Trace formulae for graph Laplacians with applications to recovering matching conditions

TL;DR

This paper derives trace formulae for finite metric graph Laplacians with delta and delta prime conditions, enabling the reconstruction of vertex matching conditions from spectral data.

Contribution

It introduces a series of trace formulae linking graph Laplacians with matching conditions, facilitating spectral reconstruction of these conditions.

Findings

Derived infinite series of trace formulae for graph Laplacians

Established methods for reconstructing matching conditions from spectra

Applied trace formulae to inverse spectral problems

Abstract

Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type. In either case, an infinite series of trace formulae which link together two different graph Laplacians provided that their spectra coincide is derived. Applications are given to the problem of reconstructing matching conditions for a graph Laplacian based on its spectrum.