  The spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain
with ferromagnetic next-nearest-neighbour (NNN) interaction is investigated.
The ground state is the Haldane phase for weak NNN interaction, and is the
ferromagnetic phase for weak antiferromagnetic interaction. We find a series of
topologically distinct spin-gap phases with various magnitudes of edge spins
for strong NNN interaction. The phase boundaries between these phases are
determined on the basis of the DMRG calculation with additional spins that
compensate the edge spins. It is found that each of the exact solutions with
short-range antiferromagnetic correlation on the ferromagnetic-nonmagnetic
phase boundary is representative of each spin gap phase.
