  The spectral and localization properties of heterogeneous random graphs are
determined by the resolvent distributional equations, which have so far
resisted an analytic treatment. We solve analytically the resolvent equations
of random graphs with an arbitrary degree distribution in the high-connectivity
limit, from which we perform a thorough analysis of the impact of degree
fluctuations on the spectral density, the inverse participation ratio, and the
distribution of the local density of states. We show that all eigenvectors are
extended and that the spectral density exhibits a logarithmic or a power-law
divergence when the variance of the degree distribution is large enough. We
elucidate this singular behaviour by showing that the distribution of the local
density of states at the center of the spectrum displays a power-law tail
determined by the variance of the degree distribution. In the regime of weak
degree fluctuations the spectral density has a finite support, which promotes
the stability of large complex systems on random graphs.
