In this thesis _rst principle calculations with in density functional theory (DFT) of electronicrnstructure of gold Au doped graphene is studied by using a plane wave pseudopotentialrnmethod. We used local density approximation (LDA) and generalized gradientrnapproximations (GGA) for the exchange correlation potential. In all calculations, therngeometry optimization option was employed in alloying the structure to fully relax. Thernconvergence of the system was checked by calculating total energy versus kinetic energyrncutt-o_ and total energy versus the size of k-points set. Convergence was attained forrngraphene starting from 41Ryd or 557:6eV , hence used to reduce computational cost. Arnuniform mesh 0f 13 _ 13 _ 1 k-points was used since it gives a good convergence at arnreasonable computational cost providing charge density results and lattice constant ofrn2:485_A. From the band structure of graphene the two bands touch each other at K-point.rnDOS plot also gives neither band gap between conduction and valence bands nor overlappingrnat the Fermi energy indicating that graphene is semi-conductor. Finally, in the bandrnstructure of Au doped graphene built in 13_13_1 super cell the _=__ band cross at Fermirnlevel. As clear evidence of the interaction between Au dopant and the graphene, the Fermirnlevel is now moved to the conduction band, and the DOS display a sharp peak at thernFermi level. Therefore, the doping of Au in the graphene makes graphene semi-metallic.