  Van der Waals-Casimir dispersion interactions between two apposed graphene
layers, a graphene layer and a substrate, and in a multilamellar graphene
system are analyzed within the framework of the Lifshitz theory. This
formulation hinges on a known form of the dielectric response function of an
undoped or doped graphene sheet, assumed to be of a random phase approximation
form. In the geometry of two apposed layers the separation dependence of the
van der Waals-Casimir interaction for both types of graphene sheets is
determined and compared with some well known limiting cases. In a multilamellar
array the many-body effects are quantified and shown to increase the magnitude
of the van der Waals-Casimir interactions.
