  Using exact numerical diagonalization, we investigate localization in two
classes of random matrices corresponding to random graphs. The first class
comprises the adjacency matrices of Erdos-Renyi (ER) random graphs. The second
one corresponds to random cubic graphs, with Gaussian random variables on the
diagonal. We establish the position of the mobility edge, applying the
finite-size analysis of the inverse participation ratio. The fraction of
localized states is rather small on the ER graphs and decreases when the
average degree increases. On the contrary, on cubic graphs the fraction of
localized states is large and tends to $1$ when the strength of the disorder
increases, implying that for sufficiently strong disorder all states are
localized. The distribution of the inverse participation ratio in localized
phase has finite width when the system size tends to infinity and exhibits
complicated multi-peak structure. We also confirm that the statistics of level
spacings is Poissonian in the localized regime, while for extended states it
corresponds to the Gaussian orthogonal ensemble.
