Atomic geometry, electronic states, and optical transitions for isolated monolayer, bilayer and trilayer graphene, and graphene grown on ultra-thin layers of hexagonal boron nitride (h-BN) have been studied theoretically by using the density functional theory and the planewave pseudopotential method. For monolayer graphene, the dispersion curve near the K point is linear with Dirac electron's speed of 0.9 x 10(6) m/s. For bilayer graphene the lowest unoccupied energy band is characterised by a mixture of linear and quadratic behaviours, with a relative effective mass of 0.023. For trilayer graphene there are overlapping electron and hole bands near the Fermi level, with a relative electron effective mass of 0.0541. For a monolayer graphene on monolayer h-BN substrate, a small band gap of 57 meV is established. At Brillouin zone centre, the theoretically obtained direct transition of 6.3 eV for graphene is reduced to 5.7 eV for graphene/h-BN. Results are also presented for the interface between graphene and a multilayer h-BN. (c) 2012 Elsevier B.V. All rights reserved.