Commuting graph of some prime order elements in symplectic and mathieu groups

This thesis concentrates on an investigation to describe the properties of groups in graph-theoretic context such as connectivity, distance and many more. One of the ways to achieve this is by studying a commuting graph. Let G be a group and X is a subset of G. The commuting graph C (G,X) is defi...

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Main Author:	Mohd Kasim, Suzila
Format:	Thesis
Language:	English
Published:	2020
Subjects:	Graph theory Group theory - Research
Online Access:	http://psasir.upm.edu.my/id/eprint/90776/1/IPM%202020%208%20%20-%20IR.pdf
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spelling	my-upm-ir.907762021-09-27T06:47:02Z Commuting graph of some prime order elements in symplectic and mathieu groups 2020-01 Mohd Kasim, Suzila This thesis concentrates on an investigation to describe the properties of groups in graph-theoretic context such as connectivity, distance and many more. One of the ways to achieve this is by studying a commuting graph. Let G be a group and X is a subset of G. The commuting graph C (G,X) is defined by taking a vertex set X, and letting two distinct vertices x, y ∈ X be adjacent whenever xy = yx. We construct the commuting graph by selecting G as a symplectic group or a Mathieu group, and X denotes a conjugacy class of G for prime order elements. The main contribution of this thesis deals with various aspects of the structure of the commuting graph C (G,X). We choose t ∈ X to be a fixed vertex of C (G,X), then we continue to determine the connectivity of the graph whether it is connected or disconnected. If C (G,X) is found to be connected, then we further the analysis in the commuting graph C (G,X) to obtain the distance of x ∈ X from t, the disc size and its diameter. However, if the commuting graph C (G,X) is disconnected, then there are more than one subgraph can be obtained. These subgraphs which are named as subgraphs D (G,X) of C (G,X) are then found to be isomorphic with each other and the diameter of D (G,X) is either 1 or 2 for the cases we consider. Consequently the number of the subgraphs can also be determined and for each subgraph D (G,X), we describe its structure as we treat the connected commuting graph C (G,X). Next, the study of the commuting graph takes naturally into consideration of the suborbits of G on X, that is the orbits under the action of a centralizer CG(t). Apart from finding the sizes of CG(t)-orbits of X, we obtain representatives x ∈ X for each of these orbits. Moreover, we also include some properties which in many cases aid speedy identification of the given orbit. We obtain the subgroup ⟨t, x⟩ generated by t and x. Then, we conduct a deeper analysis on the CG(t)-orbits by specifying the connectedness between every two orbits. We visualize the interaction of the CG(t)-orbits in a collapsed adijacency diagram, showing a line to join two circles of so-called CG(t)-orbits. We also compute how many vertices a CG(t)-orbit representative is connected to and record those data in a collapsed adjacency matrix. The entries of that matrix may indicate whether two distinct orbits are adjacent or not in the graph. Finally, we present the spectrum of the collapsed adjacency matrix of the commuting graph, that is the multiset of eigenvalues of its collapsed adjacency matrix and of course is one of many important invariants from which much information about the graph can be ascertained. The study of the spectrum of the graph, Spec(G,X) relates to the graph energy, ε(G,X). We obtain ε(G,X) > 0 if the graph is connected, otherwise ε(G,X) = 0. Graph theory Group theory - Research 2020-01 Thesis http://psasir.upm.edu.my/id/eprint/90776/ http://psasir.upm.edu.my/id/eprint/90776/1/IPM%202020%208%20%20-%20IR.pdf text en public doctoral Universiti Putra Malaysia Graph theory Group theory - Research Nawawi, Athirah
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
advisor	Nawawi, Athirah
topic	Graph theory Group theory - Research
spellingShingle	Graph theory Group theory - Research Mohd Kasim, Suzila Commuting graph of some prime order elements in symplectic and mathieu groups
description	This thesis concentrates on an investigation to describe the properties of groups in graph-theoretic context such as connectivity, distance and many more. One of the ways to achieve this is by studying a commuting graph. Let G be a group and X is a subset of G. The commuting graph C (G,X) is defined by taking a vertex set X, and letting two distinct vertices x, y ∈ X be adjacent whenever xy = yx. We construct the commuting graph by selecting G as a symplectic group or a Mathieu group, and X denotes a conjugacy class of G for prime order elements. The main contribution of this thesis deals with various aspects of the structure of the commuting graph C (G,X). We choose t ∈ X to be a fixed vertex of C (G,X), then we continue to determine the connectivity of the graph whether it is connected or disconnected. If C (G,X) is found to be connected, then we further the analysis in the commuting graph C (G,X) to obtain the distance of x ∈ X from t, the disc size and its diameter. However, if the commuting graph C (G,X) is disconnected, then there are more than one subgraph can be obtained. These subgraphs which are named as subgraphs D (G,X) of C (G,X) are then found to be isomorphic with each other and the diameter of D (G,X) is either 1 or 2 for the cases we consider. Consequently the number of the subgraphs can also be determined and for each subgraph D (G,X), we describe its structure as we treat the connected commuting graph C (G,X). Next, the study of the commuting graph takes naturally into consideration of the suborbits of G on X, that is the orbits under the action of a centralizer CG(t). Apart from finding the sizes of CG(t)-orbits of X, we obtain representatives x ∈ X for each of these orbits. Moreover, we also include some properties which in many cases aid speedy identification of the given orbit. We obtain the subgroup ⟨t, x⟩ generated by t and x. Then, we conduct a deeper analysis on the CG(t)-orbits by specifying the connectedness between every two orbits. We visualize the interaction of the CG(t)-orbits in a collapsed adijacency diagram, showing a line to join two circles of so-called CG(t)-orbits. We also compute how many vertices a CG(t)-orbit representative is connected to and record those data in a collapsed adjacency matrix. The entries of that matrix may indicate whether two distinct orbits are adjacent or not in the graph. Finally, we present the spectrum of the collapsed adjacency matrix of the commuting graph, that is the multiset of eigenvalues of its collapsed adjacency matrix and of course is one of many important invariants from which much information about the graph can be ascertained. The study of the spectrum of the graph, Spec(G,X) relates to the graph energy, ε(G,X). We obtain ε(G,X) > 0 if the graph is connected, otherwise ε(G,X) = 0.
format	Thesis
qualification_level	Doctorate
author	Mohd Kasim, Suzila
author_facet	Mohd Kasim, Suzila
author_sort	Mohd Kasim, Suzila
title	Commuting graph of some prime order elements in symplectic and mathieu groups
title_short	Commuting graph of some prime order elements in symplectic and mathieu groups
title_full	Commuting graph of some prime order elements in symplectic and mathieu groups
title_fullStr	Commuting graph of some prime order elements in symplectic and mathieu groups
title_full_unstemmed	Commuting graph of some prime order elements in symplectic and mathieu groups
title_sort	commuting graph of some prime order elements in symplectic and mathieu groups
granting_institution	Universiti Putra Malaysia
publishDate	2020
url	http://psasir.upm.edu.my/id/eprint/90776/1/IPM%202020%208%20%20-%20IR.pdf
_version_	1747813657155207168

Commuting graph of some prime order elements in symplectic and mathieu groups

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