  Complex-valued Laplacians have been shown to be powerful tools in the study
of distributed coordination of multi-agent systems in the plane including
formation shape control problems and set surrounding control problems. In this
paper, we first provide some characterizations of complex Laplacians. As an
application, we then establish some necessary and sufficient conditions to
ensure that the agents interacting on complex-weighted networks converge to
consensus in some sense. These general consensus results are used to discuss
some multi-agent coordination problems in the plane.
