  Using the Green's function method, we study the effect of an impurity
potential on the electronic structure of the honeycomb lattice in the one-band
tight-binding model that contains both the nearest neighbor ($t$) and the
second neighbor ($t'$) interactions. The model is relevant to the case of the
substitutional vacancy in graphene. If the second neighbor interaction is large
enough ($t' > t /3$), then the linear Dirac bands no longer occur at the Fermi
energy and the electronic structure is therefore fundamentally changed. With
only the nearest neighbor interactions present, there is particle-hole
symmetry, as a result of which the vacancy induces a "zero-mode" state at the
band center with its wave function entirely on the majority sublattice, i. e.,
on the sublattice not containing the vacancy. With the introduction of the
second neighbor interaction, the zero-mode state broadens into a resonance peak
and its wave function spreads into both sublattices, as may be argued from the
Lippmann-Schwinger equation. The zero-mode state disappears entirely for the
triangular lattice and if $t'$ is large for the honeycomb lattice as well. In
case of graphene, $t'$ is relatively small, so that a well-defined zero-mode
state occurs in the vicinity of the band center.
