On eigenvalues of Laplacian matrix for a class of directed signed graphs

The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. The Laplacian matrix naturally arises in a wide range of applications involving networks. First, a class of directed signed graphs is studied in which one pair of nodes (either connected or not) is perturbed with negative weights. A necessary and sufficient condition is proposed to attain the following objective for the perturbed graph: the real parts of the non-zero eigenvalues of its Laplacian matrix are positive. Under certain assumption on the unperturbed graph, it is established that the objective is achieved if and only if the magnitudes of the added negative weights are smaller than an easily computable upper bound. This upper bound is shown to depend on the topology of the unperturbed graph. It is also pointed out that the obtained condition can be applied in a recursive manner to deal with multiple edges with negative weights. Secondly, for directed graphs, a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight, the resulting Laplacian matrix will have at least one eigenvalue with negative real part. Illustrative examples are presented to show the applicability of our results.