Seminar for biomathematics and mathematical chemistry Archives - UP Famnit

Ivan Damnjanović: Reinforcement learning meets graph theory: A new framework

mario.krajoski — Wed, 01 Apr 2026 22:00:00 +0000

2026-03-26 18:00 – 19:00 Online via Zoom (See link below) Ivan Damnjanović (FAMNIT, University of Primorska, Koper, Slovenia, and Faculty of Electronic Engineering, University of Niš, Niš, Serbia) Reinforcement learning meets graph theory: A new framework Reinforcement learning (RL) is a branch of machine learning concerned with designing models that can learn effective decision-making strategies through interaction and experience. In a recent influential paper, Wagner showed that the Deep Cross-Entropy method from RL can be used to address problems in extremal graph theory by reformulating them as combinatorial optimization tasks. This idea sparked growing interest in the community, leading to a range of refinements and extensions of Wagner’s original framework, as well as the development of RL environments specialized for graph theory. As a result, several problems in extremal graph theory have already been successfully approached using RL techniques. In this work, we introduce Reinforcement Learning for Graph Theory (RLGT), a new framework that unifies and organizes these previous efforts. RLGT is designed to handle both undirected and directed graphs, optionally allowing loops and supporting an arbitrary number of edge colors. The framework provides efficient graph representations and is intended to support future research in RL-based extremal graph theory through improved computational efficiency and a modular architecture. (This is a joint work with Uroš Milivojević, Irena Ðorđević and Dragan Stevanović.) Join Zoom Meeting HERE!

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Ivan Damnjanović: On the maximum spectral radius of connected graphs with a prescribed order and size

mario.krajoski — Wed, 07 May 2025 22:00:00 +0000

2025-05-08 18:00 Online via Zoom (See link below) Ivan Damnjanović (FAMNIT, University of Primorska, Koper, Slovenia, and Faculty of Electronic Engineering, University of Niš, Niš, Serbia) On the maximum spectral radius of connected graphs with a prescribed order and size The spectral radius, or index, of a graph is the largest eigenvalue of its (0,1)-adjacency matrix. Let Cn,e be the set comprising all the connected simple graphs of order n and size n − 1 + e. We investigate the spectral radius maximization problem on Cn,e and provide the solution for the case when e ∈ {0, 1, 2, …, 130} or n ≥ e + 2 + 13√e. Join Zoom Meeting HERE!

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Niko Tratnik: Resonance graphs of plane bipartite graphs as daisy cubes

mario.krajoski — Thu, 10 Oct 2024 22:00:00 +0000

2024-10-17 16:00 Online via Zoom (See link below) Niko Tratnik, University of Maribor, Slovenia Resonance graphs of plane bipartite graphs as daisy cubes We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we show that if G is a plane elementary bipartite graph other than K2, then the resonance graph of G is a daisy cube if and only if the Fries number of G equals the number of finite faces of G. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph G is a daisy cube if and only if G is weakly elementary bipartite such that each of its elementary component Gi other than K2 holds the property that the Fries number of Gi equals the number of finite faces of Gi. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes. Join Zoom Meeting HERE!

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Ivan Damnjanović: On transmission irregular graphs — starlike and double starlike trees and long pendent paths

mario.krajoski — Thu, 10 Oct 2024 22:00:00 +0000

2023-11-30 18:00 Online via Zoom (See link below) Ivan Damnjanović, University of Niš, Serbia, and Diffine LLC, USA, and University of Primorska, Slovenia On transmission irregular graphs — starlike and double starlike trees and long pendent paths The transmission of a vertex in a connected graph is the sum of its distances to all the other vertices. A graph is transmission irregular (TI) when all of its vertices have mutually distinct transmissions. In an earlier paper, Al-Yakoob and Stevanović [Appl. Math. Comput. 380 (2020), 125257] gave the full characterization of TI starlike trees with three branches. Here, we improve these results by using a different approach to provide the complete characterization of all TI starlike trees and all TI double starlike trees. We subsequently implement the aforementioned conditions in order to find several infinite families of TI starlike trees and TI double starlike trees. Besides that, we disclose five families of unicyclic graphs with two pendent paths whose members are TI under certain conditions. As a direct consequence, we demonstrate the existence of TI chemical graphs of almost all even orders, thereby resolving a problem recently posed by Xu, Tian and Klavžar [Discrete Appl. Math. 340 (2023), 286–295]. Join Zoom Meeting HERE!

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Dragan Stevanović: Reinforcement learning on graphs: cross-entropy methods and basic graph environments

mario.krajoski — Wed, 25 Oct 2023 22:00:00 +0000

2023-11-02 18:00 ZOOM (See link below) Dragan Stevanović, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia Reinforcement learning on graphs: cross-entropy methods and basic graph environments Adam Zsolt Wagner [arXiv:2104.14516] recently showed how reinforcement learning (RL) can be applied to construct (counter)examples in graph theory. We will showcase here a more readable, more stable and significantly faster reimplementation of his approach, and illustrate its work by finding counterexamples to a few published conjectures. We will also shortly discuss ways to implement several new RL environments that will cover constructions of simple graphs and trees, their signed variants, and graphs with bounded maximum vertex degree. Join Zoom Meeting HERE!

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Štefko Miklavič: Bounding the Mostar index

mario.krajoski — Wed, 07 Dec 2022 23:00:00 +0000

2022-12-15 18:00 ZOOM (See link below) Štefko Miklavič, University of Primorska, Slovenia Bounding the Mostar index Došlić et al. defined the Mostar index of a graph G as ∑uv ∈ E(G) | nG(u, v) – nG(v, u) |, where, for an edge uv of G, the term nG(u, v) denotes the number of vertices of G that are closer to u than to v. They also conjectured that Mostar index of G, Mo(G), is less or equal to 0.148 n3. In this talk we show that Mo(G) ≤ 0.1633 n3. If, however, G is bipartite, then we show that Mo(G) ≤ √3/18 n3, and that this bound is best possible up to terms of order O(n2). This is joint work with Johannes Pardey, Dieter Rautenbach and Florian Werner from Ulm University. Join Zoom Meeting HERE!

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Jan Goedgebeur: House of Graphs 2.0: a database of interesting graphs and more

mario.krajoski — Wed, 16 Nov 2022 23:00:00 +0000

2022-11-24 18:00 ZOOM (See link below) Jan Goedgebeur, KU Leuven, Belgium House of Graphs 2.0: a database of interesting graphs and more In 2012 we announced the House of Graphs (https://houseofgraphs.org/), which was a new database of graphs. The House of Graphs hosts complete lists of graphs of various graph classes, but its main feature is a searchable database of so called "interesting" graphs, which includes graphs that already occurred as extremal graphs or as counterexamples to conjectures. An important aspect of this database is that it can be extended by users of the website. Over the years, several new features and graph invariants were added to the House of Graphs and users uploaded many interesting graphs to the website. But as the development of the original House of Graphs website started in 2010, the underlying frameworks and technologies of the website became outdated. This is why we completely rebuilt the House of Graphs using modern frameworks to build a maintainable and expandable web application that is future-proof. On top of this, several new functionalities were added to improve the application and the user experience. In this talk we will present the House of graphs and highlight the changes and new features of the new website. We will also demonstrate how users can perform queries on this database and how they can add new interesting graphs to it. Join Zoom Meeting HERE!

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Guillermo Restrepo: Chemical space: a meeting point for chemists and mathematicians

mario.krajoski — Wed, 09 Nov 2022 23:00:00 +0000

2022-11-17 18:00 ZOOM (See link below) Guillermo Restrepo, Max Planck Institute for Mathematics in the Sciences and Leipzig University, Germany Chemical space: a meeting point for chemists and mathematicians Chemical space is defined as the set of reported substances at a given time. It actually constitutes a space once the set is endowed with a notion of nearness. There are at least two options for such a nearnees: substance's resemblance and chemical reachability. The former is the basis of approaches boiling down to the concept of molecular similarity. The latter is related to chemical reaction networks. I will analyse in this talk the upper and lower bounds of the chemical space, the required memory space to store it and the possibilities for similarity studies in it. I will also discuss the central role of directed hypergraphs for modelling the space, as well as the bounds for the number of hypergraphs. Finally, I will discuss some features of the evolution of the chemical space and its implications for chemistry, as well as the opportunities for mathematics. Join Zoom Meeting HERE!

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Tomaž Pisanski: More questions than answers about nut graphs

mario.krajoski — Sun, 15 May 2022 22:00:00 +0000

2022-05-19 18:00 – 19:00 ZOOM (See link below) Tomaž Pisanski, University of Primorska, Slovenia More questions than answers about nut graphs Singular graphs of nullity one admitting a full kernel eigenvector are called nut graphs. Nut graphs constitute a fascinating family of graphs that has found many applications, in particular in mathematical chemistry. It has been promoted mostly by Irene Sciriha and her co-authors. Sciriha almost single-handedly laid down the fundamental theory of nut graphs. The emphasis of this talk is in the entries of the standard kernel eigenvector associated with a nut graph. Namely, the standard kernel eigenvector having the property that its integer entries have no non-trivial common factor is unique up to multiplication by -1. In this talk we present some questions about nut graphs that have attracted researchers of this topic. We recall some tools that enable one to construct larger nut graphs from smaller ones. They explain some answers to these questions. We also ask some questions that may be of interest to some students. This is a preparatory talk for another talk that will be delivered in Sombor, Serbia, in June 2022. Join Zoom Meeting HERE!

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Niko Tratnik: Generalized Cut Method Applied to Some Distance-Based Topological Indices

mario.krajoski — Wed, 20 Oct 2021 22:00:00 +0000

2021-11-04 18:00 – 19:00 ZOOM (See link below) Niko Tratnik, University of Maribor, Slovenia Generalized Cut Method Applied to Some Distance-Based Topological Indices The cut method has an important role in the investigation of molecular descriptors. Very often it was applied to benzenoid systems to efficiently compute distance-based topological indices, for example the Wiener index and the Szeged index. Later, the cut method was generalized such that it can be used on any connected graph by using Djoković-Winkler relation. In this talk, we present the cut method for the edge version of the Wiener index and for an infinite family of Szeged-like topological indices. In the first part, we focus on the edge-Wiener index, which is for any connected graph G defined as the Wiener index of the line graph of G. We show that the edge-Wiener index of an edge-weighted graph can be computed in terms of the three Wiener indices of weighted quotient graphs. Thus, already known analogous methods for computing the edge-Wiener index of benzenoid systems and phenylenes are generalized. In the second part, we formally introduce the concept of a general Szeged-like topological index, which includes many well known topological indices (for example Szeged, PI and Mostar indices) and also infinitely many other topological indices that can be defined in a similar way. As the main result, we provide a cut method for computing a general Szeged-like topological index for any connected strength-weighted graph. This greatly generalizes various methods known for some of the mentioned indices. Join Zoom Meeting HERE!

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Dragan Stevanović: On Hosoya’s dormants and sprouts

mario.krajoski — Mon, 11 Oct 2021 22:00:00 +0000

2021-10-14 18:00 – 19:00 ZOOM (See link below) Dragan Stevanović, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia On Hosoya’s dormants and sprouts The study of cospectral graphs is one of the traditional topics of spectral graph theory. Initial expectation by theoretical chemists Günthard and Primas in 1956 that molecular graphs are characterized by the multiset of eigenvalues of the adjacency matrix was quickly refuted by the existence of numerous examples of cospectral graphs in both chemical and mathematical literature. This work was further motivated by Fisher in 1966 in the influential study that investigated whether one can “hear” the shape of a (discrete) drum, which has led over the years to the construction of many cospectral graphs. These findings culminated in setting the ground for the Godsil-McKay local switching and the Schwenk’s use of coalescences, both of which were used to show (around the 1980s) that almost all trees have cospectral mates. Recently, enumerations of cospectral graphs with up to 12 vertices by Haemers and Spence and by Brouwer and Spence have led to the conjecture that, on the contrary, “almost all graphs are likely to be determined by their spectrum”. This conjecture paved the way for myriad of results showing that various special types of graphs are determined by their spectra. On the other hand, in a recent series of papers, Hosoya drew the attention to a particular aspect of constructing cospectral graphs by using coalescences: that cospectral graphs can be constructed by attaching multiple copies of a rooted graph in different ways to subsets of vertices of an underlying graph. After briefly surveying the history of constructing cospectral graphs, we focus on the expectations and questions about cospectrality of multiple coalescences that were raised in Hosoya's papers. In particular, we discuss the characteristic polynomial of such multiple coalescences, from which a necessary and sufficient condition for their cospectrality follows. We enumerate such pairs of cospectral multiple coalescences for a few families of underlying graphs, and show the infinitude of cospectral multiple coalescences having paths as underlying graphs, which were deemed rare by Hosoya. Join Zoom Meeting HERE!

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On circulant nut graphs

mario.krajoski — Tue, 15 Jun 2021 22:00:00 +0000

2021-06-17 18:00 – 19:00 ZOOM (See link below) Dragan Stevanović, Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia On circulant nut graphs A nut graph is a simple graph whose adjacency matrix has the eigenvalue 0 with multiplicity 1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al. [Discuss. Math. Graph Theory 40 (2020), 533–557] to determine the pairs (n, d) for which a vertex-transitive nut graph of order n and degree d exists, Bašić et al. [arXiv:2102.04418, 2021] initiated the study of circulant nut graphs. Here we first show that the generator set of a circulant nut graph necessarily contains equally many even and odd integers. Then we characterize circulant nut graphs with the generator set {x, x + 1, …, x + 2t − 1} for x, t ∈ ℕ, which generalizes the result of Bašić et al. for the generator set {1, …, 2t}. We further study circulant nut graphs with the generator set {1, …, 2t + 1} {t}, which yields nut graphs of every even order n ≥ 4t + 4 whenever t is odd such that t ≠ 1 (mod 10) and t ≠ 15 (mod 18). This fully resolves Conjecture 9 from Bašić et al. [ibid.]. We also study the existence of 4t-regular circulant nut graphs for small values of t, which partially resolves Conjecture 10 of Bašić et al. [ibid.]. While the original question is stated in terms of (spectral) graph theory, the talk will very quickly move from the setting of graph eigenvalues and eigenvectors to polynomial algebra, with most of the obtained results based on the properties of cyclotomic polynomials. This is a joint work with Ivan Damnjanović. Join Zoom Meeting HERE!

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