Electron transport properties of an azulene-like dipole molecule anchored with carbon atomic chains sandwiched between two grapheme nanofibbon (GNR) electrodes are theoretically investigated at ab initio level The molecular junctions are constructed with a strategy of modulating symmetry of Bloch wave functions.The chemical doping in an armchair-edged GNR is shown to play a significant role in determining the conductance behavior and rectifying performance of the molecular junctions.Giant rectification ratios up to 104 at low bias voltages are obtained for the molecular junctions with asymmetric arrangement of undoped zGNR and doped aGNR electrodes.The boron (aluminum) dopants in the aGNR electrode induce a better rectifying performance for the molecular junctions than the respective nitrogen (phosphorus) dopants.Moreover, the boron or nitrogen doping is more advantageous than the respective aluminum or phosphorus doping in view of improving rectifying behaviors of the molecularjunctions.Taking double doping in the aGNR electrode, we demonstrate that the double boron-doping displays an improvement of rectifying features in comparison with the single case.