  The transition-metal dichalcogenide (TMDC) in the family of $\mathrm{MX}_2$
($\mathrm{M}=\mathrm{Mo},\mathrm{W}$; $\mathrm{X}=\mathrm{S},\mathrm{Se}$) and
the graphene monolayer are atomically thin semiconductors and semimetal,
respectively. The monolayer $\mathrm{MX}_2$ have been discovered as a new class
of semiconductors for electronics and optoelectronics applications. Because of
the hexagonal lattice structure of both the materials, $\mathrm{MX}_2$ and
graphene are often combined with each other to make van der Waals
heterostructures. Here, the $\mathrm{MX}_2/\mathrm{Gr}$ heterostructures are
investigated theoretically based on the Density Functional Theory (DFT). The
electronic structure and the optical properties of four different
$\mathrm{MX}_2/\mathrm{Gr}$ heterostructures are computed. We systematically
compare these $\mathrm{MX}_2/\mathrm{Gr}$ heterostructures for their complex
permittivity, absorption coefficient, reflectivity and refractive index.
