Discrete Structures Seminars

Date: April 2, 2025 
Time: 10:30

Which Equations, Sets, (Hyper)Graphs, and Polynomials Are Extremal? Bridging Mathematical Fields Through Combinatorial Ideas

Nika Salia, King Fahd University

Abstract:

In this talk, we will explore recent advances in combinatorics and graph theory, with a focus on stability results, extremal problems, and the structural properties of discrete objects. We will introduce a stability version of Dirac’s classical theorem, providing a full characterization of near-Hamiltonian graphs, and discuss extensions of Pósa’s theorem to hypergraphs. Additionally, I will present the resolution of a longstanding conjecture by Hakimi and Schmeichel on the maximum number of pentagons in planar graphs. Further, I will highlight connections between combinatorics and algebra, including results on intersecting families of polynomials over finite fields and higher-order extensions of Schur’s theorem. These findings illustrate the deep interplay between combinatorics and other areas of mathematics, with potential applications in both theoretical and applied settings.


Date: March 5, 2025 
Time: 12:00
Location: Faculty of Applied mathematics and cybernetics, Baku State University, Educational building no.3, 3rd floor room 317

Graph Entropy, Degree Assortativity, and Hierarchical Structures in Networks

Türker Bıyıkoğlu, MRL

Abstract:

In this talk, we connect several notions relating the structural and dynamical properties of a graph. 

Among them are the topological entropy coming from a certain dynamical system, which is related to the spectral radius of the graph’s adjacency matrix, the Randi\'{c} index from chemistry, and the degree assortativity of real networks. 
We will see that a hierarchical structure from computer science so-called breadth-first search ordering with decreasing degrees characterized the maximization of these notions.
 
This is a joint work with Fatihcan Atay, Bilkent University, Ankara.
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