  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 $l^p(X)$ 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 $l^p(X)$ 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.
