In this talk, we shall present some recent works with Charles Bordenave and Antti Knowles giving estimates for the extreme eigenvalues of the centered adjacency matrices of a possibly inhomogenous Erdös-Rényi graph. In the case of the Stochastic Block Model, these works allow to assess the accuracy of the spectral clustering algorithm, whereas in the case of an homogenous Erdös-Rényi graph, they establish a transition between a compact spectrum and a diffuse one, at a threshold which coincides with the connectivity threshold.
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