  We demonstrate topological characterization of $S=\frac{1}{2}$ Kitaev quantum
spin liquids on a series of one-dimensional honeycomb nanoribbon lattices with
zigzag and armchair terminated edges. We draw their Majorana spinon phase
diagrams with varying nearest-neighbor exchange couplings not only at the
sector of the ground flux configuration but also at some sectors of excited
flux configurations. In the ground states of the zigzag and armchair
nanoribbons, there occur a single and multiple phase transitions, respectively,
the former and latter of which are insensitive and subject to the background
flux configuration, respectively. Topological phases each have a winding number
as their invariant. On each phase boundary, the Majorana spinon dispersion
relation reflects both of the change in the winding number and the background
flux configuration.
