Isospectrality for graph Laplacians under the change of coupling at graph vertices: necessary and sufficient conditions

TL;DR

This paper establishes necessary and sufficient conditions for when two graph Laplacians are isospectral, based on vertex coupling changes, showing that spectrum generally determines matching conditions for most graphs.

Contribution

It provides a complete characterization of isospectrality for graph Laplacians with delta and delta' conditions, under rational independence of edge lengths.

Findings

Necessary and sufficient conditions for isospectrality are derived.

Spectrum generally determines matching conditions for almost all graphs.

The results apply to Laplace operators with specific vertex conditions.

Abstract

Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. Assuming rational independence of edge lengths, necessary and sufficient conditions of isospectrality of two Laplacians defined on the same graph are derived and scrutinized. It is proved that the spectrum of a graph Laplacian uniquely determines matching conditions for "almost all" graphs.