the matrix tree theorem for directed graphs and demonstrates its use in calculating principal Laplacian eigenvectors.

## Key findings

- Elementary proof of the generalized matrix tree theorem for directed graphs
- Factorization of Laplacian matrices using incidence matrices
- Application to principal eigenvector calculation

## Abstract

We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the two incidence matrices that capture the graph's topology. We also point out how this result can be used to calculate principal eigenvectors of the Laplacian matrices.