Signed Laplacian matrices generally fail to be diagonally dominant and may fail to be stable. For both undirected and directed graphs, in this paper we present conditions guaranteeing the stability of signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. Our conditions are necessary and sufficient for undirected graphs, but only sufficient for digraphs, the gap between necessity and sufficiency being filled by matrices who have this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). An exception is given by weight balanced signed digraphs, where eventual positivity corresponds to positive semidefinitness of the symmetric part of the Laplacian. Analogous conditions are obtained for signed stochastic matrices.
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