  We study the effects of insulating oxides in their crystalline forms on the
energy band structure of monolayer and bilayer graphene using a \textit{first
principles} density functional theory based electronic structure method and a
local density approximation. We consider the dielectric oxides, SiO$_{2}$
($\alpha$-quartz) and Al$_{2}$O$_{3}$ (alumina or $\alpha$-sapphire) each with
two surface terminations. Our study suggests that atomic relaxations and
resulting equilibrium separations play a critical role in perturbing the linear
band structure of graphene in contrast to the less critical role played by
dangling bonds that result from cleaving the crystal in a particular direction.
We also see that with the addition of a second graphene layer, the Dirac cone
is restored for the quartz surface terminations. Alumina needs more than two
graphene layers to preserve the Dirac cone. Our results are at best
semi-quantitative for the common amorphous forms of the oxides considered.
However, crystalline oxides for which our results are quantitative provide an
interesting option for graphene based electronics, particularly in light of
recent experiments on graphene with crystalline dielectrics (hexagonal BN) that
find considerable improvements in transport properties as compared to the those
with amorphous dielectrics.
