  We revisit the theme of non-nearest-neighbor interactions in nonlinear
dynamical lattices, in the prototypical setting of the discrete nonlinear
Schrodinger equation. Our approach offers a systematic way of analyzing the
existence and stability of solutions of the system near the so-called
anti-continuum limit of zero coupling. This affords us a number of analytical
insights such as the fact that, for instance, next-nearest-neighbor
interactions allow for solutions with nontrivial phase structure in infinite
one-dimensional lattices; in the case of purely nearest-neighbor interactions,
such phase structure is disallowed. On the other hand, such
non-nearest-neighbor interactions can critically affect the stability of
unstable structures, such as topological charge S=2 discrete vortices. These
analytical predictions are corroborated by numerical bifurcation and stability
computations.
