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A number of recent papers have considered signed graph Laplacians, ageneralization of the classical graph Laplacian, where the edge weights areallowed to take either sign. In the classical case, where the edge weights areall positive, the Laplacian is positive semi-definite with the dimension of thekernel representing the number of connected components of the graph. In manyapplications one is interested in establishing conditions which guarantee thepositive semi-definiteness of the matrix. In this paper we present aninequality on the eigenvalues of a weighted graph Laplacian (where the weightsneed not have any particular sign) in terms of the first two moments of theedge weights. This bound involves the eigenvalues of the equally weightedLaplacian on the graph as well as the eigenvalues of the adjacency matrix ofthe line graph (the edge-to-vertex dual graph). For a regular graph the boundcan be expressed entirely in terms of the second eigenvalue of the equallyweighted Laplacian, an object that has been extensively studied in connectionwith expander graphs and spectral measures of graph connectivity. We presentseveral examples including Erd\H{o}s-R\'enyi random graphs in the critical andsubcritical regimes, random large $d$-regular graphs, and the complete graph,for which the inequalities here are tight.

A moment inequality and positivity for signed graph Laplacians