doktorska disertacija
Simon Brezovnik (Author), Niko Tratnik (Mentor), Petra Žigert Pleteršek (Co-mentor)

Abstract

V doktorski disertaciji se najprej ukvarjamo z resonančnimi grafi katakondenziranih sodih obročnih sistemov (CERS-ov) in njihovo povezavo z marjetičnimi kockami. V nadaljevanju razvijemo posplošeno metodo prerezov, ki omogoča izračun različnih topoloških indeksov (Wienerjevega indeksa dvojno vozliščno-uteženega grafa, Schultzevega indeksa ter indeksov tipa Szeged). V uvodnem poglavju so predstavljeni nekateri že znani rezultati v povezavi z resonančnimi grafi in posplošeno metodo prerezov. Prav tako v nekaj stavkih napovemo rezultate, ki sledijo v nadaljevanju. V drugem poglavju zapišemo osnovne definicije, ki se dotikajo področja teorije grafov in so potrebne za razumevanje osrednjega dela. V tretjem poglavju predstavimo vse obravnavane kemijske strukture in grafe, ki modelirajo te strukture. Najprej obravnavamo benzenoidne sisteme, zatem opišemo CERS-e, fenilene in koronoide. V četrtem poglavju definiramo resonančni graf in pojasnimo povezavo med Kekuléjevimi strukturami in popolnimi prirejanji grafa. Nadalje zapišemo algoritem, ki omogoča iskanje resonančnega grafa poljubnega CERS-a, temelji pa na binarnem kodiranju njegovih popolnih prirejanj. Zatem se ukvarjamo tudi z raziskovanjem CERS-ov, ki imajo izomorfne resonančne grafe. Dobljene rezultate nato uporabimo na fenilenih in tako dobimo zvezo med njihovimi resonančnimi grafi in resonančnimi grafi katakondenziranih benzenoidnih grafov. Na koncu poglavja predstavimo definicijo marjetične kocke in karakteriziramo CERS-e, katerih resonančni grafi so marjetične kocke. V petem poglavju so predstavljeni topološki indeksi, ki temeljijo na razdaljah v grafu oziroma na stopnjah vozlišč. Nato predstavimo krepko utežene grafe in na njih definiramo indekse tipa Szeged. V zaključku poglavja predstavimo model, s katerim obravnavamo odvisnost med vrelišči alkenov in alkadienov ter povezavno-uteženimi Wienerjevimi indeksi. Pri tem izvedemo nelinearno regresijsko analizo. V šestem poglavju definiramo kvocientni graf poljubnega povezanega grafa. V nadaljevanju predstavimo posplošeno metodo prerezov in dokažemo, da lahko le-to uporabimo tudi za izračun Schultzevega in Gutmanovega indeksa. Rezultate uporabimo na fenilenih in nekaterih drugih grafovskih družinah. Na koncu šestega poglavja razvijemo posplošeno metodo prerezov za topološke indekse tipa Szeged in zapišemo formulo za izračun teh indeksov za poljuben krepko uteženi graf. Nazadnje ponudimo še nekaj zgledov uporabe izpeljane metode za različne molekularne grafe.

Keywords

disertacije;Djoković-Winklerjeva relacija;resonančni graf;benzenoidni sistem;fenilen;CERS;Kekuléjeva struktura;popolno prirejanje;marjetična kocka;kvocientni graf;topološki indeks;Wienerjev indeks;Gutmanov indeks;Schultzev indeks;topološki indeksi tipa Szeged;posplošena metoda prerezov;

Data

Language: Slovenian
Year of publishing:
Typology: 2.08 - Doctoral Dissertation
Organization: UM FNM - Faculty of Natural Sciences and Mathematics
Publisher: S. Brezovnik]
UDC: 519.17:54(043.3)
COBISS: 116474115 Link will open in a new window
Views: 49
Downloads: 11
Average score: 0 (0 votes)
Metadata: JSON JSON-RDF JSON-LD TURTLE N-TRIPLES XML RDFA MICRODATA DC-XML DC-RDF RDF

Other data

Secondary language: English
Secondary title: Resonance graphs of some bipartite outerplanar graphs and the generalized cut method
Secondary abstract: In the doctoral dissertation, we first consider resonance graphs of catacondensed even ring systems (CERS) and their relation to daisy cubes. Then, we develop a generalized cut method that enables the calculation of various topological indices (Wiener index of double vertex-weighted graph, Schultz index and Szeged-like topological indices). The introductory chapter presents some already known results related to resonance graphs and a generalized cut method. In the same chapter, we briefly announce the results that follow later. In the second chapter, we write down the basic definitions from the field of graph theory that are necessary for understanding the central part. In the third chapter, we present all the considered chemical structures and graphs that model these structures. First, benzenoid systems are considered and later, we describe CERS, phenylenes, and coronoids. In the fourth chapter, we define the resonance graph and explain the connection between Kekulé structures and perfect matchings. Next, an algorithm for constructing the resonance graph of any CERS is described and it is based on the binary coding of perfect matchings of a CERS. Furthermore, we study CERS with isomorphic resonance graphs. The obtained results are applied to phenylenes and hence the relationship between their resonance graphs and the resonance graphs of catacondensed benzenoid graphs is described. At the end of the chapter, we present the definition of a daisy cube and characterize all CERS whose resonance graphs are daisy cubes. The fifth chapter presents topological indices that are based on the distances in graphs or on the degrees of vertices. Furthermore, we define strength weighted graphs and Szeged-like topological indices on these graphs. At the end of the chapter, we present a model that is used to describe the dependence between the boiling points of alkenes or alkadienes and edge-weighted Wiener indices. A nonlinear regression analysis is performed for this purpose. In the sixth chapter, the quotient graph of a connected graph is defined. Then, we present the generalized cut method and prove that it can be also used to calculate the Schultz index and Gutman index. The results are applied to phenylenes and some other graph families. At the end of the chapter, we develop the generalized cut method for Szeged-like topological indices and present the exact formula for calculating these indices for any strength weighted graph. Finally, we include some examples showing how the obtained method can be used on various molecular graphs.
Secondary keywords: dissertations;Djoković-Winkler relation;resonance graph;benzenoid system;phenylene;CERS;Kekulé structure;perfect matching;daisy cube;quotient graph;topological index;Wiener index;Gutman index;Schultz index;Szeged-like topological indices;generalized cut method;Grafične metode;Kocka;Univerzitetna in visokošolska dela;
Type (COBISS): Doctoral dissertation
Thesis comment: Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages: IX, 137 str.
ID: 14366025