Torsional rigidity on metric graphs with delta-vertex conditions

We investigate the torsion function (or landscape function) and its associated mass, known as the torsional rigidity, for Laplacians on metric graphs with δ-vertex conditions. We establish a variational characterization of torsional rigidity and derive Hadamard-type formulas, which enable the development of surgical principles. Using these principles, we obtain upper and lower bounds on torsional rigidity and identify extremal graphs within certain classes. We further study the positivity of the torsion function and reduce this question to the positivity of the spectrum of an associated discrete weighted Laplacian. Finally, we explore possible manifestations of Kohler–Jobin-type inequalities in the setting of δ-vertex conditions.