Whole benzenoid systems and relation between the Zhang-Zhang polynomial and the cube polynomial

Magistrska naloga obravnava celotne benzenoidne sisteme in njihove resonančne grafe. Izraz \u27\u27celotni benzenoidni sistem\u27\u27 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).Master thesis focuses on whole benzenoid systems and their resonance graphs. We use a term \u27\u27whole benzenoid system\u27\u27 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)