Torsional Rigidity, Isospectrality and Quantum Graphs

TL;DR

This paper demonstrates that torsional rigidity can distinguish between isospectral non-isometric planar domains and their quantum graph analogs, providing a new geometric invariant for spectral analysis.

Contribution

It proves that torsional rigidity differentiates isospectral non-isometric pairs in both graph and quantum graph settings, a novel insight in spectral geometry.

Findings

Isospectral pairs have different torsional rigidity values.

Torsional rigidity serves as a distinguishing geometric invariant.

Results apply to both classical graphs and quantum graphs.

Abstract

We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.