Graphene and single-wall carbon nanotube (SWCNT) have attracted great attention because of their ultra-high thermal conductivity. However, there are few 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) simulation to study the thermal conductivity of CNC with different apex angles, and then compare them with that of GD and SWCNT. Our results show that, different from the homogeneous thermal conductivity in SWCNT, the CNC also has a natural graded thermal conductivity which is similar to the GD. Unexpectedly, the graded rate keeps almost the same when the apex angle decreases from 180 degrees (GD) to 19 degrees, but then suddenly declines to zero when the apex angle decreases from 19 degrees to 0 degrees (SWCNT). What is more interesting, the graded effect is not diminished when the interatomic force constant is weakened and mean free path is shorten. That is, besides nanoscale, the graded effect can be observed in macroscale graphene or CNC structures. (C) 2016 Elsevier Ltd. All rights reserved.