Why electrons are relativistic in Graphene and non relativistic in vacuum? If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron.
On the other hand conduction electrons in graphene are relativistic for the same potential difference. 
Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?
 A: According to this article: http://physics.aps.org/articles/v5/24:
The statement that in graphene the "conduction electrons are massless" is because the energy levels (bands) are  proportional to their momenta. 
So the $E = \sqrt{p^2+m^2}$ relation of a free electron becomes $E\propto p$ in graphene. 
Massless particles travel all at the same speed because of the $E\propto p$ relation but this characteristic velocity in graphene is far below c though, only 0.3% of the speed of light.
The reason that the relation $E\propto p$ leads to a characteristic speed is due to the quantum mechanical wave character. $E$ is proportional to the phase changes in time, $p$ is proportional to the phase changes in space and therefor $p/E$ is proportional to the velocity. In the case that $E\propto p$ there is a characteristic velocity $v$ independent of the energy level. 

The most striking aspect of graphene is that its electronic energy
  levels, or “bands,” produce conduction electrons whose energies are
  directly proportional to their momentum. This is the energy-momentum
  relationship exhibited by photons, which are massless particles of
  light. Electrons and other particles of matter normally have energies
  that depend on the square of their momentum.
When the bands are plotted in three dimensions, the photonlike
  energy-momentum relationship appears as an inverted cone, called a
  Dirac cone. This unusual relationship causes conduction electrons to
  behave as though they were massless, like photons, so that all of them
  travel at roughly the same speed (about 0.3 percent of the speed of
  light). This uniformity leads to a conductivity greater than copper.

Hans
A: As far as I understand, electrons in graphene are not relativistic, although quasiparticles in graphene are indeed described by the massless Dirac equation. However, for graphene, the speed velocity in this equation is replaced by the Fermi velocity, which is much smaller. 
