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Carbon has four bonding (or "valence") electrons. If each electron bonds with a different atom, the four bonds form a tetrahedron ~109° apart. To chemists and physicists, these are "sigma sp3 hybrid" bonds.
Electrons are held immobile in these "sigma" bonds and as a result they will not pass electrical current.
The most basic carbon ring structure is "cyclohexane," shown here. Six carbon atoms form a ring. Hydrogen atoms complete the rest of the bonds. (Turn the model with your mouse).
To a chemist, the name cyclohexane says the following: "cyclo" = ring, "hex" = six, "ane" = carbon atoms are bonded to each other only with single non-conducting sigma bonds (i.e. this is in the class of "alkane" carbon compounds).
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