  Complex real-world phenomena are often modeled as dynamical systems on
networks. In many cases of interest, the spectrum of the underlying graph
Laplacian sets the system stability and ultimately shapes the matter or
information flow. This motivates devising suitable strategies, with rigorous
mathematical foundation, to generate Laplacian that possess prescribed spectra.
In this paper, we show that a weighted Laplacians can be constructed so as to
exactly realize a desired complex spectrum. The method configures as a non
trivial generalization of existing recipes which assume the spectra to be real.
Applications of the proposed technique to (i) a network of Stuart-Landau
oscillators and (ii) to the Kuramoto model are discussed. Synchronization can
be enforced by assuming a properly engineered, signed and weighted, adjacency
matrix to rule the pattern of pairing interactions.
