  This work is concerned with the Dirac points for the honeycomb lattice with
impenetrable obstacles arranged periodically in a homogeneous medium. We
consider both the Dirichlet and Neumann eigenvalue problems and prove the
existence of Dirac points for both eigenvalue problems at crossing of the lower
band surfaces as well as higher band surfaces. Furthermore, we perform
quantitative analysis for the eigenvalues and the slopes of two conical
dispersion surfaces near each Dirac point based on a combination of the layer
potential technique and asymptotic analysis. It is shown that the eigenvalues
are in the neighborhood of the singular frequencies associated with the Green's
function for the honeycomb lattice, and the slopes of the dispersion surfaces
are reciprocal to the eigenvalues.
