The cores of edge dislocations, edge dislocation dipoles and edge dislocation loops in planar graphene have been studied by means of periodized discrete elasticity models. To build these models, we have found a way to discretize linear elasticity on a planar hexagonal lattice using combinations of difference operators that do not involve symmetrically all the neighbors of an atom. At zero temperature, dynamically stable cores of edge dislocations may be heptagon-pentagon pairs (glide

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... s) or octagons (shuffle dislocations) depending on the choice of initial configuration. Possible cores of edge dislocation dipoles are vacancies, pentagon-octagon-pentagon divacancies, Stone-Wales defects and 7-5-5-7 defects. While symmetric vacancies, divacancies and 7-5-5-7 defects are dynamically stable, asymmetric vacancies and 5-7-7-5 Stone-Wales defects seem to be unstable.