Since the isolation of graphene in 2004, this novel material has become the major object of modern condensed matter physics. Despite of enormous research activity in this field, there are still a number of fundamental phenomena that remain unexplained and challenge researchers for further investigations. Moreover, due to its unique electronic properties, graphene is considered as a promising candidate for future nanoelectronics. Besides experimental and technological issues, utilizing graphene as a fundamental block of electronic devices requires development of new theoretical methods for going deep into understanding of current propagation in graphene constrictions.

This thesis is devoted to the investigation of the effects of electron-electron interactions, spin and different types of disorder on electronic and transport properties of graphene and graphene nanoribbons.

In paper I we develop an analytical theory for the gate electrostatics of graphene nanoribbons (GNRs). We calculate the classical and quantum capacitance of the GNRs and compare the results with the exact self-consistent numerical model which is based on the tight-binding p-orbital Hamiltonian within the Hartree approximation. It is shown that electron-electron interaction leads to significant modification of the band structure and accumulation of charges near the boundaries of the GNRs.

It's well known that in two-dimensional (2D) bilayer graphene a band gap can be opened by applying a potential difference to its layers.  Calculations based on the one-electron model with the Dirac Hamiltonian predict a linear dependence of the energy gap on the potential difference. In paper II we calculate the energy gap in the gated bilayer graphene nanoribbons (bGNRs) taking into account the effect of electron-electron interaction. In contrast to the 2D bilayer systems the energy gap in the bGNRs depends non-linearly on the applied gate voltage. Moreover, at some intermediate gate voltages the energy gap can collapse which is explained by the strong modification of energy spectrum caused by the electron-electron interactions.

Paper III reports on conductance quantization in grapehene nanoribbons subjected to a perpendicular magnetic field. We adopt the recursive Green's function technique to calculate the transmission coefficient which is then used to compute the conductance according to the Landauer approach. We find that the conductance quantization is suppressed in the magnetic field. This unexpected behavior results from the interaction-induced modification of the band structure which leads to formation of the compressible strips in the middle of GNRs. We show the existence of the counter-propagating states at the same half of the GNRs. The overlap between these states is significant and can lead to the enhancement of backscattering in realistic (i.e. disordered) GNRs.

Magnetotransport in GNRs in the presence of different types of disorder is studied in paper IV. In the regime of the lowest Landau level there are spin polarized states at the Fermi level which propagate in different directions at the same edge. We show that electron interaction leads to the pinning of the Fermi level to the lowest Landau level and subsequent formation of the compressible strips in the middle of the nanoribbon. The states which populate the compressible strips are not spatially localized in contrast to the edge states. They are manifested through the increase of the conductance in the case of the ideal GNRs. However due to their spatial extension these states are very sensitive to different types of disorder and do not significantly contribute to conductance of realistic samples with disorder. In contrast, the edges states are found to be very robust to the disorder. Our calculations show that the edge states can not be easily suppressed and survive even in the case of strong spin-flip scattering.

In paper V we study the effect of spatially correlated distribution of impurities on conductivity in 2D graphene sheets. Both short- and long-range impurities are considered. The bulk conductivity is calculated making use of the time-dependent real-space Kubo-Greenwood formalism which allows us to deal with systems consisting of several millions of carbon atoms. Our findings show that correlations in impurities distribution do not significantly influence the conductivity in contrast to the predictions based on the Boltzman equation within the first Born approximation.

In paper VI we investigate spin-splitting in graphene in the presence of charged impurities in the substrate and calculate the effective g-factor. We perform self-consistent Thomas-Fermi calculations where the spin effects are included within the Hubbard approximation and show that the effective g-factor in graphene is enhanced in comparison to its one-electron (non-interacting) value. Our findings are in agreement to the recent experimental observations.