Talk: History of Mathematical Chemistry

Abstract: The lecture offers a review of the history of what nowadays is called Mathematical Chemistry. We start with Couper and Kekulé who conceived structural formulas, and Cayley, Sylvester, and Clifford who recognized their graph-theoretical nature. Three major directions of research - isomer enumeration, applications of graph spectra, and topological indices - are considered. The most important personalities in this field of science are mentioned and commented.

Talk: Graphs with many eigenvalues close to their largest eigenvalues

Abstract: Given t>0 and a small ε>0, we discuss families of connected n-vertex graphs with bounded degrees which have positive proportion of their eigenvalues in an interval [t-ε, t] and only a bounded number of eigenvalues greater than t. Special cases when t=λ1(G) or t=λ2(G) are of particular interest.

Talk: On certain singular graphs of nullity one

Abstract: Singular graphs of nullity one admitting a full kernel eigenvector are called nut graphs. We present a gentle introduction to the class of nut graphs. We are mostly interested in the existence problem of various subfamilies of nut graphs. For instance, chemical nut graphs provide a prime example of such a family. We also show how the concept of nut graphs can be carried over to signed graphs. In this overview, we present some methods that allow construction of larger nut graphs from smaller ones. In particular, the powerful Fowler construction is explained. The talk will mostly cover topics, recently considered by the author and his co-workers. For a more comprehensive survey on nut graphs, in particular in connection with mathematical chemistry, see the recent monograph: I. Sciriha, I. Farrugia, From Nut Graphs to Molecular Structure and Conductivity, Mathematical Chemistry Monographs, University of Kragujevac, 2021. The author would like to thank his collaborators, in particular Irene Sciriha for introducing him to this exciting research topic. Special thanks to Nino Bašić for delivering this talk.

Talk: Wiener index versus eccentricity, and Wiener complexity versus eccentric complexity

Abstract: Let G be a connected graph. The eccentricity of a vertex v of G is the distance from v to a farthest vertex from v, and the eccentricity of G is the sum of the eccentricities of its vertices. In the first part of the talk, the Wiener index of G will be compared with the eccentricity of G. The Wiener complexity of G is the number of different total distances (alias transmissions) of its vertices, and the eccentric complexity of G is the number of different eccentricities of its vertices. In the second part of the talk, the Wiener complexity of G will be compared with the eccentric complexity of G.

Talk: Some problems and results on Wiener index and related graph parameters

Abstract: In the talk I will survey some results and open problems on several topics of the Wiener index that include: 1) Determining its minimum value in some classes of graphs, 2) Wiener index of directed graphs, 3) Šoltes problem, etc.

Talk: Representability of reaction networks

Abstract: Reaction networks comprise a set X of species and a set R of reactions Y to Y', each converting a multiset of educts Y⊆ X into a multiset Y' ⊆ X of products. Reaction networks therefore are equivalent to directed (multi)-hypergraphs. Not all such hypergraphs, however, admit a chemical interpretation because that may contradict fundamental principles of physics: "Futile cycles" that have no net mass conversion violate conservation of energy, and networks that allow composite reaction that describe net generation or destruction of mass obviously violate conservation of mass and atom types. The existence of a positive left kernel vector for the stoichiometric matrix and the absence of irreversible futile cycle are well known sufficient conditions. Here we show that these conditions are also necessary. More interestingly, they are also necessary and sufficient for the existence of sum formula representations that preserve atom types and for the existence of structural formular representations conforming to Lewis formulas. In the latter case, all chemical reactions in the network have an interpretation as rearrangements of bonds/electron pairs. Interestingly, therefore, molecular with separated charges or radicals do not extend the range of possible stoichometries, i.e., chemically feasible hypergraphs. The results suggest many directions for future work, in particular with regard to

the generation of "random" chemical networks.