Fractional Torsional Rigidity of Compact Metric Graphs

TL;DR

This work extends torsional rigidity to fractional Laplacians on compact metric graphs by defining the fractional torsion function $u_\alpha$ as the solution to $(-\Delta_{\mathcal{G}})^\alpha u_\alpha = 1$ with Dirichlet conditions on a vertex set $\mathcal{V}_D$, and $T_\alpha(\mathcal{G})=\int_\mathcal{G} u_\alpha$. It develops a variational framework in the Hilbert space $H_0^\alpha(\mathcal{G})$, proving existence, positivity, and a spectral representation $T_\alpha(\mathcal{G})=\sum_n |\langle 1,\varphi_n\rangle|^2/\lambda_n^\alpha$, with $u_\alpha$ as the maximizer of a quadratic functional. The authors introduce surgery principles—edge doubling, unfolding, cycle cutting, and vertex gluing—to compare $T_\alpha$ across graphs, yielding explicit universal upper and lower bounds in terms of total length and edge count; these bounds consistently recover the classical $\alpha=1$ results. The results provide nonlocal, isoperimetric-like insights for network geometries and are validated on fundamental graphs such as intervals and flowers.

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 $(-Δ_{\mathcal{G}})^αu_α= 1$ 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.

Figures (6)

• Figure 1: The interval graph of length $L$, serving as the simplest one-dimensional domain where the fractional torsion function and torsional rigidity can be computed explicitly.
• Figure 2: The flower graph, consisting of several loops attached to a common vertex. This structure highlights how cycles influence the behavior of the fractional torsion function.
• Figure 3: The graph obtained by doubling each edge: for every original connection between two vertices, a second parallel edge is added.
• Figure 4: Triangle with each edge doubled. Label the parallel edges $e_{ij}^{(1)},e_{ij}^{(2)}$. The Eulerian closed trail shown in red traverses the edges in order $e_{12}^{(1)},e_{23}^{(1)},e_{31}^{(1)},e_{12}^{(2)},e_{23}^{(2)},e_{31}^{(2)}$, i.e. $\mathsf{v}_1\to\mathsf{v}_2\to\mathsf{v}_3\to\mathsf{v}_1\to\mathsf{v}_2\to\mathsf{v}_3\to\mathsf{v}_1$.
• Figure 5: Cutting the cycle at $v_1$ splits it into two boundary points $v_1^{-},v_1^{+}$ and unfolds the loop to an interval; interior vertices (e.g. $v_2,v_3$) lie along the resulting path.

Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Definition 3.1: Fractional Torsion Function
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Definition 3.4
• Lemma 3.5
• proof