  We study the spectral properties of the adjacency matrix in the giant
connected component of Erd\"os-R\'enyi random graphs, with average connectivity
$p$ and randomly distributed hopping amplitudes. By solving the self-consistent
cavity equations satisfied by the matrix elements of the resolvent, we compute
the probability distribution of the local density of states, which governs the
scaling with the system size of the moments of the eigenvectors' amplitudes, as
well as several other observables related to the spectral statistics. For small
values of $p>1$ above the percolation threshold, we unveil the presence of an
exotic delocalized but (weakly) multifractal phase in a broad region of the
parameter space, which separates the localized phase found for $p\le1$ from the
fully-delocalized GOE-like phase expected for $p\to \infty$. We explore the
fundamental physical mechanism underlying the emergence of delocalized
multifractal states, rooted in the pronounced heterogeneity in the topology of
the graph. This heterogeneity arises from the interplay between strong
fluctuations in local degrees and hopping amplitudes, and leads to an effective
fragmentation of the graph. We further support our findings by characterizing
the level statistics and the two-point spatial correlations within the
multifractal phase, and address the ensuing anomalous transport and relaxation
properties affecting the quantum dynamical evolution.
