  The induced-proximity effects of nearly commensurate lattice structure of a
graphene layer on Ni(111) and Co(0001) substrates in the AC stacking
configuration are addressed through an analytical tight-binding approach within
the Slater-Koster method. A minimal Hamiltonian is constructed by considering
the hybridizations of the magnetic $3d$-orbitals of Ni(Co) atoms with the
$p_z$-orbitals of graphene, in addition to the atomic spin-orbit coupling and
the magnetization of the Ni(Co) atoms. A low-energy effective Hamiltonian for
graphene/Ni(Co) describing the perturbed $\pi$-bands in the vicinity of the
Dirac points is derived which enable us to get further insight on the physical
nature of the induced-effective couplings to the graphene layer. It is shown
that a magneto-spin-orbit type effect may emerge through two competing
mechanisms simultaneously present, namely the proximity induced exchange and
Rashba spin-orbit interaction. Such effects results in giant exchange
splittings and robust Rashba spin-orbit coupling transferred to the graphene
layer in agreement with recent density functional theory calculations and
experimental observations. We further analyze the physical conditions for the
appearance of intact Dirac cones in the minority spin bands as observed by
recent photoemission measurements with spin resolution.
