  We have used molecular dynamics to calculate the thermal conductivity of
symmetric and asymmetric graphene nanoribbons (GNRs) of several nanometers in
size (up to ~4 nm wide and ~10 nm long). For symmetric nanoribbons, the
calculated thermal conductivity (e.g. ~2000 W/m-K @400K for a 1.5 nm {\times}
5.7 nm zigzag GNR) is on the similar order of magnitude of the experimentally
measured value for graphene. We have investigated the effects of edge chirality
and found that nanoribbons with zigzag edges have appreciably larger thermal
conductivity than nanoribbons with armchair edges. For asymmetric nanoribbons,
we have found significant thermal rectification. Among various
triangularly-shaped GNRs we investigated, the GNR with armchair bottom edge and
a vertex angle of 30{\deg} gives the maximal thermal rectification. We also
studied the effect of defects and found that vacancies and edge roughness in
the nanoribbons can significantly decrease the thermal conductivity. However,
substantial thermal rectification is observed even in the presence of edge
roughness.
