University of Zagreb. Faculty of Science. Department of Mathematics.
Abstract
U ovom diplomskom radu bavili smo se računanjem determinanti matrica udaljenosti pojedinih klasa kaktusa. Na početku smo obradili najosnovnije pojmove teorije grafova te završili s definicijama stabla, kaktusa i matrice udaljenosti. Zatim smo spomenuli neke osnovne rezultate o stablima te pomoću njih došli do formule za determinantu matrice udaljenosti stabla. Proučavali smo kako determinanta matrice udaljenosti grafa ovisi o njegovim blokovima. Osim toga, detaljno smo proučili biciklične grafove i došli do formule za determinante njihovih matrica udaljenosti. Nakon toga smo krenuli proučavati pojedine klase kaktusa te došli do formula za računanje determinanti njihovih matrica udaljenosti ovisno o njihovoj građi. Na samom kraju smo na konkretnim primjerima kaktusa pokazali kako pomoću naših rezultata lakše računati determinante matrica udaljenosti.In this graduate thesis we have engaged in calculating the determinants of distance matrices of certain classes of cacti. At the beginning we introduced the most basic concepts related to graph theory and ended with definitions of a tree, cactus and distance matrix. After that, we mentioned some basic results about trees and used them to come up with formula for determinant of distance matrix of a tree. We studied how the determinants of distance matrices of graphs depend on their blocks. Besides that, we thoroughly studied bicyclic graphs and came up with the formula for determinants of their distance matrices. After that, we started studying certain classes of cacti and came up with formulas for calculating determinants of their distance matrices depending on their structure. At the very end, we showed on concrete examples of cacti how our results can help us calculate determinants of distance matrices
