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Graphene/MoS$_2$ van der Waals (vdW) heterostructures have promisingtechnological applications due to their unique properties and functionalities.Many experimental and theoretical research groups across the globe have madeoutstanding contributions to benchmark the properties of graphene/MoS$_2$heterostructures. Even though some research groups have already made an attemptto model the graphene/MoS$_2$ heterostructures using {\it first-principles}calculations, there exists several discrepancies in the results from differenttheoretical research groups and the experimental findings. In the present work,we revisit this problem by first principles approach and address the existingdiscrepancies about the interlayer spacing between graphene and MoS$_2$monolayers in graphene/MoS$_2$ heterostructures, and the location of Diracpoints near Fermi-level. We find that the Tkatchenko--Scheffler methodefficiently evaluates the long-range vdW interactions and accurately predictsinterlayer spacing between graphene and MoS$_2$ sheets. We further investigatethe electronic, mechanical and vibrational properties of the optimizedgraphene/MoS$_2$ heterostructures created using 5$\times$5/4$\times$4 and4$\times$4/3$\times$3 supercell geometries having different magnitudes oflattice mismatch. The effect of the varying interlayer spacing on theelectronic properties of heterostructures is discussed. Our phonon calculationsreveal that the interlayer shear and breathing phonon modes, which are verysensitive to the weak vdW interactions, play vital role in describing thethermal properties of the studied systems. The thermodynamic and elasticproperties of heterostructures are further discussed. A comparison between ourresults and the results reported from other research groups is presented.

First-principles investigation of graphene/MoS2 bilayer heterostructures using Tkatchenko-Scheffler van der Waals method