  Ordered graphene has been extensively studied. In this paper we undertake a
first density functional study of it topologically disordered analogues of
graphene, in the form of a random network, consisting predominantly of
hexagonal rings, but also including pentagons and heptagons. After some
preliminaries with crystalline material, we relax various random network models
and find that the presence of carbon pentagons induce local curvature, thus
breaking the initial planar symmetry, in some analogy with the case of
fullerenes. Using density functional theory to calculate the total energy, we
find that while the planar state is locally stable, there is a puckered state
that has lower energy. The scale of the puckering is consistent with that
expected with local maxima and minima associated with pentagons surrounded by
larger rings; forming local "buckyball domes".
