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Title:
Anihilacijsko število grafa in njegova povezava s celotnim dominantnim številom
Authors:
ID
Lužnic, Lara
(Author)
ID
Jakovac, Marko
(Mentor)
More about this mentor...
Files:
MAG_Luznic_Lara_2019.pdf
(730,62 KB)
MD5: FF2638FAB0164AE002357563303DE5A6
PID:
20.500.12556/dkum/5303eef9-002c-47e4-a8ca-a29c04fec19d
Language:
Slovenian
Work type:
Master's thesis/paper
Typology:
2.09 - Master's Thesis
Organization:
FNM - Faculty of Natural Sciences and Mathematics
Abstract:
Anihilacijsko število grafa je največje naravno število k, za katerega velja, da vsota prvih k členov v nepadajočem zaporedju stopenj grafa ne presega števila povezav tega grafa. V magistrskem delu je predstavljena definicija anihilacijskega števila, nekatere njegove lastnosti ter njegova povezava s celotnim dominantnim številom grafa. V prvem poglavju so predstavljeni osnovni pojmi in rezultati iz teorije grafov, ki jih potrebujemo za definiranje pojmov in dokazovanje v nadaljevanju. V drugem poglavju je na podlagi anihilacijskega procesa izpeljana definicija anihilacijska števila, opisana je povezava med anihilacijskim procesom in Havel-Hakimijevim algoritmom, predstavljene so nekatere lastnosti anihilacijskega števila in algoritem za iskanje le-tega. V tem delu je izpostavljena tudi povezava med anihilacijskim in neodvisnostnim številom grafa. Velja, da lahko neodvisnostno število navzgor omejimo z anihilacijskim številom. Ta meja je v nekaterih primerih natančnejša od drugih znanih mej. V zadnjem poglavju je podrobneje obravnavana povezava med anihilacijskim in celotnim dominantnim številom. Postavljena je domneva, da lahko v vsakem netrivialnem grafu celotno dominantno število navzgor omejimo z anihilacijskim številom. V magistrskem delu bo ta domneva dokazana za grafe z najmanjšo stopnjo 3, cikle, drevesa, kaktus grafe in bločne grafe.
Keywords:
anihilacijsko število
,
celotno dominantno število
,
neodvisnostno število
,
drevo
,
kaktus graf
,
bločni graf
Place of publishing:
Maribor
Publisher:
[L. Lužnic]
Year of publishing:
2019
PID:
20.500.12556/DKUM-74762
UDC:
519.17(043.2)
COBISS.SI-ID:
24866824
NUK URN:
URN:SI:UM:DK:OTTREDWM
Publication date in DKUM:
05.11.2019
Views:
1031
Downloads:
107
Metadata:
Categories:
FNM
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Vancouver
:
LUŽNIC, Lara, 2019,
Anihilacijsko število grafa in njegova povezava s celotnim dominantnim številom
[online]. Master’s thesis. Maribor : L. Lužnic. [Accessed 3 April 2025]. Retrieved from: https://dk.um.si/IzpisGradiva.php?lang=eng&id=74762
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Licences
License:
CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
Link:
http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:
The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:
05.09.2019
Secondary language
Language:
English
Title:
The annihilation number of a graph and its relation with the total domination number
Abstract:
The annihilation number of a graph is the maximum positive integer k, such that the sum of the first k terms in a non-decreasing degree sequence is less or equal to the number of edges of this graph. In this master thesis we introduce the definition of the annihilation number, some of its properties and its relation with total domination number. In the first chapter we introduce some of the basic definitions and results from graph theory, which are needed for definitions and proofs in later chapters. In the second chapter we describe the annihilation process, from which we form the definition of the annihilation number. The relation between Havel-Hakimi algorithm and some of the properties of the annihilation number are introduced, and we describe the algorithm for determination of the annihilation number. In this part we also introduce the relation between the annihilation and the independence number. It is shown that the annihilation number is a sharp upper bound for independence number. In some cases this bound is a better approximation than some other bounds. In the last chapter we describe the relation between the annihilation and the total domination number. It is conjectured that the annihilation number is an upper bound for the total domination number for every nontrivial graph. In this master thesis we prove the conjecture for graphs with minimum degree 3, cycles, trees, cactus graphs and block graphs.
Keywords:
annihilation number
,
total domination number
,
independence number
,
tree
,
cactus graph
,
block graph
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