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The signless Laplacian Q and signless edge-Laplacian S of a given graph mayor may not be invertible. The Moore-Penrose inverses of Q and S are studied. Inparticular, using the incidence matrix, we find combinatorial formulas of theMoore- Penrose inverses of Q and S for trees. Also we present combinatorialformulas of the inverses of Q and S for odd unicyclic graphs.

Moore-Penrose Inverses of the Signless Laplacian and Edge-Laplacian of Graphs