Summary and central questions: Graphene is a bidimensional material made of carbon atoms arranged in honeycomb lattice. It enjoys amazing properties in mechanics, electronics and optics. Its electronic properties were revealed more than fifteen years ago [12, 13, 17], and the proof of existence of gapless cone structure in the energy spectrum emphasized that conducting electrons move very fast, making graphene a good candidate for being used as superconductor. However, the absence of a band gap structure in the energy spectrum of the charge carriers has some severe limitations in the possible technological applications, and it is still a challenging question to open a gap in the electronic spectrum.

Despite several advances done in the spectral theory for graphene models [3, 4, 5, 6, 8, 9, 21] the mathematical analysis of spectral gaps for relevant models remains a subject with many open questions.