You are saying that carbon compounds does not conduct electricity, then how do graphite conduct electricity?

Graphite comprises of a carbon atom bonded with 3 other carbon atoms; C=C=C=C

Each carbon atom uses three of its electrons to form simple bonds to its three close neighbours. That leaves a fourth electron in the bonding level. This makes it an anion. These "spare" electrons in each carbon atom conduct electricity due to presence of ions.

Basically, every bond with a spare electron or proton will conduct electricity.