  We derive ground state wave functions of superconducting instabilities on the
honeycomb lattice induced by nearest-neighbor attractive interactions. They
reflect the Dirac nature of electrons in the low-energy limit. For the order
parameter that is the same irrespective of the direction to any of the nearest
neighbors we find weak pairing (slowly decaying) behavior in the orbital part
of the Cooper pair with no angular dependence. At the neutrality point, in the
spin-singlet case, we recover a strong pairing behavior. We also derive ground
state wave functions for the superconductivity on the bilayer honeycomb
lattice, with strong interlayer coupling, induced by attractive interactions
between sites that participate in a low-energy description. Without these
interactions, free electrons are described by a Dirac equation with a quadratic
dispersion. This unusual feature, similarly to $^3$He - B phase, leads to the
description with two kinds of Cooper pairs, with $p_x + i p_y$ and $p_x - i
p_y$ pairing, in the presence of the attractive interactions. We discuss the
edge modes of such a spin-singlet superconductor and find that it represents a
trivial topological superconductor.
