The unique physical properties of graphene, a two-dimensional hexagonal crystal of carbon atoms, have led to both theoretical and experimental observation of many fascinating physical phenomena. Furthermore, graphene has immense potential to be a key ingredient in many new devices, such as single molecule gas sensors, touch screens, solar cells, nanoscale transistors, and spintronic devices. In this chapter, we review recent advances in the understanding of the physical and chemical properties low-dimensional graphene derivatives. We do this from a theorist.s perspective while discussing several examples which illustrate the powerful predictive capabilities of state-of-the-art computational methodologies, mainly in the field of density functional theory. We focus on graphene nanoribbons, quasi-one-dimensional confined structures made of graphene. The role of the sublattice symmetry, edge geometry, and the dimensions of the nanoribbon on the electronic and magnetic properties of the system are discussed. Furthermore, the effects of substrate interactions, adsorbed atoms, and doping on the electronic structure of finite-sized graphene systems are briefly reviewed.
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