In this thesis, we explore different properties of Graphene, a material with a single layer of carbon atoms. In the first section, we propose a extended tight-binding model including spin-orbit and magnetization effects in order to investigate spintronics effects on the equilibrium currents of the material. We found that there is a interplay between the spin-locking due to Rashba spin-orbit coupling and the external magnetization proposed in the model. This interplay makes the spin-currents developed in the material to have specific properties, which is highlighted on the section on nanoribbons. In the following section, we investigate the role of nonsymmorphic symmetries, a class of symmetries that can only be realized with reflection or rotation and translation, on the electronic properties of bidimensional materials. We proposed a method to produce these symmetries on any lattice by performing vacancy engineering, i.e. removing sites at certain positions. We show that when theses symmetries are present in the structure of the material, the spectra must be restricted in such a way that every two bands must touch at a point in the Brillouin Zone, in particular there must be nodal lines along the boundaries of the Brillouin Zone. Exploring further on the idea of vacancy engineering, we propose a method for using it to create zero-energy flat-bands for bipartite lattices. This is a realization that the Hamiltonian of bipartite lattices takes a special form, namely anti-block-diagonal, and vacancy engineering consists of removing rows and columns from the tight-binding Hamiltonian. We analyze the role of the zero-energy flat-band on the electronic correlations such as formation of a superconductivity phase. The last two sections are dedicated to quantum metric and applications thereof, in particular in the study of topological order. We give examples and show that for continuum systems there is a closed formula for dimension D. As an application, we show that the opacity of Graphene is directly related to the topological charge.