An unusual combination of ultralight weight and outstanding mechanical properties of graphene aerogel made it popular for a wide range of applications in the fields of material science, energy, and technology. In the present work, the mechanical properties and fracture behavior of graphene aerogels, which are highly brittle in nature, are investigated using molecular dynamics (MD) simulations. In tensile tests, elastic modulus and tensile strength exhibit a power law dependence on the density with their exponents predicted to be 2.95 +/- 0.05 and 1.61 +/- 0.04, respectively, which are in an excellent agreement with the reported works in the literature. In the compression simulations, Lennard-Jones contribution in the AIREBO potential is vital to predicting early densification. Moreover, in the compression loading-unloading simulations, as the density decreases, the dissipation energy increases. At the onset of the crack propagation, as the crack length to height ratio increases, the fracture strength decreased. However, for a considered range of the ratios, the fracture toughness values were nearly constant for all densities. Furthermore, the fracture toughness shows a power law dependence on the density, with the exponent estimated to be 1.41 +/- 0.04. The outcome of this work is a vital step in the in-depth understanding of nanomechanics while designing advanced, highly porous materials.