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The pseudo-relativistic theory of graphene arises from the fact that, in low temperatures, the mobile electrons of graphene behave as if they were massless relativistic fermions. Therefore, their states are described by spinors, instead of the non-relativistic wave functions.

Effective Dirac equation

Honeycomb hexagonal lattice.

Non-relativistic electrons such as those found in graphene follow Schrödinger equation. However, proper analysis shows that it is possible to derive an effective Dirac equation -- usually used to describe relativistic quantum particles.

Graphene has a two-dimensional honeycomb crystal lattice. Its primitive vectors are $\mathbf {a_{1}} =a\mathbf {\hat {x}}$ and $\mathbf {a_{2}} ={\frac {a}{2}}\left({\sqrt {3}}\mathbf {\hat {x}} +\mathbf {\hat {y}} \right)$. Therefore, its reciprocal lattice primitive vectors are: $\mathbf {b_{1}} ={\frac {2\pi }{a}}\left(\mathbf {\hat {x}} -{\sqrt {3}}\mathbf {\hat {y}} \right)$ and $\mathbf {b_{2}} ={\frac {4\pi }{a}}\mathbf {\hat {y}}$.