20.500.12556/DKUM-42693 Resonančni grafi nekaterih nanocevk in njihova struktura The structure of resonance graphs of some carbon nanotubes Lucasove kocke so bile vpeljane kot nov model komunikacijskega omrežja. Množica vozlišč Lucasove kocke Λn 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. 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. ogljikova nanocevka 1-faktor Kekuléjeva struktura resonančni graf Z-transformirani graf resonantna množica Lucasova kocka. carbon nanotube 1-factor Kekulé structure resonance graph Z-transformation graph resonant set Lucas cube. true true false M. Berlič] Slovenski jezik Angleški jezik Doktorsko delo/naloga 2013-10-07 09:15:26 2013-10-17 12:28:53 2022-05-08 03:05:24 2021-04-14 03:42:55 2013 0 [Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17:66.017-022.532(043.3) 269330176 URN:SI:UM:DK:EY01ZU14 DR_Berlic_Martina_2013.pdf DR_Berlic_Martina_2013.pdf 1 CBBFEA71A81115E03D3CB4F3EA61A1B9 25fc65d46d1042762b83ffe66ee9075df804d8fd10e440fd98836a2d78cd75a8 d2e82f68-7c07-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=59933 Fakulteta za naravoslovje in matematiko 0 0 0