On torsional rigidity and ground-state energy of compact quantum graphs

TL;DR

This paper extends the concept of torsional rigidity to metric graphs with Dirichlet vertices, establishing inequalities and bounds that relate torsional rigidity to ground-state energy, simplifying spectral analysis of quantum graphs.

Contribution

It introduces a variational framework for torsional rigidity on metric graphs and derives isoperimetric inequalities and sharp bounds linking torsional rigidity to ground-state energy.

Findings

Established isoperimetric inequalities for metric graphs.

Derived sharp bounds on ground-state energy using torsional rigidity.

Simplified spectral analysis by reducing eigenvalue computation to matrix inversion.

Abstract

We develop the theory of torsional rigidity -- a quantity routinely considered for Dirichlet Laplacians on bounded planar domains -- for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to P\'olya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin Inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph's vertices and is, thus, much easier than computing eigenvalues.