  Generation mechanism of energy gaps between conductance and valence bands is
at the centre of the study of graphene material. Recently Chamon, Jackiw, et
al. proposed a mechanism of using a Kekul\'{e} distortion background field $%
\varphi $ and its induced gauge potential $A_{i}$ to generate energy gaps. In
this paper various vortex structures inhering in this model are studied.
Regarding $\varphi $ as a generic background field rather than a fixed
Nielson-Oleson type distribution, we have found two new types of vortices on
the graphene surface --- the velocity field vortices and the monopole-motion
induced vortices --- from the inner structure of the potential $A_{i}$. These
vortex structures naturally arise from the motion of the Dirac fermions instead
of from the background distortion field.
