  We report a systematic study on strong enhancement of spin-orbit interaction
(SOI) in graphene driven by transition-metal dichalcogenides (TMDs). Low
temperature magnetotoransport measurements of graphene proximitized to
different TMDs (monolayer and bulk WSe$_2$, WS$_2$ and monolayer MoS$_2$) all
exhibit weak antilocalization peaks, a signature of strong SOI induced in
graphene. The amplitudes of the induced SOI are different for different
materials and thickness, and we find that monolayer WSe$_2$ and WS$_2$ can
induce much stronger SOI than bulk ones and also monolayer MoS$_2$. The
estimated spin-orbit (SO) scattering strength for the former reaches $\sim$ 10
meV whereas for the latter it is around 1 meV or less. We also discuss the
symmetry and type of the induced SOI in detail, especially focusing on the
identification of intrinsic and valley-Zeeman (VZ) SOI via the dominant spin
relaxation mechanism. Our findings offer insight on the possible realization of
the quantum spin Hall (QSH) state in graphene.
