Selected topics in spectral graph theory
Document TypePhD thesis
Access StatusOnly available to University of Melbourne staff and students, login required
© 2015 Dr. Xiaogang Liu
The purpose of this thesis is to study the spectra and several spectral properties of several classes of graphs. Apart from the introduction in Chapter 1, the first part of the main body of the thesis consists of Chapters 2 and 3. In this part we determine the spectra of some classes of Cayley graphs, including unitary Cayley graphs and quadratic unitary Cayley graphs. We also investigate some spectral properties of these graphs, such as spectral moments, energies and hyperenergeticities of such graphs, and classify those which are Ramanujan. In the second part, which consists of Chapters 4 and 5, we determine the spectra of graphs obtained by some graph operations, including neighbourhood corona, subdivision-vertex neighbourhood coronae and subdivision-edge neighbourhood coronae. By using the spectra, we construct infinitely many pairs of cospectral graphs and new expanders from known ones. In the last part, Chapters 6-8, we investigate spectral characterizations of some joins and some bicyclic graphs including propeller graphs, dumbbell graphs and theta graphs. We prove that all these graphs are determined by their spectra.
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