Graphene grown by chemical vapor deposition has excellent mechanical, thermal and electrical properties, and has become the preferred material for a new generation of nanodevices. Theoretical research on the electronic properties of graphene will prompt the development and application of nanodevices. Based on the combination of density functional theory and non-equilibrium Green's function, the electronic structure characteristics of graphene and graphene/boron nitride with AA vertical stack structures were studied systematically in this paper. The results show that the band gap is zero at the high symmetry K point. In the range of 50 K to 400 K, the mobility of graphene decreases significantly with increasing temperature due to the electron-phonon interaction on Fermi surface. In addition, the characteristic analysis of energy band structures, density of states and electron density of graphene/boron nitride with different interlamellar space indicates that with the increase of layer spacing, the energy band gap decreases, and the energy difference between the conduction band and the valence band decreases. And increasing the number of atoms, the regularity of the change of the band gap opening between supercell and protocell structure of graphene/boron nitride is consistent, this can serve an guideline for the structural design of graphene-based devices.