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We investigate topological properties of models that describe graphene onrealistic substrates which induce proximity spin-orbit coupling in graphene. A$\mathbb{Z}_2$ phase diagram is calculated for the parameter space of(generally different) intrinsic spin-orbit coupling on the two graphenesublattices, in the presence of Rashba coupling. The most fascinating case isthat of staggered intrinsic spin-orbit coupling which, despite beingtopologically trivial, $\mathbb{Z}_2 = 0$, does exhibit edge states protectedagainst time-reversal scattering for zigzag ribbons as wide as micrometers. Wecall these states pseudohelical as their helicity is locked to the sublattice.The spin character and robustness of the pseudohelical modes is best exhibitedon a finite flake, which shows that the edge states have zero $g$-factor, carrya finite spin current in the crossection of the flake, and exhibit spin-flipreflectionless tunneling at the armchair edges.

Protected pseudohelical edge states in proximity graphene ribbons and flakes