  Topology concepts have significantly deepened of our understanding in recent
years of the electronic properties of one-dimensional (1D) nano structures such
as the graphene nanoribbons. Controlling topological electronic properties of
GNRs has been demonstrated in both theoretical studies and experimental
realization. Most previous works rely on classification theory requiring both
time reversal and spatial symmetry of a unit cell in the 1D bulk material that
is commensurate to its boundary. To access boundary structures that lead to
unit cell with no spatial symmetry and to generalize the theory, we propose
here another classification scheme, using chiral symmetry, to arrive at a Z
classification that is not only applicable to GNRs with arbitrary terminations,
but also to any general 1D chiral structures. This theory, combining with
Lieb's theorem, moreover enables access to the electron's spin degree of
freedom, allowing for investigation of spin physics.
