Spectral Stability of Graph Laplacians Under Structured Edge Perturbations

The spectrum of a graph Laplacian reflects fundamental properties of a graph, including its connectivity and overall structure. Existing results in spectral graph theory largely describe the effects of random perturbations or isolated edge modifications, while comparatively little attention has been given to changes that follow deterministic structural rules. Many evolving networks gain or lose edges according to mechanisms that depend on their existing organization rather than by chance alone. Whether these structured perturbations produce predictable changes in Laplacian spectra across different graph families remains an open mathematical question.

This project asks whether structured edge perturbations lead to consistent patterns of spectral change that are independent of the underlying graph construction. Graph families including Erdős-Rényi, Barabási-Albert, Watts-Strogatz, and selected deterministic graphs will undergo edge additions and deletions according to predefined structural rules. For each perturbation, the Laplacian spectrum will be recomputed and compared with the original graph. The central question is whether changes in spectral gap and eigenvalue distribution can be explained by the structure of the perturbation itself rather than by the properties of an individual graph family. The group will compare the observed spectral behavior across graph classes while examining the extent to which existing theoretical bounds describe the measured changes. Computational experiments will be repeated over a range of graph sizes to determine whether consistent relationships emerge as networks become larger.