  We compute analytic expressions for the edge states in a zigzag Kane-Mele
nanoribbon (KMNR) by solving the eigenvalue equations in presence of intrinsic
and Rashba spin-orbit couplings. Owing to the P-T symmetry of the Hamiltonian
the edge states are protected by topological invariance and hence are found to
be robust. We have done a systematic study for each of the above cases, for
example, a pristine graphene, graphene with an intrinsic spin-orbit coupling,
graphene with a Rashba spin-orbit coupling, a Kane-Mele nanoribbon and
supported our results on the robustness of the edge states by analytic
computation of the electronic probability amplitudes, the local density of
states (LDOS), band structures and the conductance spectra.
