  We consider a deformed flat-band Hubbard model under a periodic boundary
condition in arbitrary dimensions. We show that the ground state is only
all-spin-up or -down state. We obtain upper and lower bounds of the one-magnon
spin-wave energy with an arbitrary momentum. This dispersion relation is the
same as that in the XXZ model in the certain parameter region. Therefore the
spin wave has a finite energy gap. These results suggests the our model and the
XXZ model.
