  We present systematic study of zero modes and gaps by introducing effects of
anisotropy of hopping integrals for a tight-binding model on the honeycomb
lattice in a magnetic field. The condition for the existence of zero modes is
analytically derived. From the condition, it is found that a tiny anisotropy
for graphene is sufficient to open a gap around zero energy in a magnetic
field. This gap behaves as a non-perturbative and exponential form as a
function of the magnetic field. The non-analytic behavior with respect to the
magnetic field can be understood as tunneling effects between energy levels
around two Dirac zero modes appearing in the honeycomb lattice, and an explicit
form of the gap around zero energy is obtained by the WKB method near the
merging point of these Dirac zero modes. Effects of the anisotropy for the
honeycomb lattices with boundaries are also studied. The condition for the
existence of zero energy edge states in a magnetic field is analytically
derived. On the basis of the condition, it is recognized that anisotropy of the
hopping integrals induces abrupt changes of the number of zero energy edge
states, which depend on the shapes of the edges sensitively.
