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

2026/07/22 by XiaoYang Liu, Sudipta Mallik, Anh Tang
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Abstract

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).

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