  We propose new mathematical optimization models for generating sparse
dynamical graphs, or networks, that can achieve synchronization. The
synchronization phenomenon is studied using the Kuramoto model, defined in
terms of the adjacency matrix of the graph and the coupling strength of the
network, modelling the so-called coupled oscillators. Besides sparsity, we aim
to obtain graphs which have good connectivity properties, resulting in small
coupling strength for synchronization. We formulate three mathematical
optimization models for this purpose. Our first model is a mixed integer
optimization problem, subject to ODE constraints, reminiscent of an optimal
control problem. As expected, this problem is computationally very challenging,
if not impossible, to solve, not only because it involves binary variables but
also some of its variables are functions. The second model is a continuous
relaxation of the first one, and the third is a discretization of the second,
which is computationally tractable by employing standard optimization software.
We design dynamical graphs that synchronize, by solving the relaxed problem and
applying a practical algorithm for various graph sizes, with randomly generated
intrinsic natural frequencies and initial phase variables. We test robustness
of these graphs by carrying out numerical simulations with random data and
constructing the expected value of the network's order parameter and its
variance under this random data, as a guide for assessment.
