Seminar za biomatematiko in matematično kemijo - Arhiv

Došlić et al. defined the Mostar index of a graph G as ∑_{uv ∈ E(G)} | n_G(u, v) - n_G(v, u) |, where, for an edge uv of G, the term n_G(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.

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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the original House of Graphs website started in 2010, the underlying frameworks and technologies of the website became outdated.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.

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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.

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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.

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