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Cayleyjev graf končnega kolobarja : delo diplomskega seminarja
ID
Trdin, Meta
(
Author
),
ID
Dolžan, David
(
Mentor
)
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Abstract
Naj bo $K$ končen komutativen kolobar z enico $1 \ne 0$. Cayleyjev graf končnega kolobarja $K$ je graf, kjer so vozlišča vsi elementi kolobarja $K$. Vozlišči $a$ in $b$ pa sta sosedni, če je $a - b$ obrnljiv element, torej če je $a - b \in K^\ast$. Tak graf je očitno neusmerjen. V nadaljevanju jih bomo označili z $G_K$. Pokazali bomo, da vedno dobimo regularen graf, v primeru lokalnega kolobarja z maksimalnim idealom $I$ pa tudi poln večdelen graf, katerega deli so ekvivalenčni razredi v njegovem kvocientnem kolobarju. Obravnavali bomo tudi strukturo grupe njegovih avtomorfizmov in izračunali, da so vsi kolobarji, ki podajajo ravninski graf, izomorfni eni izmed štirih oblik.
Language:
Slovenian
Keywords:
Cayleyjev graf
,
lokalen kolobar
,
klika
,
regularnost
Work type:
Final seminar paper
Typology:
2.11 - Undergraduate Thesis
Organization:
FMF - Faculty of Mathematics and Physics
Year:
2022
PID:
20.500.12556/RUL-140816
UDC:
519.17
COBISS.SI-ID:
122452739
Publication date in RUL:
18.09.2022
Views:
571
Downloads:
53
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Language:
English
Title:
Unitary Cayley graph of a finite ring
Abstract:
Let $K$ be finite commutative ring with unity $1 \ne 0$. Vertices of the Cayley graph of a finite ring $K$ are the elements of $K$. Two vertices $a$ and $b$ are adjacent if and only if $a - b$ is unit in $K$ so if $a - b \in K^\ast$. Obviously that graph is undirected. Throughout this paper we label Cayley graph with $G_K$. We show that the graph is always regular and if we take local ring for $K$ with maximal ideal $I$ we get a complete multipartite graph whose partite sets are the cosets of $I$. We also determine the structure of its automorphism group. We compute all planar graphs and show that there are only four forms for $K$ in that case.
Keywords:
Cayley graph
,
local ring
,
clique
,
regularity
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