  The ground state phase diagram of the spin-1/2 Heisenberg frustrated
$\Delta$-chain with a ferromagnetic main chain is investigated. In addition to
the ferromagnetic phase, various nonmagnetic ground states are found. If the
ferromagnetic coupling between apical spins and the main chain is strong, this
model is approximated by a spin-1 bilinear-biquadratic chain and the spin
quadrupolar phase with spin-2 gapless excitation is realized in addition to the
Haldane and ferromagnetic phases. In the regime where the coupling between the
apical spins and the main chain is weak, the numerical results which suggest
the possibility of a series of phase transitions among different nonmagnetic
phases are obtained. Physical pictures of these phases are discussed based on
the numerical results.

Strongly Correlated Electrons;Statistical Mechanics