  The magnetic structures and interedge magnetic couplings of Fe, Co and Ni
transition-metal terminated graphene nanoribbons with zigzag (ZGNR) and
armchair (AGNR) edges are studied by first-principles calculations. Fe-ZGNR is
found to show antiferromagnetic (AF) coupling between two edges, while the
interedge coupling of Co-ZGNR is ferromagnetic (FM). For Fe-AGNRs and Co-AGNRs,
increasing the interedge distance we follow oscillatory transitions from FM to
AF coupling with a period of about 3.7 {\AA}. The damped oscillatory behavior
indicates a Ruderman-Kittel-Kasuya-Yosida type interedge magnetic coupling and
the oscillation period is determined by the critical spanning vector which
connects two inequivalent Dirac points in the graphene Brillouin zone. The two
edges in Ni-ZGNR are decoupled independent of the ribbon width and Ni-AGNRs are
found to be nonmangetic.
