  We present a detailed description of the nature of the wavefunction and spin
distribution of the zero energy modes of zigzag graphene ribbons (ZGRs) in the
presence of the intrinsic spin_orbit (I-SO) interaction. These states
characterize the quantum spin Hall (QSH) phase in graphene ribbons. We provide
analytic expressions for wavefunctions and show how these evolve as the
strength of the interaction and the ribbon width are changed. For odd-width
ribbons, we show that its insulating nature precludes the existence of a QSH
phase. For these systems the I-SO interaction is predicted to have a stronger
effect as shown by the enhancement of the gap as the interaction strength is
turned on.
