Fractional Torsional Rigidity of Compact Metric Graphs

TL;DR

This paper extends the concept of torsional rigidity to fractional powers of the Laplacian on compact metric graphs, establishing bounds and geometric dependence using variational methods and surgery principles.

Contribution

It introduces the fractional torsional rigidity for metric graphs, providing variational characterizations and bounds, a novel extension of classical torsional rigidity to nonlocal operators.

Findings

Explicit bounds for fractional torsional rigidity on graphs

Comparison results with interval and flower graphs

Methods adapted for nonlocal fractional Laplacian

Abstract

This paper investigates fractional torsional rigidity on compact, connected metric graphs, a novel extension of the classical concept to nonlocal operators. The fractional torsional rigidity is defined as the $L^1$-norm of the fractional torsion function, which is the unique solution to the boundary value problem $(-\Delta_{\mathcal{G}})^\alpha u_\alpha = 1$ on a graph $\mathcal{G}$ with zero boundary conditions at Dirichlet vertices. We establish a variational characterization for this quantity, which serves as a powerful tool to prove a series of results on its geometric dependence. By applying surgery principles, we derive explicit upper and lower bounds, indicating that the interval serves as an upper comparison case and the flower graph as a lower one among graphs of fixed total length. These findings mirror the classical case, yet the methods required are substantially different due to the nonlocal nature of the fractional Laplacian.