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.
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