Cospectral quantum graphs

TL;DR

This paper investigates the spectral properties of quantum graphs, focusing on how eigenvalue asymptotics can uniquely determine the graph's shape or substructure, with implications for spectral graph theory.

Contribution

It identifies conditions under which the first two eigenvalue asymptotic terms uniquely determine the graph's shape or interior subgraph.

Findings

Eigenvalue asymptotics can determine graph shape

Unique determination of interior subgraph from spectral data

Conditions for spectral uniqueness established

Abstract

Spectral problems are considered generated by the Sturm-Liouville equation on connected simple equilateral graphs with the Neumann and Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. We highlight the cases where the first and the second terms of the asymptotics of the eigenvalues uniquely determine the shape of the graph or of its interior subgraph.