  We study the ferromagnetic random-field Ising model on random graphs of fixed
connectivity z (Bethe lattice) in the presence of an external magnetic field
$H$. We compute the number of single-spin-flip stable configurations with a
given magnetization m and study the connection between the distribution of
these metastable states in the H-m plane (focusing on the region where the
number is exponentially large) and the shape of the saturation hysteresis loop
obtained by cycling the field between $-\infty$ and $+\infty$ at T=0. The
annealed complexity $\Sigma_A(m,H)$ is calculated for z=2,3,4 and the quenched
complexity $\Sigma_Q(m,H)$ for z=2. We prove explicitly for z=2 that the
contour $\Sigma_Q(m,H)=0$ coincides with the saturation loop. On the other
hand, we show that $\Sigma_A(m,H)$ is irrelevant for describing, even
qualitatively, the observable hysteresis properties of the system.
