Mathematical Research Seminar - Archive
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Doctoral dissertation topic presentation:
The proposed doctoral dissertation addresses three independent research directions in spectral and extremal graph theory. The common theme of these directions is the study of graph invariants arising from eigenvalues, symmetry and optimization. Each part is intended to form a substantial and self-contained contribution.
The first part focuses on nut graphs, that is, graphs on at least two vertices whose adjacency matrix has a simple eigenvalue zero with a corresponding eigenvector having no zero entries. The main objective is to investigate nut graphs with a high degree of symmetry, including vertex-transitive nut graphs and Cayley nut graphs. In particular, we aim to determine the pairs (n, d) for which there exists a d-regular vertex-transitive (resp. Cayley) nut graph, and to investigate the existence of infinitely many regular (resp. Cayley) nut graphs with a prescribed degree. This part also examines the realizability of cubic ℓ-circulant nut graphs for given ℓ ∈ ℕ, and aims to characterize all quartic 2-circulant nut graphs. In addition, we study which pairs (ov(G), oe(G)) of vertex and edge orbit counts can be attained by a nut graph G.
The second part concerns the spectral radius maximization over connected graphs with a prescribed order and size. Let Cn,e denote the set of all the connected graphs with n vertices and n − 1 + e edges. The goal is to solve the spectral radius maximization problem on Cn,e in the case e ≤ 130 or n ≥ e + 2 + 13√e.
The third part explores the applications of reinforcement learning (RL) methods to extremal graph theory problems. Building on recent developments in which graphs are constructed through sequential decision-making processes, we aim to develop a modular and reusable software framework that enables the application of various RL techniques to the optimization of graph invariants. The long-term objective is to facilitate future graph theory research and to use this framework to generate potential examples or counterexamples that could lead to new theoretical results.
