Algebraic approach to several families of chemical graphs

doctoral thesis

Nino Bašić (Author), Tomaž Pisanski (Mentor), Patrick W. Fowler (Co-mentor)

Abstract

To delo je poskus vzpostavitve krepkejše povezave med matematiko in kemijo. Polega tega poskušamo diskretne strukture, kot so npr. zemljevidi, uveljaviti v matematični kemiji. Najprej predstavimo Hücklovo teorijo molekulskih orbital, posebno pozornost pa namenimo konceptu proste valence. V literaturi je pogosto predpostavljeno, da je največje ▫$\pi$▫ vezno število (tj. skupna vsota redov ▫$\pi$▫ vezi, ki izhajajo iz nekega ▫$sp^2$▫ ogljikovega atoma), ki ga je mogoče teoretično doseči (pri poljubnem atomu v poljubnem ▫$sp^2$▫ ▫$\pi$▫ sistemu), največ ▫$\sqrt{3}$▫. Vendar vse kaže, da ni bila ta domneva nikoli formalno dokazana. V tem delu smo uspeli dobiti nekatere delne rezultate. Poleg tega postrežemo z izračuni obnašanja ▫$\pi$▫ veznega števila kot funkcije števila atomov ▫$n$▫ v družini kemijskih grafov in opišemo družino grafov, ki dosežejo lokalne maksimume za manjše vrednosti parametra ▫$n$▫. Leta 2013 je skupina znanstvenikov pod vodstvom Romana Jerale uspešno izdelala samosestavljiv polipeptid, ki se je zložil v tetraeder. Najprej podamo matematični model, ki je primeren za opis samosestavljanja. Nato predstavimo algoritem, ki s pomočjo dinamičnega programiranja našteje krepke obhode, tj. dvojne obhode, ki imajo še neke dodatne lastnosti. Leta 2012 so Cruz in sodelavci uvedli zanimivo družino konveksnih benzenoidov. V tem delu predstavimo več ekvivalentnih definicij konveksnih benzenoidov in nekatere njihove lastnosti. V enciklopediji OEIS zaporedje z oznako A116513, ki ga je definiral A. C. Wechsler, predstavlja njihovo enumeracijo. S. Reynolds je preštel in poiskal vse primerke, ki imajo največ 250 šestkotnikov. Naša študija neodvisno potrdi pravilnost njihove enumeracije. Konveksne benzenoide razdelimo v tako imenovane fundamentalne družine, generiranje pa opravimo v vsaki družini posebej. S takšnim pristopom z lahkoto preštejemo vse konveksne benzenoide, ki imajo do ▫$10^6$▫ šestkotnikov. V tem delu se posvetimo tudi koronoidom, še posebej večkratnim koronoidom. Predstavimo matematično formalizacijo teorije koronoidnih ogljikovodikov, ki temelji zgolj na sosednosti med šestkotniki neskončne šestkotniške mreže v ravnini. Nekaj pozornosti namenimo še naluknjanim obližem, ki posplošijo koronoide. Poleg šestkotniških smejo imeti tudi lica drugačnih dolžin. Tako kot lahko koronoide obravnavamo kot benzenoide z luknjami, lahko tudi naluknjane obliže obravnavamo kot obliže z luknjami. Na enih in na drugih lahko naredimo posplošeno operacijo altan, ki poteka na več luknjah hkrati. Izpeljemo formulo, ki prešteje Kekulejeve strukture posplošenega altana, če je število Kekulejevih struktur originalnega grafa že znano. Tudi Paulingov red vezi lahko enostavno izračunamo za altan, če že od prej poznamo njihove vrednosti v osnovnem grafu.

Keywords

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

Data

Language:	English
Year of publishing:	2016
Source:	Ljubljana
Typology:	2.08 - Doctoral Dissertation
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[N. Bašić]
UDC:	519.17(043.3)
COBISS:	17740889
Views:	1130
Downloads:	563
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Algebrajski pristop k obravnavi nekaterih družin kemijskih grafov
Secondary abstract:	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.
Secondary keywords:	matematika;teorija grafov;kemija;altan;posplošeni altan;iterirani altan;benzenoidi;koronoidi;obliži;naluknjani obliži;Kekulejeva struktura;Paulingov red vezi;petkotniška incidenčna particija;obhod zemljevida;Teorija grafov;Disertacije;
Type (COBISS):	Doctoral dissertation
Thesis comment:	Univ. Ljubljana, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 3. stopnja
Pages:	VIII, 261 str.
ID:	10910524