Acoustic diffraction problems on periodic graphs

We consider acoustic diffraction by graphs Γ embedded in ℝ² and periodic with respect to an action of the group ℤⁿ, n = 1, 2. The diffraction problem is described by the Helmholtz equation with variable nonperiodic bounded coefficients and nonperiodic transmission conditions on the graph Γ. We introduce single and double layer potentials on Γ generated by the Schwartz kernel of the operator inverse to the Helmholtz operator on ℝ² and reduce the diffraction problem to a boundary pseudodifferential equation on the graph. Necessary and sufficient conditions for the boundary operators to be Fredholm are obtained.