  In this article we derive the lattice Green Functions (GFs) of graphene using
a Tight Binding Hamiltonian incorporating both first and second nearest
neighbour hoppings and allowing for a non-orthogonal electron wavefunction
overlap. It is shown how the resulting GFs can be simplified from a double to a
single integral form to aid computation, and that when considering off-diagonal
GFs in the high symmetry directions of the lattice this single integral can be
approximated very accurately by an algebraic expression. By comparing our
results to the conventional first nearest neighbour model commonly found in
literature, it is apparent that the extended model leads to a sizeable change
in the electronic structure away from the linear regime. As such, this article
serves as a blueprint for researchers who wish to examine quantities where
these considerations are importa
