We use ab initio density functional calculations to determine the interaction of a graphene monolayer with the Si(111) surface. We found that graphene forms strong bonds to the bare substrate and accommodates the 12% lattice mismatch by forming a wavy structure consisting of free-standing conductive ridges that are connected by ribbon-shaped regions of graphene, which bond covalently to the substrate. We perform quantum transport calculations for different geometries to study changes in the

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... sport properties of graphene introduced by the wavy structure and bonding to the Si substrate. Our results suggest that wavy graphene combines high mobility along the ridges with efficient carrier injection into Si in the contact regions.