The Moore–Penrose inverse of the normalized graph Laplacian

We prove a formula that relates the Moore–Penrose inverses of two matrices A,BA,B such that A=N−1BM−1A=N−1BM−1 and discuss some applications, in particular to the representation of the Moore–Penrose inverse of the normalized Laplacian of a graph. The Laplacian matrix of an undirected graph is symmetric and is strictly related to its connectivity properties. However, our formula applies to asymmetric matrices, so that we can generalize our results for asymmetric Laplacians, whose importance for the study of directed graphs is increasing.