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.