Fractional Torsional Rigidity of Compact Metric Graphs

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.