Nekaj metričnih lastnosti grafovskih produktov

Rus, Gregor

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Title:	Nekaj metričnih lastnosti grafovskih produktov
Authors:	ID Rus, Gregor (Author) ID Klavžar, Sandi (Mentor) More about this mentor...
Files:	DOK_Rus_Gregor_2022.pdf (965,92 KB) MD5: 4778CC2F77B5ABAF74867636F0647178
Language:	Slovenian
Work type:	Doctoral dissertation
Typology:	2.08 - Doctoral Dissertation
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Doktorska disertacija obravnava koncepta množice vozlišč v splošni legi v grafih in l-razdaljno-uravnoteženost grafov. Oba koncepta sta bila v tej obliki vpeljana nedavno, splošna lega leta 2018 v članku avtorjev Manuela in Klavžarja, l-razdaljna uravnoteženost pa v doktorski diseratciji Freliha leta 2014. V disertaciji so predstavljeni novi rezultati, ki so večinoma povezani z različnimi grafovskimi produkti. Dokazana je točna vrednost gp-števila v kartezičnem produktu poljubnega števila poti, natančneje, da velja $\gp(P^{\cp,n}) = 2^{2^{n-1}}$. Dokazana je točna vrednost gp-števila v produktu poti in cikla in produkta dveh ciklov. Dokazana je tudi točna vrednost gp-števila v nekaterih Kneserjevih grafih. V razdelku, ki se ukvarja z l-razdaljno-uravnoteženostjo, je pokazan pogoj, kdaj je leksikografski produkt grafov $G[H]$ $\ell$-razdaljno-uravnotežen za poljuben $\ell \in \{3,\ldots,\diam(G)\}$. Prav tako je dokazano, kdaj je $\ell$-razdaljno-uravnotežen korona produkt. Določimo pa tudi pogoj, kdaj je $\ell$-razdaljno uravnotežen kartezični produkt $G\cp K_n.$
Keywords:	teorija grafov, množica vozlišč v splošni legi, gp-število, grafovski produkti, poti, cikli, razdaljno-uravnoteženi grafi, l-razdaljno-uravnoteženi grafi
Place of publishing:	Maribor
Publisher:	[G. Rus]
Year of publishing:	2022
PID:	20.500.12556/DKUM-81826
UDC:	519.17(043.3)
COBISS.SI-ID:	124417283
Publication date in DKUM:	07.10.2022
Views:	982
Downloads:	79
Metadata:
Categories:	FNM
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Licences

License:	CC BY 4.0, Creative Commons Attribution 4.0 International

Link:	http://creativecommons.org/licenses/by/4.0/
Description:	This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.
Licensing start date:	06.06.2022

Secondary language

Language:	English
Title:	Some metric properties of graph products
Abstract:	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 $G[H]$ 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.
Keywords:	graph theory, general position set, gp-number, graph products, paths, cycles, distance-balanced graphs, l-distance-balanced graphs

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