An Elementary Proof of a Matrix Tree Theorem for Directed Graphs

We present an elementary proof of a generalization of Kirchhoff'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 involves only 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.