The equilibrium atomic geometry, electronic properties, electronic orbital states, and optical properties of graphene and graphene based systems have been comprehensively investigated using the density functional theory (DFT) in the framework of plane wave pseudopotential. These properties have been compared with recently found experimental results and theoretical works done by tight-binding and DFT methods. In Chapter 1, a brief theoretical overview about the structural properties of graphene based systems has been provided. The basic concepts of semiconductor surfaces have been introduced for the study of graphene on InAs(111) surfaces. In Chapter 2, explanations of the essential ingredients of the density functional theory and the plane wave pseudopotential theory have been provided. The main concepts of geometry optimisation in all calculations have been explained. The theory of scanning tunnelling microscopy which gives very detailed information about geometrical structures and electronic states has been described . In Chapter 3, atomic geometry, electronic structures, and interband optical transitions for isolated monolayer graphene, bilayer graphene, ABA-stacked trilayer graphene, and graphite systems have been studied. The electron velocity and effective mass were estimated using the in-plane electronic band calculation. The modifications in the electronic properties due to increasing the number of graphene layers have been discussed. The changes in interband optical transitions have also been presented for bilayer graphene, trilayer graphene, and graphite with respect to the monolayer graphene. In Chapter 4, a detailed theoretical study of the electronic structures of ABC-stacked trilayer and N-layer graphene has been presented. The nature of the trigonal warping of energy bands slightly above the Fermi level for different layer thicknesses was examined. The orbital natures of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) states have been investigated. In Chapter 5, the equilibrium geometry and electronic structure of multilayer graphene deposited on hexagonal boron nitride substrate have been studied. It has been found that the graphene sheet is weakly adsorbed on the boron nitride substrate. Using the in-plane electronic band calculations the carrier velocity and effective mass were estimated. The importance of the interlayer interaction and stacking patterns of multilayer graphene/boron nitride has been explained for band gap and effective mass tuning in multilayer graphene. The interband transition energies for all systems were also calculated. In Chapter 6, the atomic geometry and electronic structure of graphene on the most stable In-vacancy InAs(111)A surface have been investigated. The effect of the substrate on electronic charge re-distribution around the graphene sheet was examined. The transparency effect of graphene has also been investigated by simulating scanning tunnelling microscopy. Finally, the summary of results and the future works have been outlined in Chapter 7.

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