Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract

Abstract One interesting phenomenon of graphene is the presence of the conical singularity
or Dirac points. Using the quantum graph model, we show that there exist three
classes of possible Dirac points for all of the periodic quantum graphs associated with
Archimedean tilings, when the potentials are identical and even. They occur at the
periodic eigenvalues, anti-periodic eigenvalues and other double eigenvalues of the dispersion
relations respectively. We also characterize their associated potentials. Moreover,
we show that there are no other possible Dirac points. Finally, we solve an
inverse spectral problem for the potential, given the knowledge of the pure point and
absolutely continuous spectra.