  We formulate a model of relativistic fermions moving in two Euclidean
dimensions based on a tight-binding model of graphene. The eigenvalue spectrum
of the resulting Dirac operator is solved numerically in smooth U(1) gauge
field backgrounds carrying an integer-valued topological charge Q, and it is
demonstrated that the resulting number of zero-eigenvalue modes is in accord
with the Atiyah-Singer index theorem applied to two continuum flavors. A
bilinear but gauge non-invariant chirality operator appropriate for
distinguishing the topological zero modes is identified. When this operator is
used to calculate Q, it is found that the maximum topological charge capable of
being measured in this fashion scales with the perimeter of the lattice. Some
concluding remarks compare these results to what is known for staggered lattice
fermions.
