Networks of identical linear time-invariant single-input single-output appear, for instance, in models for gene regulatory networks. A given set of node systems is called network stabilizable, if there exists an interconnection structure such that the network is input-output stable. Using frequency domain analysis and the Hermite-Fujiwara theorem, we reformulate this problem into a linear matrix inequality with nonlinear constraint. Equivalently, we cast the problem as a rank-constrained linear matrix inequality, which can be solved by a Newton method. Similar techniques are applied to the synchronizability problem.