Nekaj metričnih lastnosti grafovskih produktov : doktorska disertacija

slovenian

doktorska dela

2022

cycles   distance-balanced graphs   doctoral theses   general position set   gp-number   graph products   graph theory   l-distance-balanced graphs   paths

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

The dissertation focuses on two concepts: the general position problem and the l-distance-balancness property in graphs. The general position problem was introduced lately in paper by Manuel and Klavžar from 2018, while l-distance-balanced graphs were first studied in Frelih's doctoral thessis in 2014. In the disseration we present new results, which are mainly connected to different graph products. We determine the exact value of the gp-number in the Cartesian product of arbitrary number of path graphs, precisely that ?$\gp(P^{\cp,n}) = 2^{2^{n-1}}$? hold. We also determine the exact value of the gp-number in the Cartesian product of a path and a cycle and in the Cartesian product of two cycle graphs. Also the gp-value in some Kneser graphs is presented. In the second part we prove when the lexicographic product of ?$GH$? is an ?$\ell$?-distance-balanced graph, for any ?$\ell \in \{3,\ldots,\diam(G)\}$?. A similar condition is derived to test whether the corona product is ?$\ell$?-distance-balanced. We also study and characterize when the Cartesian product ?$G\cp K_n$? is l-distance-balanced.

08.08.2023