I asked my teacher about this and he says that each carbon only make 3 bonds and has a lone pair. These lone pairs are de-localised hence, that is why buckminsterfullerene is able to conduct electricity.
However, I am doubtful about this as how can a lone pair on an atom be de-localised. Would the electrons be in their orbitals, and not be able to freely move around the compound. I try searching the internet for any information, but there is no mention on how many bonds each carbon atoms makes.
I am guessing that there is alternating double bonds between each carbon so that each carbon has 4 bonds. It is these pi bonds which along the compound that makes it a hyper-conjugated compound. This would explain why buckminsterfullerene and graphite are able to conduct electricity. Am I correct?