On the p-torsional rigidity of compact metric graphs: a sharp Kohler--Jobin inequality

What the math gives to ML

The paper develops torsional rigidity as a variational functional of a nonlinear graph p-Laplacian and proves sharp comparisons with the first eigenvalue under fixed total length. The transferable asset is the positive solution generated by a uniform source: it provides a geometry-aware notion of how strongly each location is connected to Dirichlet boundaries, while the nonlinear exponent p controls sensitivity to bottlenecks and uneven edge weights. In graph neural networks, this can become a deterministic positional encoding or feature gate computed once from the input graph, supplying global geometry that ordinary local message passing may require many layers to discover. The strongest first test is a torsion-augmented GNN on graphs with bottlenecks, long branches, or varying boundary structure, compared against degree, Laplacian eigenvectors, and diffusion positional encodings.