  The main goal of the paper is to prove central limit theorems for the
magnetization rescaled by $\sqrt{N}$ for the Ising model on random graphs with
$N$ vertices. Both random quenched and averaged quenched measures are
considered. We work in the uniqueness regime $\beta>\beta_c$ or $\beta>0$ and
$B\neq0$, where $\beta$ is the inverse temperature, $\beta_c$ is the critical
inverse temperature and $B$ is the external magnetic field. In the random
quenched setting our results apply to general tree-like random graphs (as
introduced by Dembo, Montanari and further studied by Dommers and the first and
third author) and our proof follows that of Ellis in $\mathbb{Z}^d$. For the
averaged quenched setting, we specialize to two particular random graph models,
namely the 2-regular configuration model and the configuration model with
degrees 1 and 2. In these cases our proofs are based on explicit computations
relying on the solution of the one dimensional Ising models.
