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Title:Celotni benzenoidni sistemi ter povezava med Zhang-Zhang-ovim polinomom in polinomom kock
Authors:Tratnik, Niko (Author)
Žigert Pleteršek, Petra (Mentor) More about this mentor... New window
Files:.pdf MAG_Tratnik_Niko_2014.pdf (3,05 MB)
 
Language:Slovenian
Work type:Master's thesis/paper (mb22)
Typology:2.09 - Master's Thesis
Organization:FNM - Faculty of Natural Sciences and Mathematics
Abstract:Magistrska naloga obravnava celotne benzenoidne sisteme in njihove resonančne grafe. Izraz ''celotni benzenoidni sistem'' uporabljamo kot skupno ime za benzenoidne sisteme in odprte ogljikove nanocevke. Benzenoidni sistemi so v kemijski teoriji grafov zanimivi za proučevanje, saj predstavljajo kemijske spojine, imenovane benzenoidni ogljikovodiki. Ogljikove nanocevke si lahko predstavljamo kot vložitev benzenoidnega sistema na plašč valja. Osnovni pogoj za kemijsko stabilnost benzenoidnega ogljikovodika je, da ima Kekuléjeve strukture, ki ponazarjajo dvojne vezi v benzenoidnem ogljikovodiku. Resonančni graf celotnega benzenoidnega sistema pa predstavlja interakcije med njegovimi Kekuléjevimi strukturami. V prvem delu je navedenih nekaj definicij in pomembnih rezultatov teorije grafov, ki jih potrebujemo v nadaljevanju. V drugem delu definiramo celotni benzenoidni sistem in pokažemo povezavo med Kekuléjevimi strukturami in popolnimi prirejanji celotnega benzenoidnega sistema. Definiciji resonančnega grafa in resonantne množice sta predstavljeni v tretjem delu. V zadnjem poglavju definiramo Zhang-Zhang-ov polinom (Clarov polinom) celotnega benzenoidnega sistema, ki šteje strukture, imenovane Clarova pokritja. Kot glavni rezultat dokažemo, da je Zhang-Zhang-ov polinom celotnega benzenoidnega sistema B enak polinomu kock njegovega resonančnega grafa R(B), tako da definiramo bijekcijo med Clarovimi pokritji celotnega benzenoidnega sistema B in hiperkockami v R(B).
Keywords:celotni benzenoidni sistem, popolno prirejanje, resonančni graf, resonantna množica, Clarovo pokritje, Zhang-Zhang-ov polinom, polinom kock.
Year of publishing:2014
Publisher:[N. Tratnik]
Source:Maribor
UDC:519.17:54(043.2)
COBISS_ID:20803848 Link is opened in a new window
NUK URN:URN:SI:UM:DK:6RLXXAXE
Views:1093
Downloads:125
Metadata:XML RDF-CHPDL DC-XML DC-RDF
Categories:FNM
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Secondary language

Language:English
Title:Whole benzenoid systems and relation between the Zhang-Zhang polynomial and the cube polynomial
Abstract:Master thesis focuses on whole benzenoid systems and their resonance graphs. We use a term ''whole benzenoid system'' for either a benzenoid system or a carbon nanotube (without caps). Benzenoid systems are investigated in chemical graph theory since they represent the chemical compounds known as benzenoid hydrocarbons. Carbon nanotubes can be seen as an embedding of a benzenoid system to a surface of a cylinder. A necessary condition for a benzenoid hydrocarbon to be chemically stable is that it possesses Kekulé structure, which describes double bonds in a benzenoid hydrocarbon. The resonance graph of a whole benzenoid system models interactions among its Kekulé structures. In the first part, we introduce some definitions and important results of graph theory which are needed in the following chapters. In the second part, we define a whole benzenoid system and show correspondence between Kekulé structures and perfect matchings of a whole benzenoid system. The concepts of a resonance graph and a resonant set are introduced in the third part. In the last chapter, we define the Zhang-Zhang polynomial (Clar covering polynomial) of a whole benzenoid system as a counting polynomial of resonant structures called Clar covers. As the main result, we prove that the Zhang-Zhang polynomial of a whole benzenoid system B coincides with the cube polynomial of its resonance graph R(B) by establishing a bijection between the Clar covers of B and the hypercubes in R(B).
Keywords:whole benzenoid system, perfect matching, resonance graph, resonant set, Clar cover, Zhang-Zhang polynomial, cube polynomial


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