  We have performed an ab initio investigation of the electronic properties of
the graphene sheet adsorbed by Ru adatoms (Ru/graphene). For a particular set
of triangular arrays of Ru adatoms, we find the formation of four
(spin-polarized) Dirac cones attributed to a suitable overlap between two
hexagonal lattices: one composed by the C sites of the graphene sheet, and the
other formed by the surface potential induced by the Ru adatoms. Upon the
presence of spin-orbit coupling (SOC) nontrivial band gaps take place at the
Dirac cones promoting several topological phases. Depending on the Ru
concentration, the system can be topologically characterized among the phases
i) Quantum Spin Hall (QSH), ii) Quantum Anomalous Hall (QAH), iii) metal iv) or
trivial insulator. For each concentration, the topological phase is
characterized by the ab-initio calculation of the Chern number.
