Signed graphs and inverses of their incidence matrices

The Laplacian matrix $L$ of a signed graph $G$ may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of $L$. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of $G$. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas.