Graphene and single-wall carbon nanotube (SWCNT) have attracted greatattention because of their ultra-high thermal conductivity. However, there arefew works exploring the relations of their thermal conductivity quantitatively.The carbon nanocone (CNC) is a graded structure fall in between graphene disk(GD) and SWCNT. We perform non-equilibrium molecular dynamics (NEMD) simulationto study the thermal conductivity of CNC with different apex angles, and thencompare them with that of GD and SWCNT. Our results show that, different fromthe homogeneous thermal conductivity in SWCNT, the CNC also has a naturalgraded thermal conductivity which is similar to the GD. Unexpectedly, thegraded rate keeps almost the same when the apex angle decreases from 180{\deg}(GD) to 19{\deg}, but then suddenly declines to zero when the apex angledecreases from 19{\deg} to 0{\deg} (SWCNT). What is more interesting, thegraded effect is not diminished when the interatomic force constant is weakenedand mean free path is shorten. That is, besides nanoscale, the graded effectcan be observed in macroscale graphene or CNC structures.
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