The structural and electronic properties of graphene and graphene-related materials have been intensively investigated using the plane wave based periodic density func- tional theory (DFT). The Vienna ab initio simulation package (VASP) code employing the generalized gradient approximation (GGA) for the exchange correlation potential was used. In all calculations, the geometry optimization option was employed in allow- ing the structure to fully relax. Hydrogen adatoms were adsorbed on C, Si and SiC in the graphene structure in-volving (1x1),(2x2),(3x3) and (4x4) two dimensional unit cells. The density of states reveals that the adsorption of 50% hydrogen makes the system metallic but 100% coverage at the on top sites generates a band gap. Our results show that SiC in the graphene structure is a plausible structure with a wide band gap. For adsoption of lithium adatoms, we considered various configurations involving the (1x1), (2x1) and (2x2) two-dimensional unit cells, and we consider the isolated Li dimer on graphene. We consider more detailed configurations than have been studied before, and our results compare favourably with previously calculated results where such results exist. For 100% coverage, we have new results for Li on the on-top site, which suggests a staggered configuration for the lowest energy structure for which the Li adatoms are alternately pushed into and pulled out of the graphene layer. For 50% coverage, Li favours the hollow site. We discovered that a careful relaxation of the system also shows a staggered configuration, a result that has not been investigated before.