  The comparison studies of theoretical approaches to the description of the
Casimir interaction in layered systems including graphene is performed. It is
shown that at zero temperature the approach using the polarization tensor leads
to the same results as the approach using the longitudinal density-density
correlation function of graphene. An explicit expression for the
zero-temperature transverse density-density correlation function of graphene is
provided. We further show that the computational results for the Casimir free
energy of graphene-graphene and graphene-Au plate interactions at room
temperature, obtained using the temperature-dependent polarization tensor,
deviate significantly from those using the longitudinal density-density
correlation function defined at zero temperature. We derive both the
longitudinal and transverse density-density correlation functions of graphene
at nonzero temperature. The Casimir free energy in layered structures including
graphene, computed using the temperature-dependent correlation functions, is
exactly equal to that found using the polarization tensor.
