Forest Matrices Around the Laplaeian Matrix

We study the matrices of in-forests of a weighted digraph Γ and their connections with the Laplaeian matrix of Γ. The () entry of is the total weight of spanning converging forests () with arcs such that belongs to a tree rooted at . The forest matrices, , can be calculated recursively and expressed by polynomials in the Laplaeian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of . The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplaeian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices are the matrix coefficients in the polynomial expansion of adj(λ + ). Thereby they are precisely Faddeev's matrices for −