  We consider a network of identical agents, coupled through linear asymmetric
coupling. An important case is when each agent has an asymptotically stable
periodic orbit, so that the full network inherits a synchronous periodic orbit,
but also chaotic trajectories are of interest. In this work, we will restrict
to "nearest-neighbor" type of couplings.
  The Master Stability Function (MSF) is a powerful tool to establish local
stability of the synchronous orbit, in particular a negative MSF implies
asymptotic stability. But not every network structure gives a negative MSF.
Moreover, there are many situations where in order to obtain a negative MSF,
symmetric networks need a coupling strength so large, that the model bears
little physical interest. We make two main contributions: (i) Given a
tridiagonal nearest neighbor topology, we show how it is possible to choose
appropriate coupling so that the synchronous orbit is stable, and (ii) we show
that this stability comes without the need of a large coupling strength if the
structure is not symmetric. Our construction is based on solving inverse
eigenvalue problems. We will see that the coupling of the agents cannot always
be chosen to be symmetric so that the underlying graph structure is that of a
directed graph with edges having different weights. We provide numerical
implementation of our technique on networks of van der Pol and of chaotic
R\"ossler oscillators, where the standard symmetric nearest neighbor coupling
fails to give stability of the synchronous orbit.
