Generally carbon compounds do not conduct electricity because they do not contain free electrons because carbon is capable of forming many allotropes due to its valency. Well-known forms of carbon include diamond and graphite. In recent decades many more allotropes and forms of carbon have been discovered and researched including ball shapes such as buckminsterfullerene and sheets such as graphene. Larger scale structures of carbon include nanotubes, nanobuds and nanoribbons. Other unusual forms of carbon exist at very high temperature or extreme pressures.
Carbon compounds in molten state don't conduct electricity, bcoz they lack of free electrons. Free electrons are necessary for conducting electricity.
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