  Spin transport in graphene carries the potential of a long spin diffusion
length at room temperature. However, extrinsic relaxation processes limit the
current experimental values to 1-2 um. We present Hanle spin precession
measurements in gated lateral spin valve devices in the low to high (up to
10^13 cm^-2) carrier density range of graphene. A linear scaling between the
spin diffusion length and the diffusion coefficient is observed. We measure
nearly identical spin- and charge diffusion coefficients indicating that
electron-electron interactions are relatively weak and transport is limited by
impurity potential scattering. When extrapolated to the maximum carrier
mobilities of 2x10^5 cm^2/Vs, our results predict that a considerable increase
in the spin diffusion length should be possible.
