  The thermal conductivity of low-dimensional materials and graphene
nanoribbons in particular, is limited by the strength of line-edge-roughness
scattering. One way to characterize the roughness strength is the dependency of
the thermal conductivity on the channel width in the form W^{\beta}. Although
in the case of electronic transport this dependency is very well studied,
resulting in W^6 for nanowires and quantum wells and W^4 for nanoribbons, in
the case of phonon transport it is not yet clear what this dependence is. In
this work, using lattice dynamics and Non-Equilibrium Greens Function
simulations, we examine the width dependence of the thermal conductivity of
ultra-narrow graphene nanoribbons under the influence of line edge-roughness.
We show that the exponent {\beta} is in fact not a single well-defined number,
but it is different for different parts of the phonon spectrum depending on
whether phonon transport is ballistic, diffusive, or localized. The exponent
{\beta} takes values {\beta} < 1 for semi-ballistic phonon transport, values
{\beta} >> than 1 for sub-diffusive or localized phonons, and {\beta} = 1 only
in the case where the transport is diffusive. The overall W^{\beta} dependence
of the thermal conductivity is determined by the width-dependence of the
dominant phonon modes (usually the acoustic ones). We show that due to the long
phonon mean-free-paths, the width-dependence of thermal conductivity becomes a
channel length dependent property, because the channel length determines
whether transport is ballistic, diffusive, or localized.
