  Electronic structures of graphene sheet with different defective patterns are
investigated, based on the first principles calculations. We find that
defective patterns can tune the electronic structures of the graphene
significantly. Triangle patterns give rise to strongly localized states near
the Fermi level, and hexagonal patterns open up band gaps in the systems. In
addition, rectangular patterns, which feature networks of graphene nanoribbons
with either zigzag or armchair edges, exhibit semiconducting behaviors, where
the band gap has an evident dependence on the width of the nanoribbons. For the
networks of the graphene nanoribbons, some special channels for electronic
transport are predicted.
