Spectral clustering is one of the most popular, yet still incompletely understood, methods for community detection on graphs. This article studies spectral clustering based on the Bethe-Hessian matrix H r = (r 2 − 1)I n + D − rA for sparse heterogeneous graphs (following the degree-corrected stochastic block model) in a two-class setting. For a specific value r = ζ, clustering is shown to be insensitive to the degree heterogeneity. We then study the behavior of the informative eigenvector of H ζ and, as a result, predict the clustering accuracy. The article concludes with an overview of the generalization to more than two classes along with extensive simulations on synthetic and real networks corroborating our findings.