Question

why non-metals do not conduct electricity except graphite (allotrope of carbon)

Answer 1

In case of metals the free electrons present are responsible for the conduction of electricity .They consist of free electrons because of there electropositive nature.
But in case of non metals because of there highly electronegative nature they do not losse there outer electrons rather they have a tendency to accept very easily or gain electrons.
Since these free electrons are absent in non metals therefore they do not conduct electricity.