  Kane and Mele predicted that in presence of spin-orbit interaction graphene
realizes the quantum spin Hall state. However, exceptionally weak intrinsic
spin-orbit splitting in graphene ($\approx 10^{-5}$ eV) inhibits experimental
observation of this topological insulating phase. To circumvent this problem,
we propose a novel approach towards controlling spin-orbit interactions in
graphene by means of covalent functionalization of graphene edges with
functional groups containing heavy elements. Proof-of-concept first-principles
calculations show that very strong spin-orbit coupling can be induced in
realistic models of narrow graphene nanoribbons with tellurium-terminated
edges. We demonstrate that electronic bands with strong Rashba splitting as
well as the quantum spin Hall state spanning broad energy ranges can be
realized in such systems. Our work thus opens up new horizons towards
engineering topological electronic phases in nanostructures based on graphene
and other materials by means of locally introduced spin-orbit interactions.
