arXiv:2607.09390 2026 Stability 1 ideas extracted · analyzed Aug 30, 2026
The paper gives a sharp local-combinatorial certificate for controlling the largest Laplacian eigenvalue: forbidding an independent set of size k inside every vertex neighborhood forces \(\mu(G)\le (2-2/k)(\Delta+1)\). This can transfer to graph neural networks and sparse attention graphs, where the Laplacian spectral radius controls the stability of diffusion, smoothing, and residual propagation. The transferable asset is the conversion of a local neighborhood constraint into a global spectral-norm bound. A practical adaptation is to regularize or construct routing graphs so that their neighborhoods contain few large independent sets, then use the theorem-derived step-size ceiling for certified stable propagation.
Constrain a learned binary graph or sparse attention-routing graph so that every node neighborhood has no independent set of size k. This local anti-star condition gives an explicit upper bound on the graph Laplacian spectral radius, allowing a larger but certified stable diffusion step or residual propagation coefficient.