A comprehensive first-principles study of the correlation between zero-energy states and the tunability of the spin-selective semiconducting properties of zigzag-edged bowtie-shaped graphene nanoflakes under an electric field is presented for the first time. We demonstrate that the spin degenerate semiconducting ground state can be lifted by the electric field. In particular, we find that the number of zero-energy states ('the nullity') defined by the structural configuration determines the complexity and efficiency of the tunability of spin polarization. The fine-tuning of spin-dependent properties by the electric field originates from the manipulation of spin-polarized molecular orbital energies. We expect this study to aid the design of more effective and controllable low-dimensional molecular spintronics.