We consider graphene deposited on monolayers of such transition-metal dichalcogenides like MoSe \(\phantom{0}_2\) , WSe \(\phantom{0}_2\) , MoS \(\phantom{0}_2\) , and WS \(\phantom{0}_2\) . Our key objective in this paper is to study the impact of relative twist angle between the monolayers on the proximity-induced spin-orbit interaction and orbital phenomena in graphene. To do this we use an effective model Hamiltonian for low-energy states, taken from the available literature. The linear response theory and Green function formalism are used to calculate analytical formulas for the spin Hall effect and nonequilibrium current-induced spin polarization in the systems. In addition, we also evaluate the valley Hall effect and nonequilibrium valley polarization, and focus especially on their dependence on the twist angle. We show that the valley Hall conductivity can achieve the quantum value equal to \(\pm 2 e^2/h\) .
