  By using the density functional theory (DFT) combined with the molecular
dynamics (MD) simulations, structural and electronic properties of mono-vacancy
(MV) defect in M\"obius strip formed from graphene are investigated. Two kinds
of MV are observed depending on the local structures around defects. At static
condition, in the curved areas of M\"obius strip, MV has the configuration of
one pentagon and one nonagon ring (59-type), which is similar to that of carbon
nanotubes and graphene. the most stable MV appear in the twisted areas and has
a profile of two pentagon and two hexagon rings (5566-type) with one sp3
hybridized carbon at the central site. While DFT-MD simulations prove that the
5566-type MV is an unstable configuration at room temperature and will
transform into a 59-type MV. Additionally, the melting behavior of graphene
based M\"obius strips are investigated through empirical potential MD
simulations, and we find that their melting temperature is about 2750 K, which
is lower than that of carbon nanotubes and graphene.
