Abstract
We built graphene nanoflakes doped or not with $C$ atoms in the $sp^3$ hybridization or with $Si$ atoms. These nanoflakes are isolated, i.e. are not connected to any object (substrate or junction). We used a modified tight binding method to compute the $\pi$ and $\sigma$ density of states. The nanoflakes are semiconducting (due to the armchair geometry of their boundaries) when their are pure but the become conducting when doped because doping removes the degeneracy of the density of states levels. Moreover, we showed that the $\pi$ Fermi level and the Fermi level of both $\pi$ and $\sigma$ electrons are not superimposed for small isolated nanoflakes.