In this dissertation, we analyze characteristics of eigenfunctions of the Schr?dinger operator on graphs. In particular, we are interested in the zeros of the eigenfunctions and their relation to the spectrum of the magnetic Schr?dinger operator.
We begin by studying the nodal count on finite quantum graphs, analyzing both the number and location of the zeros of eigenfunctions. This question was completely solved by Sturm in one dimension. In higher dimensions (including domains and graphs), we only know bounds for the nodal count. We discover more information about the nodal count on quantum graphs while analyzing eigenvalues of the magnetic Schr?dinger operator. In particular, we show a relation between the stability of eigenvalues of the magnetic Schr?dinger operator with respect to magnetic flux and the number of zeros of the corresponding eigenfunctions. We also study the location of the zeros of eigenfunctions while analyzing partitions. Specifically, we show that the critical points of the energy functional are the nodal partitions corresponding to zeros of an eigenfunction and that the stability of these critical points is related to the nodal count.
Then using Floquet-Bloch theory, we study the spectrum of the Schr?dinger operator on infinite periodic graphs by analyzing the eigenvalues of the magnetic Schr?dinger operator on a fundamental domain. Here we consider both discrete and quantum graphs. We find a characterization of critical points of the dispersion relation that occur inside the Brillouin zone under certain conditions on the graph. In particular, we show that if the fundamental domain is a tree, then the eigenfunction corresponding to an interior critical point must be zero on a vertex.
Finally, we use the results from infinite periodic graphs to study the magnetic Schr?dinger operator on a finite quantum d-mandarin graph. We find that extremal points of the dispersion surface occur inside the Brillouin zone where two surfaces touch and the corresponding eigenfunction is zero on a vertex.