  We investigate charge transport in C$_{60}$-based single-molecule junctions
with graphene electrodes employing a combination of density functional theory
(DFT) electronic structure calculations and Landauer transport theory. In
particular, the dependence of the transport properties on the conformation of
the molecular bridge and the type of termination of the graphene electrodes is
investigated. Furthermore, electron pathways through the junctions are analyzed
using the theory of local currents. The results reveal, in agreement with
previous experiments, a pronounced dependence of the transport properties on
the bias polarity, which is rationalized in terms of the electronic structure
of the molecule. It is also shown that the edge states of zigzag-terminated
graphene induce additional transport channels, which dominate transport at
small voltages. The importance of the edge states for transport depends
profoundly on the interface geometry of the junctions.
