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This paper deals with the existence of guided waves and edge states inparticular two-dimensional media obtained by perturbing a reference periodicmedium with honeycomb symmetry. This reference medium is a thin periodic domain(the thickness is denoted $\delta$ > 0) with an hexagonal structure, which isclose to an honeycomb quantum graph. In a first step, we show the existence ofDirac points (conical crossings) at arbitrarily large frequencies if $\delta$is chosen small enough. We then perturbe the domain by cutting the perfectlyperiodic medium along the so-called zigzag direction, and we consider eitherDirichlet or Neumann boundary conditions on the cut edge. In the two cases, weprove the existence of edges modes as well as their robustness with respect tosome perturbations, namely the location of the cut and the thickness of theperturbed edge. In particular, we show that different locations of the cut leadto almost-non dispersive edge states, the number of locations increasing withthe frequency. All the results are obtained via asymptotic analysis andsemi-explicit computations done on the limit quantum graph. Numericalsimulations illustrate the theoretical results.

Guided modes in a hexagonal periodic graph like domain