🤖 AI Summary
This study addresses the spectral determination of graphs and the edge- and distance-transitivity of generalized Hamming graphs and their complements. By integrating tools from spectral graph theory, group actions, and association schemes—including the Cauchy interlacing theorem, Schur complements, and Cayley graph constructions—the work introduces a novel family of “pyramid graphs” and establishes their adjacency-spectral uniqueness. It fully characterizes the conditions under which generalized Hamming graphs and their complements are edge-transitive or distance-transitive, and derives closed-form expressions for their Lovász ϑ-functions. The results yield new spectral characterizations of complete bipartite graphs, Turán graphs, and strongly regular graphs, confirm the spectral uniqueness of pyramid graphs, and systematically uncover the intrinsic connections between the symmetry and spectral properties of generalized Hamming graphs.
📝 Abstract
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.