  This article deals with the spectra of Laplacians of weighted graphs. In this
context, two objects are of fundamental importance for the dynamics of complex
networks: the second eigenvalue of such a spectrum (called algebraic
connectivity) and its associated eigenvector, the so-called Fiedler vector.
Here we prove that, given a Laplacian matrix, it is possible to perturb the
weights of the existing edges in the underlying graph in order to obtain simple
eigenvalues and a Fiedler vector composed of only non-zero entries. These
structural genericity properties with the constraint of not adding edges in the
underlying graph are stronger than the classical ones, for which arbitrary
structural perturbations are allowed. These results open the opportunity to
understand the impact of structural changes on the dynamics of complex systems.
