  The two dimensionality plus the linear band structure of graphene leads to
new behavior of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which is
the interaction between two magnetic moments mediated by the electrons of the
host crystal. We study this interaction from linear response theory. There are
two equivalent methods both of which may be used for the calculation of the
susceptibility, one involving the integral over a product of two Green's
functions and the second that involves the excitations between occupied and
unoccupied states, which was followed in the original work of Ruderman and
Kittel. Unlike the $J \propto (2k_FR)^{-2} \sin (2k_FR) $ behavior of an
ordinary two-dimensional (2D) metal, $J$ in graphene falls off as $1/R^3$,
shows the $1 + \cos ((\bm{K}-\bm{K'}).\bm{R})$-type of behavior, which contains
an interference term between the two Dirac cones, and it oscillates for certain
directions and not for others. Quite interestingly, irrespective of any
oscillations, the RKKY interaction in graphene is always ferromagnetic for
moments located on the same sublattice and antiferromagnetic for moments on the
opposite sublattices, a result that follows from particle-hole symmetry.
