arXiv:2611.08246v1 Announce Type: new Abstract: Pure point spectral convergence is generally unavailable for sparse graph sequences, where Laplacian eigenvalues may remain unstable under graph growth. This poses challenges for analyzing spectral operators underlying many graph-based learning methods, including graph neural networks. We identify a class of sparse bipartite graph sequences for which eigenvalue-level spectral convergence can be recovered. Focusing on graphs with strong degree heterogeneity, we exploit the convergence properties of their line-graph sequences. Using a spectral correspondence between bipartite graph Laplacians and line-graph adjacency operators, we transfer convergence guarantees from dense line-graphs back to the original sparse graphs. This framework yields eigenvalue-level spectral convergence for a class of deterministic sparse graph sequences. We further analyze superlinear preferential attachment trees, establishing convergence to an infinite-dimensional limit operator. Numerical experiments empirically support our spectral convergence results.