Algebraic approach to several families of chemical graphs : doctoral thesis

english

doktorska dela

2016

benzenoids   chemistry   coronoids   generalised altan   graph theory   iterated altan   Kekulé structure   map trace   mathematics   patches   Pauling bond order   pentagonal incidence partition   perforated patches

Univerza v Ljubljani, Fakulteta za matematiko in fiziko
(Obvezni izvod spletne publikacije)

This work is an attempt to establish a stronger link between mathematics and chemistry and also to introduce discrete structures, e. g. maps, into the field of mathematical chemistry. We present the Hückel Molecular-Orbital theory and focus our attention to the notion of free valence. It is assumed in the literature that the maximum ?$\pi$? bond number (i.e., the total ?$\pi$? bond order around a ?$sp^2$? carbon atom) that can be theoretically obtained (on any centre in any ?$sp^2$? ?$\pi$? system) is no larger than ?$\sqrt{3}$?. This statement does not appear to have been formally proved. We obtained some partial results. We also provide empirical evidence on the behaviour of maximum ?$\pi$? bond number as a function of vertex count, ?$n$?, of chemical graphs and describe the family of graphs that realises local maxima for small ?$n$?. In 2013, a group of scientists led by Roman Jerala successfully designed a self-assembled tetrahedral polypeptide. We describe a suitable mathematical model for self-assembly of polypeptide structures. We also provide a dynamic programming algorithm for enumeration of strong traces, i. e., double traces of a graph that have additional properties. In 2012 the interesting family of convex benzenoids was introduced by Cruz et al. We present several equivalent definitions of convex benzenoids and explain some of their properties. In OEIS the sequence A116513 by A. C. Wechsler represents their enumeration. S. Reynolds enumerated and listed them all up to 250 hexagons. Our study independently verifies their enumeration. Furthermore, we stratify their generation into what we call the fundamental families of convex benzenoids. We provide an algorithm which extends the table up to ?$10^6$? hexagons. In this work we also revisit coronoids, in particular multiple coronoids. We consider a mathematical formalisation of the theory of coronoid hydrocarbons that is solely based on incidence between hexagons of the infinite hexagonal grid in the plane. We also consider perforated patches, which generalise coronoids: in addition to the hexagons of any benzenoid, other polygons may also be present. Just as coronoids may be considered as benzenoids with holes, perforated patches are patches with holes. Both cases, coronoids and perforated patches, admit a generalisation of the altan operation that can be performed at several holes simultaneously. A formula for the number of Kekulé structures of a generalised altan can be derived easily if the number of Kekulé structures is known for the original graph. Pauling bond brders for generalised altans are also easy to derive from those of the original graph.

12.05.2023