This dissertation includes work done on graphene and related materials, examining their electronic and topological properties using first-principles methods. Ab-initio computational methods, like density functional theory (DFT), have become increasingly popular in condensed matter and material science. Motivated by the search for novel materials that would help us devise fast, low-power, post-CMOS transistors, we explore the properties of some of these promising materials. We begin by studying graphene and its interaction with dielectric oxides. Graphene has recently inspired a flurry of research activity due to its interesting electronic and mechanical properties. For the device community, graphene's high charge carrier mobility and continuous gap tunability can have immense use in novel transistors. In Chapter 3 we examine the properties of graphene placed on two oxides, namely quartz and alumina. We find that oxygen-terminated quartz is a useful oxide for the purpose of graphene based FETs. Inspired by a recent surge of interest in topological insulators, we then explore the topological properties of two-dimensional materials. We conduct a theoretical study to examine the relationship between crystal space group symmetry and the electric polarization of a two-dimensional crystal. We show that the presence of symmetry restricts the polarization values to a small number of distinct groups. There groups in turn are topologically inequivalent, making polarization a topological index. We also conduct density functional theory calculations to obtain actual polarization values of materials belonging to C3 symmetry and show that our results are consistent with our theoretical analysis. Finally we prove that any transformation from one class of polarization to another is a topological phase transition. In Chapter 5 we use density functional theory to examine the electronic properties of graphene intercalation compounds. Bilayer pseudospin field effect transistor (BiSFET) has been proposed as an interesting low-power, efficient post-CMOS switch. In order to implement this device we need bilayer graphene with reduced interlayer interaction. One way of achieving that is by inserting foreign molecules between the layers, a process which is called intercalation. In this chapter we examine the electronic properties of bilayer graphene intercalated with iodine monochloride and iodine monobromide molecules. We find that intercalation of graphene indeed makes it promising for the implementation of BiSFET, by reducing interlayer interaction. As an interesting side problem, we also use hybrid, more extensive approaches in DFT, to examine the electronic and optical properties of dilute nitrides. Dilute nitrides are highly promising and interesting materials for the purposes of optoelectronic applications. Together, we hope this work helps in elucidating the electronic properties of promising material systems as well as act as a guide for experimentalists. / text