  We use a tight-binding Bogoliubov-de Gennes (BdG) formalism to
self-consistently calculate the proximity effect, Josephson current, and local
density of states in ballistic graphene SNS Josephson junctions. Both short and
long junctions, with respect to the superconducting coherence length, are
considered, as well as different doping levels of the graphene. We show that
self-consistency does not notably change the current-phase relationship derived
earlier for short junctions using the non-selfconsistent Dirac-BdG formalism
but predict a significantly increased critical current with a stronger junction
length dependence. In addition, we show that in junctions with no Fermi level
mismatch between the N and S regions superconductivity persists even in the
longest junctions we can investigate, indicating a diverging Ginzburg-Landau
superconducting coherence length in the normal region.
