In this paper the transmission properties of a defective one-dimensional dielectric-graphene periodic structure have been investigated using the transfer matrix method. The structure with a central defect consists of a stack of graphene nanolayers which are separated alternatively by the same dielectric layers. The results show that the suggested structure possesses a graphene induced photonic band gap (GIPBG) at low THz frequencies. Also, one defect mode is created in the GIPBG due to the breaking of the periodicity of the graphene lattice. So, we call it the graphene induced defect mode (GIDM). The dependence of the GIDM on the relative permittivity of dielectric layers and the thickness between graphene nanolayers has been studied at normal incidence. Since the permittivity of graphene nanolayer depends on its chemical potential, the frequency of the GIDM can be controlled via a gate voltage. Also, the electromagnetic field profile at the frequency of the GIDM inside the structure has been plotted.