20.500.12556/DKUM-70163 Računanje Wienerjevega indeksa uteženega grafa z združevanjem ?*-razredov Computing the Wiener index of a weighted graph with the unification of ?*-classes Wienerjev indeks igra pomembno vlogo pri poznavanju kemijskih in fizikalnih lastnosti različnih spojin. Predstavlja vsoto razdalj med vsemi neurejenimi pari vozlišč znotraj grafa. Uteženi graf je graf skupaj s funkcijo, ki vsakemu vozlišču predpiše realno število, imenovano utež. Magistrsko delo obravnava računanje Wienerjevega indeksa uteženega grafa s pomočjo reduciranja na posebno skupino grafov, tj. kvocientne grafe in nadaljnje redukcije kvocientnih grafov na enostavnejše grafe. V prvem delu predstavimo nekaj osnovnih definicij in ugotovitev teorije grafov. Zapišemo osnovno definicijo Wienerjevega indeksa in njegovo razširitev na utežene grafe. Spoznamo Djoković-Winklerjevo relacijo in njeno tranzitivno zaprtje. Ob koncu prvega dela spoznamo definicijo delne kocke in zapišemo njeno novo karakterizacijo. Osrednji del magistrske naloge podaja novi metodi za izračun Wienerjevega indeksa nekaterih uteženih grafov. Glavni izrek povezuje izračun Wienerjevega indeksa uteženega grafa z vsoto Wienerjevih indeksov uteženih kvocientnih grafov prvotnega grafa po vseh Θ^∗-razredih, kjer Θ^∗ predstavlja tranzitivno zaprtje Djoković-Winklerjeve relacije. V zadnjem delu predstavimo uporabo zgoraj omenjenega izreka na posebni družini grafov G_n, na benzenoidnih sistemih ter na linearnih fenilenih F_n. The Wiener index plays an important role in understanding the chemical and physical properties of different compounds. It represents the sum of the distances between all the unordered pairs of vertices in the graph. A weighted graph is a graph along with a function that specifies a real number called a weight for each vertex. The thesis deals with the calculation of the Wiener index of a weighted graph by means of reduction to a specific group of graphs, i.e. the quotient graphs and further reduction of the quotient graphs to simpler graphs. In the first part, we present some basic definitions and findings of the theory of graphs. We write the basic definition of the Wiener index and its extension on weighted graphs. We become acquainted with the Djoković-Winkler’s relation and its transitive closure. At the end of the first part, we get to know the definition of a partial cube and write down its new characterization. The main part of the thesis presents two new methods for calculating the Wiener index of some weighted graphs. The main theorem links the calculation of the Wiener index of a weighted graph with the sum of Wiener indices of weighted quotient graphs of the primary graph for all Θ^∗-classes, where Θ^∗ represents the transitive closure of Djoković-Winkler’s relation. In the last part, we present the use of the above-mentioned theorem on a special family of graphs G_n, on the benzenoid systems and on the linear phenylenes F_n. Wienerjev indeks delna kocka uteženi graf kvocientni graf Djoković-Winklerjeva relacija tranzitivno zaprtje Wiener index partial cube weighted graph quotient graph Djoković-Winkler relation transitive closure true true false [S. Brezovnik] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2018-04-12 12:06:34 2018-09-24 14:55:40 2022-08-01 18:53:04 2019-07-11 17:57:57 2018 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17(043.2) 23868424 URN:SI:UM:DK:LQGRF7IQ MAG_Brezovnik_Simon_2018.pdf MAG_Brezovnik_Simon_2018.pdf 1 81A0BEAE1E73A572B78737C8A4559E34 bb68265e2d5dae9ac8890d6998e5ee5cd4fccd6baa2808e7cc9a337df0ecc9a2 e2019293-7c0f-11eb-bb7a-00155d0001ca 20.500.12556/dkum/34f8ba97-7be4-4239-afe8-58e7e691b47b https://dk.um.si/Dokument.php?lang=slv&id=123838 Fakulteta za naravoslovje in matematiko 0 0 0