  Complex real-world phenomena across a wide range of scales, from aviation and
internet traffic to signal propagation in electronic and gene regulatory
circuits, can be efficiently described through dynamic network models. In many
such systems, the spectrum of the underlying graph Laplacian plays a key role
in controlling the matter or information flow. Spectral graph theory has
traditionally prioritized unweighted networks. Here, we introduce a
complementary framework, providing a mathematically rigorous weighted graph
construction that exactly realizes any desired spectrum. We illustrate the
broad applicability of this approach by showing how designer spectra can be
used to control the dynamics of various archetypal physical systems.
Specifically, we demonstrate that a strategically placed gap induces chimera
states in Kuramoto-type oscillator networks, completely suppresses pattern
formation in a generic Swift-Hohenberg model, and leads to persistent
localization in a discrete Gross-Pitaevskii quantum network. Our approach can
be generalized to design continuous band gaps through periodic extensions of
finite networks.
