Spectral Geometry of Graphs

- 1. Very Personal Introduction. - 2. How to Define Differential Operators on Metric Graphs. - 3. Vertex Conditions. - 4. Elementary Spectral Properties of Quantum Graphs. - 5. The Characteristic Equation. - 6. Standard Laplacians and Secular Polynomials. - 7. Reducibility of Secular Polynomials. - 8. The Trace Formula. - 9. Trace Formula and Inverse Problems. - 10. Arithmetic Structure of the Spectrum and Crystalline Measures. - 11. Quadratic Forms and Spectral Estimates. - 12. Spectral Gap and Dirichlet Ground State. - 13. Higher Eigenvalues and Topological Perturbations. - 14. Ambartsumian Type Theorems. - 15. Further Theorems Inspired by Ambartsumian. - 16. Magnetic Fluxes. - 17. M-Functions: Definitions and Examples. - 18. M-Functions: Properties and First Applications. - 19. Boundary Control: BC-Method. - 20. Inverse Problems for Trees. - 21. Boundary Control for Graphs with Cycles: Dismantling Graphs. - 22. Magnetic Boundary Control I: Graphs with Several Cycles. - 23. Magnetic Boundary Control II: Graphs on One Cycle and Dependent Subtrees. - 24. Discrete Graphs.