Abstract: This talk is based on joint work with David Gontier and Thaddeus Roussigné. We consider a tight-binding model for graphene, where distortion of the honeycomb lattice is allowed, but penalized by a quadratic energy. We prove that the optimal 3-periodic lattice configuration has Kekulé O-type symmetry, and that for a sufficiently small elasticity parameter, the minimizer is not translation-invariant. Conversely, we prove that for a large elasticity parameter the translation-invariant configuration is the unique energy minimizer. We also investigate a related continuum model involving the Dirac operator.