Theoretically supported techniques are given for clustering the nodes of edge-weighted graphs via non-backtracking spectra when the number of nodes is large and the skeleton graph is sparse. If the graph comes from a sparse stochastic block model, the structural real eigenvalues, out of the bulk of the spectrum, of the non-backtracking matrix are aligned with those of the expected adjacency matrix if it is of low rank. However, only the unweighted or weighted non-backtracking matrix is at our disposal. We show how the corresponding eigenvectors of the non-backtracking matrix and lower order companion matrices can be used to find assortative clusters of the nodes even in the case, when the expected adjacency matrix does not have a reduced rank, but it has a low-rank approximation. The paper gives the theoretical background and tools for sparse spectral clustering in very general frameworks. Application to sparse quantum chemistry networks is also presented.