Topological insulators are two or three dimensional materials which have insulting bulk and metallic surface states . The metallic surface states of topological insulators have gapless electronic states that are protected by time reversal symmetry . Topological insulators are appropriate to use in spintronic and nano electronic devices , because of spin-orbit coupling and novel electronic and magnetic properties . Graphene is widely considered as a start point in recent carbon-based nanostructure studies . As a result of light atom the spin-orbit coupling in graphene is weak . As we deal with heavier atoms in the structure it may lead to stronger spin-orbit coupling . The strong spin-orbit coupling plays an essential role in the formation of topological phases , therefore adatom absorption could be an effective way to achieve the goal . Topological insulators was first proposed by Kane and Mele on graphene lattice . In this thesis , functional method in Kane-Mele model is used to strengthen the topological phases . Additionally , in this thesis gap structure in Kane-Mele model is studied . The spin-orbit coupling of a 11*11 supercell which made by graphene lattice with one fluorine atom in top position , is calculated . Finally , the Kane-Mele model and functionalizing state are compared . As a result of comparing , we realized that graphene atoms with fluorine atom in top has stronger spin-orbit coupling . These changes have important consequences on the local spectroscopic properties of the adatom and its surrounding atoms and will certainly modify the charge and spin traort properties of the system .