  We discuss the magnetic phases of the Hubbard model for the honeycomb lattice
both in two and three spatial dimensions. A ground state phase diagram is
obtained depending on the interaction strength
  U and electronic density n. We find a first order phase transition between
ferromagnetic regions where the spin is maximally polarized (Nagaoka
ferromagnetism) and regions with smaller magnetization (weak ferromagnetism).
When taking into account the possibility of spiral states, we find that the
lowest critical U is obtained for an ordering momentum different from zero. The
evolution of the ordering momentum with doping is discussed. The magnetic
excitations (spin waves) in the antiferromagnetic insulating phase are
calculated from the random-phase-approximation for the spin susceptibility. We
also compute the spin fluctuation correction to the mean field magnetization by
virtual emission/absorpion of spin waves. In the large $U$ limit, the
renormalized magnetization agrees qualitatively with the Holstein-Primakoff
theory of the Heisenberg antiferromagnet, although the latter approach produces
a larger renormalization.
