  We present theoretical description of the Casimir interaction in graphene
systems which is based on the Lifshitz theory of dispersion forces and the
formalism of the polarization tensor in (2+1)-dimensional space-time. The
representation for the polarization tensor of graphene allowing the analytic
continuation to the whole plane of complex frequencies is given. This
representation is used to obtain simple asymptotic expressions for the
reflection coefficients at all Matsubara frequencies and to investigate the
origin of large thermal effect in the Casimir force for graphene. The developed
theory is shown to be in a good agreement with the experimental data on
measuring the gradient of the Casimir force between a Au-coated sphere and a
graphene-coated substrate. The possibility to observe the thermal effect for
graphene due to a minor modification of the already existing experimental setup
is demonstrated.
