  Graphene is a monoatomic layer of graphite with Carbon atoms arranged in a
two dimensional honeycomb lattice configuration. It has been known for more
than sixty years that the electronic structure of graphene can be modelled by
two-dimensional massless relativistic fermions. This property gives rise to
numerous applications, both in applied sciences and in theoretical physics.
Electronic circuits made out of graphene could take advantage of its high
electron mobility that is witnessed even at room temperature. In the
theoretical domain the Dirac-like behavior of graphene can simulate high energy
effects, such as the relativistic Klein paradox. Even more surprisingly,
topological effects can be encoded in graphene such as the generation of
vortices, charge fractionalization and the emergence of anyons. The impact of
the topological effects on graphene's electronic properties can be elegantly
described by the Atiyah-Singer index theorem. Here we present a pedagogical
encounter of this theorem and review its various applications to graphene. A
direct consequence of the index theorem is charge fractionalization that is
usually known from the fractional quantum Hall effect. The charge
fractionalization gives rise to the exciting possibility of realizing graphene
based anyons that unlike bosons or fermions exhibit fractional statistics.
Besides being of theoretical interest, anyons are a strong candidate for
performing error free quantum information processing.
