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We have investigated the quantum $J_1-J_2-J_3$ model on the honeycomb latticewith exact diagonalizations and linear spin-wave calculations for selectedvalues of $J_{2}/J_{1}$, $J_{3}/J_{1}$ and antiferromagnetic ($J_{1}>0$) orferromagnetic ($J_{1}<0$) nearest neighbor interactions. We found a variety ofquantum effects: "order by disorder" selection of a N{\'e}el orderedground-state, good candidates for non-classical ground-states with dimer longrange order or spin-liquid like. The purely antiferromagnetic Heisenberg modelis confirmed to be N{\'e}el ordered. Comparing these results with thoseobserved on the square and triangular lattices, we enumerate some conjectureson the nature of the quantum phases in the isotropic models.Comment: 14 pages, 22 Postscript figures, uses svjour.cls and svepj.clo,  submitted to European Physical Journal B: condensed matter physi

An investigation of the quantum J 1 - J 2 - J 3 model on the honeycomb lattice