The leitmotif of my proposed research is the extraction of geometric information about graphs from the spectra of the graph's Schrödinger operators, and from the distribution of zeros of the corresponding eigenfunctions. The spectral geometric point of view shows intrigue links between quantum and combinatorial graphs, which go over towards higher dimensional domains. Therefore, the proposed research investigates spectral geometry for both quantum and combinatorial graphs and also makes connections between quantum graphs which are one dimensional objects and separable domains which are of higher dimension. In addition, I consider the mathematical physics point of view which spreads out of the pure spectral geometry.
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