Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by Berkolaiko, Kennedy, Kurasov, and Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by Kac and Pivovarchik in 2011, we characterize the graphs exhibiting maximally degenerate eigenvalues which are the figure-of-eight graph, the 3-watermelon graph, and the lasso trees-namely, trees decorated with lasso graphs.

Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic χ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured. In the present work we analyse sufficient hypotheses which guarantee the successful recovery of χ. We also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required. Finally, we compare our findings with numerical examples-surprisingly, just a few dozens of eigenvalues can be enough.

This thesis consists of four papers concerning topics in the spectral theory of quantum graphs, which are differential operators on metric graphs.

In paper I we present a family of graphs with an arbitrary number of cycles for which a certain eigenvalue upper estimate is sharp. This result disproves that such estimate could be improved as it was conjectured in the paper where it was originally derived.

In paper II we study the problem of maximizing the first eigenvalue—also called ground-state energy—of the Schrödinger operator on a fixed metric graph with delta-type vertex conditions subject to integral constraints on the potential and coupling constant. Depending on whether an optimal solution exists or not we either characterize the optimal potential and coupling constant or we discuss the asymptotic behaviour. Remarkably, it appears that the solution is independent of the topology of the graph. In particular, for strong potential the ground-state is given as a function of the distance from the nearest vertex.

Paper III deals with the inverse problem of recovering the number of independent cycles of a graph from a limited number of the smallest eigenvalues of the standard Laplacian. The mathematical analysis of the method is supported by numerical simulations inspired by a recent experiment where the spectrum is obtained by measuring resonances in a microwave network.

In paper IV we present a class of graphs for which both upper and lower estimates, recently established, are sharp on the same infinite sequence of eigenvalues. This is possible due to the presence of multiple eigenvalues.

Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it exists is a constant function on its support formed by a set of intervals separated from the vertices. In the case where the optimal configuration does not exist explicit optimising sequences are presented.

We prove that the upper spectral estimate for quantum graphs due to Berkolaiko-Kennedy-Kurasov-Mugnolo [5] is sharp.

The magnetic Schrödinger operator on the 8-shaped graph is studied. It is shown that for specially chosen vertex conditions the spectrum of the magnetic operator is independent on the flux through one of the loops, provided the flux through the other loop is zero. Topological reasons for this effect are explained.

The magnetic Schrodinger operator was studied on a figure 8-shaped graph. It is shown that for specially chosen vertex conditions, the spectrum of the magnetic operator is independent of the flux through one of the loops, provided the flux through the other loop is zero. Topological reasons for this effect are explained.

Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by I. Kac and V. Pivovarchik in 2011 we characterize the family of graphs exhibiting maximally degenerate eigenvalues which we call lasso trees, namely graphs constructed from trees by attaching lasso graphs to some of the vertices.