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

摘要

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