  Following the recent proposal by Weeks et al., which suggested that indium
(or thallium) adatoms deposited on the surface of graphene should turn the
latter into a quantum spin Hall (QSH) insulator characterized by a sizeable
gap, we perform a systematic study of the transport properties of this system
as a function of the density of randomly distributed adatoms. While the samples
are, by construction, very disordered, we find that they exhibit an extremely
stable QSH phase with no signature of the spatial inhomogeneities of the adatom
configuration. We find that a simple rescaling of the spin-orbit coupling
parameter allows us to account for the behaviour of the inhomogeneous system
using a homogeneous model. This robustness opens the route to a much easier
experimental realization of this topological insulator. We additionally find
this material to be a very promising candidate for thermopower generation with
a target temperature tunable from 1 to 80K and an efficiency ZT close to 1.
