Amzee Zamri, Siti Norziahidayu (2018) The conjugation degree on a set of metacyclic 3groups and 5groups with their related graphs. PhD thesis, Universiti Teknologi Malaysia.

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Abstract
The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group commute. In this research, the presentation of metacyclic pgroups where p is an odd prime is used. Meanwhile, the set considered is a set of an ordered pair of commuting elements in the metacyclic pgroups, where p is equal to three and five, satisfying certain conditions. The conjugation degree on the set is obtained by dividing the number of orbits with the size of the set. Hence, the results are obtained by finding the elements of the group that follow the conditions of the ordered set, followed by the computation of the number of orbits of the set. In the second part of this research, the obtained results of the conjugation degree on a set are then associated with graph theory. The corresponding orbit graph, generalized conjugacy class graph, generalized commuting graph and generalized noncommuting graph are determined where a union of complete and null graphs, one complete and null graphs, one complete and null graphs with one empty and null graphs are found. Accordingly, several properties of these graphs are obtained, which include the degree of the vertices, the clique number, the chromatic number, the independence number, the girth, as well as the diameter of the graph. Furthermore, some new graphs are introduced, namely the orderly set graph, the order class graph, the generalized coprime order graph, and the generalized non coprime order graph, which resulted in the finding of one complete or empty graphs, a union of two complete or one complete graphs, a union of complete and empty graphs and a complete or empty graphs. Finally, several algebraic properties of these graphs are determined.
Item Type:  Thesis (PhD) 

Uncontrolled Keywords:  algebraic properties, conjugation degree, chromatic number, 
Subjects:  Q Science > Q Science (General) 
Divisions:  Science 
ID Code:  81407 
Deposited By:  Narimah Nawil 
Deposited On:  23 Aug 2019 04:07 
Last Modified:  23 Aug 2019 04:07 
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