Gaps in the spectrum of a periodic quantum graph with periodically distributed -type interactions

We consider a family of quantum graphs , where Γ is a -periodic metric graph and the periodic Hamiltonian is defined by the operation on the edges of Γ and either -type conditions or the Kirchhoff conditions at its vertices. Here is a small parameter. We show that the spectrum of has at least m gaps as ( is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of Γ and of coupling constants involved in -type conditions.