05C50 Online Seminar: Milica Anđelić

The Laplacian matrix L of a signed graph G may or may not be invertible. We present an entrywise combinatorial formula for the Moore-Penrose inverse of L, obtained by deriving a combinatorial expression for the Moore-Penrose inverse of an incidence matrix of G. Our results extend known formulas for incidence and (signless) Laplacian matrices of unsigned graphs. We also highlight several particular cases in which the computation of the Moore-Penrose inverse can be considerably simplified by applying alternative, non-combinatorial methods. Finally, we indicate directions for further generalizations and applications, including the resistance matrix and the development of suitable partition-based approaches.