  We study the exchange interaction $J$ between two magnetic impurities in
graphene (the RKKY interaction) by directly computing the lattice Green's
function for the tight-binding band structure for the honeycomb lattice. The
method allows us to compute $J$ numerically for much larger distances than can
be handled by finite-lattice calculations as well as for small distances. %
avoids the use of a cutoff function often invoked in the literature to curtail
the diverging contributions from the linear bands and yields results that are
valid for all distances. In addition, we rederive the analytical long-distance
behavior of $J$ for linearly dispersive bands and find corrections to the
oscillatory factor that were previously missed in the literature. The main
features of the RKKY interaction in graphene are that unlike the $J \propto
(2k_FR)^{-2} \sin (2k_FR) $ behavior of an ordinary 2D metal in the
long-distance limit, $J$ in graphene falls off as $1/R^3$, shows the $1 + \cos
((K-K').R)$-type oscillations with additional phase factors depending on the
direction, and exhibits a ferromagnetic interaction for moments on the same
sublattice and an antiferromagnetic interaction for moments on the opposite
sublattices as required by particle-hole symmetry. The computed $J$ with the
full band structure agrees with our analytical results in the long-distance
limit including the oscillatory factors with the additional phases.
