Let (\cX, \rho) be a discrete metric space. We suppose that the group
\sZ^n acts freely on X and that the number of orbits of X with respect to
this action is finite. Then we call X a \sZ^n-periodic discrete metric
space. We examine the Fredholm property and essential spectra of band-dominated
operators on lp(X) where X is a \sZ^n-periodic discrete metric space.
Our approach is based on the theory of band-dominated operators on \sZ^n and
their limit operators.
In case X is the set of vertices of a combinatorial graph, the graph
structure defines a Schr\"{o}dinger operator on lp(X) in a natural way. We
illustrate our approach by determining the essential spectra of Schr\"{o}dinger
operators with slowly oscillating potential both on zig-zag and on hexagonal
graphs, the latter being related to nano-structures