We investigate the origin of non-ideal transfer characteristics in graphene-based three-branch nano-junction (TBJ) devices. Fabricated graphene TBJs often show asymmetric nonlinear voltage transfer characteristic, although symmetric one should appear ideally. A simple model considering the contact resistances in two input electrodes is deduced and it suggests that the non-ideal characteristic arises from inequality of the metal-graphene contact resistances in the inputs. We fabricate a graphene TBJ device with electrically equal contacts by optimizing the contact formation process and almost ideal nonlinear characteristic was successfully demonstrated.