  The antiferromagnetism (AFM) and chiral-d wave superconductivity (SC) in a
honeycomb lattice close to an antiferromagnetic (AF) Mott state at half band
filling are studied with a t-J model and slave boson mean field theory. The
order parameters and single particle dispersion relations at different band
filling fractions are investigated. It is found that the AFM enhances the SC,
and leads to nodal single particle dispersion relations at two band filling
fractions. These unexpected nodal dispersion relations out of nodeless chiral-d
wave superconducting order are discussed. A comparison between AF chiral-d wave
states and AF extended-s wave states is also given to highlight the pertinent
features in chiral-d wave superconducting states. This study may be related to
the honeycomb lattice materials such as the In$_3$Cu$_2$VO$_9$ compound.
