The two-dimensional graphene structure is composed of the isotropic honeycomb lattice of carbon atoms. The graphene structure has several specific electronic properties. For instance, there are no gaps in energy bands of the Fermi level and electrons behave similar to the Dirac fermions. In this structure, anisotropy can be created, by electrochemical pressure exertion, so that the geometry and the parameters of lattice change. In this thesis, the effects of anisotropy in one direction in the electronic nano-structure of graphene, using Hubbard model, known as tight-binding method, are studied. Therefore the n-states, explaining the electronic properties are considered, and the electronic bands of n-states are compared with the electronic bands of isotropy honeycomb lattice in graphene. Consequently, the spectral function and density states are studied for both states of the honeycomb lattice of graphene. According to the results of the study, the gap in electronic bands will be created in the Fermi level with pressure or tension exertion. In the next step, strongly correlated materials, due to the finding high-temperature superconductors and the metal insulator phase transition in these metals, are studied. One of the simplest models to study these metals is the Hubbard model. In this model, in addition to the kinetic energy of electrons, the interaction between electrons are considered and the large interaction and half filling state in this model using dynamical mean field theory are studied.
