Combinatorial formula for the Moore-Penrose inverse of the complex signless Laplacian of an oriented graph

We find necessary and sufficient conditions for the rank of the signless incidence matrix of a weakly connected oriented graph with non-zero complex edge weights.

We use this to find the combinatorial formulas for the Moore-Penrose inverse of the complex signless incidence and complex signless Laplacian matrix of a weakly connected oriented graph with non-zero complex edge weights.

This resolves the open problem posed in the concluding remarks of Barik et. al.

(Discrete Mathematics 349 (9), 115117, 2026).