We designed acene molecules attached to two semi-infinite metallic electrodes to explore the source-drain current of graphene and the gate leakage current of the gate dielectric material in the field-effect transistors (FETs) device using the first-principles density functional theory combined with the non-equilibrium Green's function formalism. In the acene-based molecular junctions, we modify the connection position of the thiol group at one side, forming different electron transport routes. The electron transport routes besides the shortest one are defined as the cross channels. The simulation results indicate that electron transport through the cross channels is as efficient as that through the shortest one, since the conductance is weakly dependent on the distance. Thus, it is possible to connect the graphene with multiple leads, leading the graphene as a channel utilized in the graphene-based FETs in the mesoscopic system. When the conjugation of the cross channel is blocked, the junction conductance decreases dramatically. The differential conductance of the BA-1 is nearly 7 (54.57 μS) times as large as that of the BA-4 (7.35 μS) at zero bias. Therefore, the blocked graphene can be employed as the gate dielectric material in the top-gated graphene FET device, since the leakage current is small. The graphene-based field-effect transistors fabricated with a single layer of graphene as the channel and the blocked graphene as the gate dielectric material represent one way to overcome the problem of miniaturization which faces the new generation of transistors.