  We investigate the spin diffusion and transport in a graphene monolayer on
SiO$_2$ substrate by means of the microscopic kinetic spin Bloch equation
approach. The substrate causes a strong Rashba spin-orbit coupling field $\sim
0.15$ meV, which might be accounted for by the impurities initially present in
the substrate or even the substrate-induced structure distortion. By surface
chemical doping with Au atoms, this Rashba spin-orbit coupling is further
strengthened as the adatoms can distort the graphene lattice from $sp^2$ to
$sp^3$ bonding structure. By fitting the Au doping dependence of spin
relaxation from Pi {\sl et al.} [Phys. Rev. Lett. {\bf 104}, 187201 (2010)],
the Rashba spin-orbit coupling coefficient is found to increase approximately
linearly from 0.15 to 0.23 meV with the increase of Au density. With this
strong spin-orbit coupling, the spin diffusion or transport length is
comparable with the experimental values. In the strong scattering limit
(dominated by the electron-impurity scattering in our study), the spin
diffusion is uniquely determined by the Rashba spin-orbit coupling strength and
insensitive to the temperature, electron density as well as scattering. With
the presence of an electric field along the spin injection direction, the spin
transport length can be modulated by either the electric field or the electron
density. (The remaining is omitted due to the limit of space)
