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Given a graph Laplacian with positively and negatively weighted edges we areinterested in characterizing the set of weights that give a particular spectralindex, i.e.~give a prescribed number of positive, zero, and negativeeigenvalues. One of the main results of this paper is that the set of signedLaplacians that exhibit multiple zero eigenvalues is "small", and thateigenvalue crossings are nongeneric --- specifically, eigenvalues repel eachother near zero in a sense that can be made precise. We exhibit an algebraicdiscriminant that measures the level of repulsion, and show that thisdiscriminant admits a combinatorial interpretation. Conversely, we exhibit aconstructive method for finding the sets of Laplacians that exhibit a largedegree of degeneracy (many eigenvalues at or near zero) in terms of thesediscriminants.

The spectral index of signed Laplacians and their structural stability