AI 中文总结
研究代数图论中谱方法在图的谱确定及广义汉明图传递性方面的应用，通过多种方法给出完全二部图等谱特征新证，分类广义汉明图传递参数并给出函数闭式，展示谱方法对理解图结构的作用及后续研究方向。
AI 中文摘要
本文研究代数图论中的两个核心方向，重点是谱方法：图的谱确定以及广义汉明图及其补图的传递性。第一部分聚焦由相关矩阵谱确定的图，研究邻接、拉普拉斯等矩阵的谱确定，给出完全二部图等谱特征新证明，引入金字塔图并证其由邻接谱确定。第二部分研究广义汉明图及其补图，通过多种方法分类其边传递或距离传递参数，还给出边传递时Lovász $\vartheta$-函数闭式。结果表明谱方法是理解图结构和对称性的有力工具并指出后续研究方向。
英文摘要
This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Turán graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called \emph{the graphs of pyramids}, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lovász $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.
Comments126 pages, 5 chapters
