  The equation of state of the electron degenerate gas in a white dwarf is
usually treated by employing the ideal dispersion relation. However, the effect
of quantum gravity is expected to be inevitably present and when this effect is
considered through a non-commutative formulation, the dispersion relation
undergoes a substantial modification. In this paper, we take such a modified
dispersion relation and find the corresponding equation of state for the
degenerate electron gas in white dwarfs. Hence we solve the equation of
hydrostatic equilibrium and find that this leads to the possibility of the
existence of excessively high values of masses exceeding the Chandrasekhar
limit although the quantum gravity effect is taken to be very small. It is only
when we impose the additional effect of neutronization that we obtain white
dwarfs with masses close to the Chandrasekhar limit with nonzero radii at the
neutronization threshold. We demonstrate these results by giving the numerical
estimates for the masses and radii of $^4He$ and $^{12}C$, and $^{16}O$ white
dwarfs.
