We have investigated single- and bi-layer graphene as source-drain electrodes for n-type MoS2 transistors. Ti-MoS2-graphene heterojunction transistors using both single-layer MoS2 (1M) and 4-layer MoS2 (4M) were fabricated in order to compare graphene electrodes with commonly used Ti electrodes. MoS2-graphene Schottky barrier provided electron injection efficiency up to 130 times higher in the subthreshold regime when compared with MoS2-Ti, which resulted in VDS polarity dependence of device parameters such as threshold voltage (VTH) and subthreshold swing (SS). Comparing single-layer graphene (SG) with bi-layer graphene (BG) in 4M devices, SG electrodes exhibited enhanced device performance with higher on/off ratio and increased field-effect mobility (μFE) due to more sensitive Fermi level shift by gate voltage. Meanwhile, in the strongly accumulated regime, we observed opposing behavior depending on MoS2 thickness for both SG and BG contacts. Differential conductance (σd) of 1M increases with VDS irrespective of VDS polarity, while σd of 4M ceases monotonic growth at positive VDS values transitioning to ohmic-like contact formation. Nevertheless, the low absolute value of σd saturation of the 4M-graphene junction demonstrates that graphene electrode could be unfavorable for high current carrying transistors.