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Title:Strukturne lastnosti resonančnih grafov tubulenov in fulerenov
Authors:Tratnik, Niko (Author)
Žigert Pleteršek, Petra (Mentor) More about this mentor... New window
Files:.pdf DOK_Tratnik_Niko_2017.pdf (1,40 MB)
 
Language:Slovenian
Work type:Doctoral dissertation (mb31)
Typology:2.08 - Doctoral Dissertation
Organization:FNM - Faculty of Natural Sciences and Mathematics
Abstract:Doktorska disertacija obravnava predvsem resonančne grafe tubulenov in fulerenov. V prvem poglavju so predstavljeni nekateri že znani rezultati o resonančnih grafih, prav tako pa je podana struktura doktorske disertacije. V naslednjem poglavju so definirani nekateri osnovni pojmi teorije grafov, ki jih potrebujemo v preostalih poglavjih. V tretjem poglavju so predstavljene tri pomembne družine kemijskih struktur, to so benzenoidni sistemi, tubuleni in fulereni. Omenjene družine predstavljajo molekule, ki jih imenujemo benzenoidni ogljikovodiki, ogljikove nanocevke in fulereni. V četrtem poglavju je najprej pokazana povezava med Kekuléjevimi strukturami določene molekule ter popolnimi prirejanji ustreznega kemijskega grafa. V nadaljevanju poglavja je definiran resonančni graf benzenoidnega sistema, tubulena in fulerena. Glavni namen tega koncepta je modeliranje interakcij med posameznimi Kekuléjevimi strukturami molekule. Nato se lotimo raziskovanja osnovnih lastnosti resonančnih grafov. Pokazano je, da je resonančni graf tubulena ali fulerena dvodelni graf, vsaka njegova povezana komponenta pa je bodisi pot bodisi graf z ožino štiri. Prav tako dokažemo, da je 2-jedro vsake povezane komponente resonančnega grafa širokega tubulena ali fulerena, ki ni pot, vedno 2-povezan graf. Nato podamo primer neskončne družine tubulenov, katerih resonančni grafi niso povezani. Na koncu poglavja definiramo resonančni graf za katerikoli graf, ki je vložen na zaprto ploskev. Dokažemo tudi, da so taki resonančni grafi inducirani podgrafi hiperkock. V petem poglavju definiramo Zhang-Zhangov polinom, ki je namenjen štetju posebnih struktur, imenovanih Clarova pokritja. Dokazano je, da je Zhang-Zhangov polinom grafa, vloženega na zaprto ploskev, enak polinomu kock ustreznega resonančnega grafa. Ta rezultat posplošuje podobne rezultate za benzenoidne sisteme, tubulene in fulerene. Na koncu se ukvarjamo s strukturo distributivne mreže resonančnih grafov. Dokazano je, da je vsaka povezana komponenta resonančnega grafa tubulena graf pokritja neke distributivne mreže. Prav tako pokažemo, da je vsaka povezana komponenta resonančnega grafa tubulena medianski graf, njen graf blokov pa je pot. Nazadnje podamo primer fulerena, katerega resonančni graf ni graf pokritja nobene distributivne mreže.
Keywords:benzenoidni sistem, ogljikova nanocevka, tubulen, fuleren, resonančni graf, Z-transformirani graf, Clarovo pokritje, Zhang-Zhangov polinom, polinom kock, distributivna mreža, medianski graf, graf blokov, grafi na ploskvah
Year of publishing:2017
Publisher:N. Tratnik]
Source:[Maribor
UDC:519.17:54(043.3)
COBISS_ID:23571720 Link is opened in a new window
NUK URN:URN:SI:UM:DK:7YDVFP7F
Views:535
Downloads:114
Metadata:XML RDF-CHPDL DC-XML DC-RDF
Categories:FNM
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Secondary language

Language:English
Title:Structural properties of resonance graphs of tubulenes and fullerenes
Abstract:The doctoral dissertation focuses mostly on the resonance graphs of tubulenes and fullerenes. In the first chapter, already known results about the resonance graphs are mentioned and the structure of the dissertation is described. In the next chapter, the basic definitions from graph theory, which are needed in the following chapters, are presented. The third chapter introduces three important families of chemical structures, these are benzenoid systems, tubulenes, and fullerenes. The mentioned families represent chemical compounds known as benzenoid hydrocarbons, carbon nanotubes, and fullerenes. In the fourth chapter, the relationship between a Kekulé structure of a chemical compound and a perfect matching of the corresponding chemical graph is described. Next, the resonance graph of a benzenoid system, tubulene, or fullerene is defined. The main goal of this concept is to model interactions among the Kekulé structures of a molecule. Then, the basic properties of resonance graphs are investigated. It is shown that the resonance graph of a tubulene or a fullerene is bipartite and every connected component of the resonance graph is either a path or a graph of girth four. Moreover, it is shown that the 2-core of every connected component of the resonance graph of a thick tubulene or a fullerene, which is not a path, is 2-connected. Furthermore, an infinite family of tubulenes with a disconnected resonance graph is given. Finally, the concept of the resonance graph is defined for any graph embedded on a closed surface. It is shown that such resonance graphs are induced subgraphs of hypercubes. The fifth chapter introduces the Zhang-Zhang polynomial, whose purpose is to count Clar covers in a chemical graph. It is shown that the Zhang-Zhang polynomial of a graph embedded on a closed surface is equal to the cube polynomial of the corresponding resonance graph. This result generalizes the corresponding results for benzenoid systems, tubulenes, and fullerenes. Finally, the distributive lattice structure of resonance graphs is considered. It is proved that every connected component of the resonance graph of a tubulene is the covering graph of some distributive lattice. Using this result, we also show that every connected component of the resonance graph of a tubulene is a median graph and that its block graph is a path. Moreover, an example of a fullerene is given such that its resonance graph is not the covering graph of some distributive lattice.
Keywords:benzenoid system, carbon nanotube, tubulene, fullerene, resonance graph, Z-transformation graph, Clar cover, Zhang-Zhang polynomial, cube polynomial, distributive lattice, median graph, block graph, graphs on surfaces


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