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Free valence electrons: 4 of them.. 
In the graphite crystal structure there are strong bonds between the neighbouring carbon atoms.. They are all quite well packed and arranged in a ring like structure.

There is one valence electron which can move from one atom to another. so that allows a good amount of electricity to pass through.

  In Graphite the energy gap between the existing electron state and conduction band is ver y  small.  much smaller than  for semiconductor and a little higher than metals.

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