Torsional Rigidity on Metric Graphs with Delta-Vertex Conditions

Abstract

We investigate the torsion function or landscape function and its integral, the torsional rigidity, of Laplacians on metric graphs subject to $δ$-vertex conditions. A variational characterization of torsional rigidity and Hadamard-type formulas are obtained, enabling the derivation of surgical principles. We use these principles to prove upper and lower bounds on the torsional rigidity and identify graphs maximizing and minimizing torsional rigidity among classes of graphs. We also investigate the question of positivity of the torsion function and reduce it to positivity of the spectrum of a particular discrete, weighted Laplacian. Additionally, we explore potential manifestations of Kohler-Jobin-type inequalities in the context of $δ$-vertex conditions.

Figures (7)

• Figure 1: The metric graph from Example \ref{['exa:2.6']}. The sum of strengths is zero but the torsion function is not positive definite.
• Figure 2: The metric graph from Example \ref{['exa:2.7']}. The sum of strengths is positive but the infimum of the spectrum can be negative.
• Figure 3: The interval graph $\mathcal{J}$ from Example \ref{['exa:interval_graph']} (left) and the flower graph $\mathcal{F}$ from Example \ref{['exa:flower']} (right).
• Figure 4: The process of inserting $\mathcal{G}'$ into $\mathcal{G}$ at a vertex $\mathsf{v}_0 \in \mathcal{G}$ is not unique. Both $\tilde{\mathcal{G}_1}$ and $\tilde{\mathcal{G}_2}$ can be obtained.
• Figure 5: Inserting an interval graph $\mathcal{G}'$ in an interval graph $\mathcal{G}$ to obtain the graph $\tilde{\mathcal{G}}$ as in Example \ref{['exa:inserting']}

Theorems & Definitions (59)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Remark 2.5
• proof : Proof of Proposition \ref{['postor']}
• Example 2.6
• Example 2.7
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:varchar']}