Resonančni grafi nekaterih dvodelnih zunajravninskih grafov in posplošena metoda prerezov : doktorska disertacija

slovenian

doktorska dela

2022

benzenoid system   daisy cube   dissertations   Djoković-Winkler relation   generalized cut method   Gutman index   Kekulé structure   perfect matching   phenylene   quotient graph   resonance graph   Schultz index   Szeged-like topological indices   topological index   Wiener index

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

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.

08.08.2023