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The eigenvalues of the Laplacian matrix for a class of directed graphs withboth positive and negative weights are studied. First, a class of directedsigned graphs is investigated in which one pair of nodes (either connected ornot) is perturbed with negative weights. A necessary condition is proposed toattain the following objective for the perturbed graph: the real parts of thenon-zero eigenvalues of its Laplacian matrix are positive. A sufficientcondition is also presented that ensures the aforementioned objective for theunperturbed graph. It is then highlighted the case where the condition becomesnecessary and sufficient. Secondly, for directed graphs, a subset of pairs ofnodes are identified where if any of the pairs is connected by an edge withinfinitesimal negative weight, the resulting Laplacian matrix will have atleast one eigenvalue with negative real part. Illustrative examples arepresented to show the applicability of our results.

On Eigenvalues of Laplacian Matrix for a Class of Directed Signed Graphs