English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this thesis, we study Laplacian eigenfunctions on metric graphs, alsoknown as quantum graphs. We restrict the discussion to standard quantum graphs.These are finite connected metric graphs with functions that satisfy Neumannvertex conditions.  The first goal of this thesis is the study of the nodal count problem. Thatis the number of points on which the $n$th eigenfunction vanishes. We provide aprobabilistic setting using which we are able to define the nodalcount\textquoteright s statistics. We show that the nodal count statisticsadmit a topological symmetry by which the first Betti number of the graph canbe obtained. We revise a conjecture that predicts a universal Gaussian behaviorof the nodal count statistics for large graphs and we prove it for a certainfamily of graphs.  The second goal is to formulate and study the Neumann count, which is thenumber of local extrema of the $n$th eigenfunction. This counting problem ismotivated by the Neumann partitions of planar domains, a novel concept inspectral geometry. We provide uniform bounds on the Neumann count andinvestigate its statistics. We show that the Neumann count providescomplimentary geometrical information to that obtained from the nodal count. Weshow that for a certain family of growing tree graphs the Neumann countstatistics approaches a Gaussian distribution.  The third goal is a genericity result, which justifies the generality of theNeumann count discussion. To this day it was known that generically,eigenfunctions do not vanish on vertices. We generalize this result toderivatives at vertices as well. That is, generically, the derivatives of aneigenfunction on interior vertices do not vanish.

Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics