Metrična dimenzija grafa deliteljev niča

delo diplomskega seminarja

Tadeja Možina (Author), David Dolžan (Mentor)

Abstract

V diplomski nalogi preučujemo metrično dimenzijo grafa deliteljev niča kolobarja. Za kolobar $R$ definiramo njegov graf deliteljev niča $\Gamma(R)$ kot enostaven neusmerjen graf, katerega vozlišča so delitelji niča, med dvema različnima vozliščema pa je povezava natanko tedaj, ko se zmnožita v 0. Metrična dimenzija takega grafa je velikost najmanjše urejene podmnožice njegovih vozlišč za katero velja, da imata poljubni vozlišči v grafu različna vektorja razdalj do elementov te množice. Najprej raziščemo kako omejiti metrično dimenzijo grafa. V ta namen definiramo množice dvojčkov grafa, to so podmnožice vozlišč grafa, kjer so vozlišča v isti množici dvojčkov, če imajo enake soseščine. Nato si podrobneje ogledamo metrični dimenziji grafov deliteljev niča kolobarja ostankov po danem modulu in kolobarja matrik nad danim poljem.

Keywords

metrična dimenzija;graf deliteljev niča;kolobarji;rešljiva množica;množica dvojčkov;

Data

Language:	Slovenian
Year of publishing:	2024
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[T. Možina]
UDC:	519.17
COBISS:	206042883
Views:	28
Downloads:	6
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Metric dimension of a zero-divisor graph
Secondary abstract:	In this thesis, we study the metric dimension of a zero-divisor graph of a ring. For a ring $R$ we define its zero-divisor graph $\Gamma(R)$ as a simple undirected graph whose vertices are zero-divisors and two distinct vertices are adjacent if and only if their product is 0. Metric dimension of such graph is the size of the smallest ordered subset of its vertices for which two distinct vertices in graph have distinct vectors of distances to elements of this subset. Firstly, we study how to limit the metric dimension of a graph, mainly with twin-sets of a graph, subsets of vertices of a graph where vertices are in a same twin-set if they have the same neighbourhoods. Then we closely study the metric dimension of a zero-divisor graph of the ring of integers modulo n and of the ring of matrices over a field.
Secondary keywords:	metric dimension;zero-divisor graph;rings;resolving set;twin-set;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	26 str.
ID:	24866603