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## Irregularity Strength of Graphs

### Marcin Anholcer

#### Abstract

One of the well-known facts about simple graphs is that in every such graph G there are at least two vertices of the same degree (it follows from the Pigeonhole Principle).The situation changes when we assign weights (being positive integers) either to the edges of G or to its edges and vertices and consider weighted degrees of vertices instead of the ordinary ones. Three graph parameters are considered in the thesis: irregularity strength s(G), total vertex irregularity strength tvs(G) and product irregularity strength ps(G). In the first chapter of the thesis upper bounds on tvs(G) of arbitrary graph G ar given. The author presents also the exact values of tvs(F), where F is a forest with no vertices of degree 2, and tvs(Cnk), where Cnk is the k-th power of cycle Cn.The second chapter contains the results on the irregularity strength, s(G). In particular, the exact value of s(Cnk) has been determined.In the last chapter the facts on the product irregularity strength are presented. The main results are the upper bounds on ps(G) for G being either cycle or grid of sufficiently many vertices.
Record ID
UAM7e75dd23565c4b4585d3ac20d30e06c9
Diploma type
Doctor of Philosophy
Author
Marcin Anholcer Marcin Anholcer,, Undefined Affiliation
Title in Polish
Siła nieregularności grafów
Title in English
Irregularity Strength of Graphs
Language
pol (pl) Polish
Certifying Unit
Faculty of Mathematics and Computer Science (SNŚ/WMiI/FoMaCS)
Discipline
mathematics / (mathematics domain) / (physical sciences)
Scientific discipline (2.0)
6.3 mathematics
Status
Finished
Defense Date
28-12-2010
Title date
28-12-2010
Supervisor
URL
http://hdl.handle.net/10593/782 Opening in a new tab
Keywords in English
Irregularity strength, Total vertex, Irregularity strength, Product irregularity strength, Irregular weighting
Abstract in English
One of the well-known facts about simple graphs is that in every such graph G there are at least two vertices of the same degree (it follows from the Pigeonhole Principle).The situation changes when we assign weights (being positive integers) either to the edges of G or to its edges and vertices and consider weighted degrees of vertices instead of the ordinary ones. Three graph parameters are considered in the thesis: irregularity strength s(G), total vertex irregularity strength tvs(G) and product irregularity strength ps(G). In the first chapter of the thesis upper bounds on tvs(G) of arbitrary graph G ar given. The author presents also the exact values of tvs(F), where F is a forest with no vertices of degree 2, and tvs(Cnk), where Cnk is the k-th power of cycle Cn.The second chapter contains the results on the irregularity strength, s(G). In particular, the exact value of s(Cnk) has been determined.In the last chapter the facts on the product irregularity strength are presented. The main results are the upper bounds on ps(G) for G being either cycle or grid of sufficiently many vertices.
Thesis file
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Uniform Resource Identifier
https://researchportal.amu.edu.pl/info/phd/UAM7e75dd23565c4b4585d3ac20d30e06c9/
URN
urn:amu-prod:UAM7e75dd23565c4b4585d3ac20d30e06c9

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