doktorska disertacija
Martina Berlič (Author), Petra Žigert Pleteršek (Mentor)

Abstract

Lucasove kocke so bile vpeljane kot nov model komunikacijskega omrežja. Množica vozlišč Lucasove kocke %n je množica vseh binarnih nizov dolžine n brez zaporednih enic ter enice na prvem in zadnjem mestu. Dve vozlišči Lucasove kocke sta sosedni, če se razlikujeta na natanko enem mestu. Ogljikove nanocevke so odkrili pred dvajsetimi leti in imajo zelo zanimivo kemijsko strukturo in lastnosti. Predstavili bomo izvirne rezultate o resonančnih grafih odprtih, enoslojnih ogljikovih nanocevk. Resonančni graf aromatskih ogljikovodikov odraža strukturo njegovih 1-faktorjev, oziroma modelira interakcijo med vsemi obstoječimi Kekuléjevimi strukturami ustrezne kemijske molekule. Najprej se omejimo na nanocevke, imenovane ciklični polifenantreni in jihove resonančne grafe. Nato rezultat razširimo in vpeljemo tako imenovane ciklične fibonacene. Izkaže se, da so pripadajoči resonančni grafi izomorfni Lucasovim kockam (skupaj z izoliranima vozliščema v sodem primeru). Slednje prinese nov rezultat o bijektivnem odnosu med maksimalnimi resonantnimi množicami cikličnega fibonacena in maksimalnimi hiperkockami njegovega resonančnega grafa, ki omogoča vpogled v strukturo resonančnih grafov cikličnih fibonacenov in s tem v strukturo Lucasove kocke. Nazadnje se posvetimo ogljikovim nanocevkam imenovanim ciklični polipireni in pojasnimo strukturo njihovih resonančnih grafov; to je unija amalgama dveh Lucasovih kock s kartezičnim produktom n kopij P3 in izoliranim vozliščem.

Keywords

ogljikove nanocevke;1-faktor;Kekuléjeva struktura;resonančni grafi;Z-transformirani graf;resonantne množice;Lucasova kocka;disertacije;

Data

Language: Slovenian
Year of publishing:
Typology: 2.08 - Doctoral Dissertation
Organization: UM FNM - Faculty of Natural Sciences and Mathematics
Publisher: M. Berlič]
UDC: 519.17:66.017-022.532(043.3)
COBISS: 269330176 Link will open in a new window
Views: 1605
Downloads: 132
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Other data

Secondary language: English
Secondary title: The structure of resonance graphs of some carbon nanotubes
Secondary abstract: Lucas cubes were introduced as model for interconnection networks. The vertex set of a Lucas cube n is the set of all binary strings of length n without consecutive 1's and 1 in the first and the last bit. Two vertices of the Lucas cube are adjacent if their strings differ in exactly one bit. Carbon nanotubes were discovered 20 years ago and have unusual chemical structure and properties. We introduce new results on the resonance graph of open-ended single-walled nanotubes. The resonance graph of an aromatic hydrocarbon reflects the structure of its perfect matchings or we can say that it models the interaction between Kekulé structures of the corresponding chemical molecule. First we restrict our attention to carbon nanotubes, called cyclic polyphenanthrenes and their resonance graphs. Next we extend result and introduce so called cyclic fibonacenes and it turns out that their resonance graphs are isomorphic to Lucas cubes (together with two isolated vertices in the even case). This gives new result about the one-to-one correspondence betwen maximal resonant set of a cyclic fibonacene and the maximal hypercubes of its resonance graph, which enables the insight into the structure of the resonance graphs of cyclic fibonaccene and therefore into the structure of Lucas cubes. Finally we focus on carbon nanotubes called cyclic polypyrenes and explains the structure of its resonance graph; it is the union of the amalgam of two Lucas cubes together with the cartesian product of n copies P3 and one isolated vertex.
Secondary keywords: carbon nanotube;1-factor;Kekule structuré;resonance graph;Z-transformation graph;resonant set;Lucas cube;dissetations;Nanocevke;Disertacije;Grafi;
URN: URN:SI:UM:
Type (COBISS): Doctoral dissertation
Thesis comment: Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za računalništvo in matematiko
Pages: XI, 87 str.
ID: 8722329