  We study the spin edge states in the quantum spin-Hall (QSH) effect on a
single-atomic layer graphene ribbon system with both intrinsic and Rashba
spin-orbit couplings. The Harper equation for solving the energies of the spin
edge states is derived. The results show that in the QSH phase, there are
always two pairs of gapless spin-filtered edge states in the bulk energy gap,
corresponding to two pairs of zero points of the Bloch function on the
complex-energy Riemann surface (RS). The topological aspect of the QSH phase
can be distinguished by the difference of the winding numbers of the spin edge
states with different polarized directions cross the holes of the RS, which is
equivalent to the Z2 topological invariance proposed by Kane and Mele [Phys.
Rev. Lett. 95, 146802 (2005)].
