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

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. Our work is systematic and rigorous. Finally, we solve an inverse spectral problem for the potential, given the knowledge of the pure point and absolutely continuous spectra.