In this dissertation, we analyze the spectrum of the Laplace operator on graphs. In particular, we are interested in generic eigenpairs. We consider a wide range of vertex conditions on vertices of a quantum graph. Furthermore, we also investigate the eigenfunctions, showing that generically they do not vanish on vertices, unless this is unavoidable due to presence of looping edges. In the proof, the simplicity of eigenvalues and non-vanishing of eigenvalues are tightly interconnected; each property is assisting in the proof of the other (the proof is done by induction). The proof is geometric in nature and uses local modifications of the graph to reduce it to previously considered cases. We also consider an application of the result to the study of the secular manifold of a graph, showing that for large classes of graphs, the set of smooth points of the manifold has exactly two connected components. The spectrum of a symmetric quantum graph is also considered. We aim to give explicit and computation-oriented formulas for extracting the part of a Schr?dinger operator on a graph which corresponds to a particular irreducible representation of the graph's symmetry. Starting with a representation of the symmetry by its action on the space of directed bonds of the graph, we find a basis which block-diagonalizes both the representation and the bond scattering matrix of the graph. The latter leads to a factorization of the secular determinant into factors that correspond to irreducible representation of the symmetry group.
In this dissertation, we analyze the spectrum of the Laplace operator on graphs. In particular, we are interested in generic eigenpairs.
We consider a wide range of vertex conditions on vertices of a quantum graph. Furthermore, we also investigate the eigenfunctions, showing that generically they do not vanish on vertices, unless this is unavoidable due to presence of looping edges. In the proof, the simplicity of eigenvalues and non-vanishing of eigenvalues are tightly interconnected; each property is assisting in the proof of the other (the proof is done by induction). The proof is geometric in nature and uses local modifications of the graph to reduce it to previously considered cases. We also consider an application of the result to the study of the secular manifold of a graph, showing that for large classes of graphs, the set of smooth points of the manifold has exactly two connected components.
The spectrum of a symmetric quantum graph is also considered. We aim to give explicit and computation-oriented formulas for extracting the part of a Schr?dinger operator on a graph which corresponds to a particular irreducible representation of the graph's symmetry. Starting with a representation of the symmetry by its action on the space of directed bonds of the graph, we find a basis which block-diagonalizes both the representation and the bond scattering matrix of the graph. The latter leads to a factorization of the secular determinant into factors that correspond to irreducible representation of the symmetry group.