delo diplomskega seminarja
Luka Sajovic (Author), David Dolžan (Mentor)

Abstract

Za kolobar $R$ definiramo komutirajoči graf $\Gamma(R)$ kot graf, v katerem so vozlišča necentralni elementi kolobarja $R$, dve vozlišči pa sta povezani natanko tedaj, ko pripadajoča elementa komutirata v $R$. Pokažemo, da je za kolobarje matrik nad poljem in $n \ge 3$, komutirajoči graf $\Gamma (M_n(F))$ povezan natanko tedaj, ko ima vsaka $F$-razširitev stopnje $n$ pravo vmesno polje. Nadalje pokažemo, da je $\Gamma (M_n(\mathbb{Q}))$ nepovezan $n \ge 2$. Dokažemo, da če je $\Gamma (M_n(F)))$ povezan, potem je njegov premer vsaj 4 in največ 6. Poiščemo nekaj primerov komutirajočih grafov s premerom 4. Dokažemo še, da če je $F$ končno polje in $n$ ni praštevilo ali kvadrat praštevila, je ${\rm diam}\,\Gamma (M_n(F)) \le 5$.

Keywords

matematika;komutirajoči graf;linearna algebra;matrike;Galoisova teorija;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [L. Sajovic]
UDC: 512
COBISS: 101306115 Link will open in a new window
Views: 851
Downloads: 39
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Other data

Secondary language: English
Secondary title: The missing field
Secondary abstract: We define the commuting graph of ring $R$ as the graph $\Gamma(R)$ in which vertices are non-central elements of ring $R$. Two vertices are adjacent if and only if the corresponding elements commute in $R$. We show that for the ring of matrices over a field where $n \ge 3$ the commuting graph $\Gamma (M_n(F))$ is connected if and only if for every $F$-extension of degree $n$ exists a proper intermediate field. We also show that $\Gamma (M_n(\mathbb{Q}))$ is not connected for $n \ge 2$. We prove that if $\Gamma (M_n(F))$ is connected then $4 \le {\rm diam}\,\Gamma (M_n(F)) \le 6$. We find some examples of commuting graphs with diameter 4. We also prove that ${\rm diam}\,\Gamma (M_n(F)) \le 5$ if $F$ is a finite field and $n$ is not a prime nor square of a prime.
Secondary keywords: mathematics;commuting graph;linear algebra;matrices;Galois theory;
Type (COBISS): Final seminar paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages: 29 str.
ID: 14785285
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