Raziskovalni matematični seminar - Arhiv

We investigate how different classes of protein binding sites, particularly conserved water binding sites, affect the pathogenicity of single-nucleotide polymorphisms (SNPs) in human proteins. Our analysis is grounded in a graph-theoretic framework implemented within the ProBiS platform, where protein surface regions are represented as similarity graphs and structurally conserved binding sites are identified via maximum-clique computations. In this work, we employ and further develop new efficient maximum-clique algorithms tailored to large-scale protein structural comparison, enabling the processing of a comprehensive dataset of human protein structures from the Protein Data Bank (PDB).

Conserved water molecules, which correspond to highly stable and topologically coherent vertices within these similarity graphs, are essential for maintaining protein stability, mediating ligand recognition, and shaping conformational dynamics. Despite their central biochemical role, the relationship between water binding sites and the pathogenicity of nearby genetic variants has remained largely uncharacterized.

Our results demonstrate that SNPs located within or proximal to conserved water binding sites exhibit significantly elevated pathogenicity relative to SNPs in other structural environments. This previously unreported association provides new mechanistic insights into how perturbations of water-mediated interaction networks contribute to human disease. To facilitate further research, we provide an openly accessible dataset containing over 40,000 SNPs mapped to conserved water binding sites and more than 500,000 SNPs associated with other binding site categories. Coupled with our improved maximum-clique methodology, this dataset forms a foundation for advancing mathematical and computational approaches to predicting the pathogenicity of newly observed genetic variants and may ultimately support the rational design of targeted therapeutic strategies.

Evra, Feigon, Maurischat, and Parzanchevksi defined a bipartite extension of Caley graphs. When viewed as hypergraphs, their definition coincides with the restriction of the Cayley hypergraphs of Shee, Lee and Kwon, or Jajcay and Jajcayová to linear, uniform, regular hypergraphs. In this talk, we explore that connection, as well as some basic properties and examples of Cayley incidence graphs. This is joint work with Arnbjörg Soffía Árnadóttir, Alexey Gordeev, Tovohery Randrianarisoa, and Joannes Vermant.

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 (o_v(G), o_e(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 C_n,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 C_n,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.