Buckminsterfullerene double bonds
All carbon atoms in fullerene are part of one "double bond", as you draw it. In quantum chemical reality, the structure of the fullerene is made up by fixed "single" bonds between those 60 sp2 hybridised carbon atoms, with the residual thirty pairs on pi electrons being somewhat delocalised.

In a sheet of graphene, the pi electrons are delocalised, and every bond lying between two six-rings gets the same share in the electron cloud. 

In fullerene, they are as delocalised, but not equally distributed. The bonds between two six-rings are "more equal" (more aromatic) than those between five- and six-rings, and grab themselves a larger part of the electron density.

Problem of the five-rings is that the bond angles are not 120°, as they should be for a proper sp2 hybridised carbon atom. They become more sp2.4 or something, which let's the pi electrons move out of there a bit.