  Motivated by the rich physics of honeycomb magnetic materials, we obtain the
phase diagram and analyze magnetic properties of the spin-1/2 and spin-1
J1-J2-J3 Heisenberg model on the honeycomb lattice. Based on the SU(2) and
SU(3) symmetry representations of the Schwinger boson approach, which treats
disordered spin liquids and magnetically ordered phases on an equal footing, we
obtain the complete phase diagrams in the (J2,J3)plane. This is achieved using
a fully unrestricted approach which does not assume any pre-defined Ansatze.
For S=1/2, we find a quantum spin liquid (QSL) stabilized between the N\'eel,
spiral and collinear antiferromagnetic phases in agreement with previous
theoretical work. However, by increasing S from 1/2 to 1, the QSL is quickly
destroyed due to the weakening of quantum fluctuations indicating that the
model already behaves as a quasi-classical system. The dynamical structure
factors and temperature dependence of the magnetic susceptibility are obtained
in order to characterize all phases in the phase diagrams. Moreover, motivated
by the relevance of the single-ion anisotropy, D, to various S=1 honeycomb
compounds, we have analyzed the destruction of magnetic order based on a SU(3)
representation of the Schwinger bosons. Our analysis provides a unified
understanding of the magnetic properties of honeycomb materials realizing the
J1-J2-J3 Heisenberg model from the strong quantum spin regime at S=1/2 to the
S=1 case. Neutron scattering and magnetic susceptibility experiments can be
used to test the destruction of the QSL phase when replacing S=1/2 by S=1
localized moments in certain honeycomb compounds.
