Graphene has a honeycomb lattice (in the absence of defects and impurities). By considering the low-energy limit of the half-filled Hubbard model used to model the strongly interacting electron gas we find that the low-energy quasiparticles obey the dispersion relation for massless fermions. These details are all covered very nicely in a paper by Gonzalez, Guniea and Vozmediano (reference) among others.
It might seem like I'm answering the question. I'm following this line of exposition because I don't want to assume that this is a topic something commonly known or understood outside the condensed matter community. Any answers which elaborated on these basics would be very useful as they would help make the discussion more broadly accessible.
My primary question is more about the implications this fact has for high-energy physics, in particular the question of emergent-matter in theories of quantum gravity. The case of graphene is a canonical example in that regard where one obtains relativistic, massless excitations in the low-energy corner of an otherwise non-relativistic system - the 2D electron gas (2DEG).
Obviously I have my own beliefs in this regard and I will try to outline them in an answer. But I also want to solicit the communities views in this regard.