The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups

The energy of a simple graph is defined as the summation of the absolute value of the eigenvalues of the adjacency matrix of the graph. It was motivated by the Huckel Molecular Orbital theory. The theory was used by chemists to estimate the energy associated with n-electron orbitals of molecules whi...

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Main Author:	Ahmad Fadzil, Amira Fadina
Format:	Thesis
Language:	English
Published:	2021
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/102102/1/AmiraFadinaAhmadPFS2021.pdf
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spelling	my-utm-ep.1021022023-08-07T08:01:48Z The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups 2021 Ahmad Fadzil, Amira Fadina QA Mathematics The energy of a simple graph is defined as the summation of the absolute value of the eigenvalues of the adjacency matrix of the graph. It was motivated by the Huckel Molecular Orbital theory. The theory was used by chemists to estimate the energy associated with n-electron orbitals of molecules which is called as conjugated hydrocarbons. Meanwhile, the Seidel energy is defined as the summation of the absolute value of the eigenvalues of the Seidel matrix of the graph. Besides, a Cayley graph associated to a finite group is defined as a graph where its vertices are the elements of a group and two vertices g and h are joined with an edge if and only if h is equal to the product of x and g for some elements x in the subset X of the group. This research combines the topics in graph theory with group theory, namely on the energy and Seidel energy for Cayley graphs, with some finite groups, namely dihedral groups, alternating groups, and symmetric groups. The results are obtained by finding the isomorphism of the Cayley graphs with respect to the subsets of order one and two, and the generating set associated to the groups. The respected Cayley graphs are found and represented as the union of complete graphs, cycle graphs, and complete bipartite graphs. The obtained graphs are then mapped onto their adjacency matrix and Seidel matrix respectively to obtain the eigenvalues and Seidel eigenvalues of the graphs. Some group theory concepts and properties of special graphs are also used to find the generalizations of the eigenvalues of the Cayley graphs. Finally, the energy and the Seidel energy for the Cayley graphs associated to the dihedral groups, alternating groups, and symmetric groups are obtained by using the eigenvalues and the Seidel eigenvalues of the graphs, respectively. The results show that the Seidel energy of Cayley graphs with respect to subsets of order one associated to the groups are equal to their energy. It is also found that the Seidel energy of Cayley graphs with respect to some subsets of order two and the generating sets associated to the groups are larger than their energy. 2021 Thesis http://eprints.utm.my/id/eprint/102102/ http://eprints.utm.my/id/eprint/102102/1/AmiraFadinaAhmadPFS2021.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:146256 phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Ahmad Fadzil, Amira Fadina The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
description	The energy of a simple graph is defined as the summation of the absolute value of the eigenvalues of the adjacency matrix of the graph. It was motivated by the Huckel Molecular Orbital theory. The theory was used by chemists to estimate the energy associated with n-electron orbitals of molecules which is called as conjugated hydrocarbons. Meanwhile, the Seidel energy is defined as the summation of the absolute value of the eigenvalues of the Seidel matrix of the graph. Besides, a Cayley graph associated to a finite group is defined as a graph where its vertices are the elements of a group and two vertices g and h are joined with an edge if and only if h is equal to the product of x and g for some elements x in the subset X of the group. This research combines the topics in graph theory with group theory, namely on the energy and Seidel energy for Cayley graphs, with some finite groups, namely dihedral groups, alternating groups, and symmetric groups. The results are obtained by finding the isomorphism of the Cayley graphs with respect to the subsets of order one and two, and the generating set associated to the groups. The respected Cayley graphs are found and represented as the union of complete graphs, cycle graphs, and complete bipartite graphs. The obtained graphs are then mapped onto their adjacency matrix and Seidel matrix respectively to obtain the eigenvalues and Seidel eigenvalues of the graphs. Some group theory concepts and properties of special graphs are also used to find the generalizations of the eigenvalues of the Cayley graphs. Finally, the energy and the Seidel energy for the Cayley graphs associated to the dihedral groups, alternating groups, and symmetric groups are obtained by using the eigenvalues and the Seidel eigenvalues of the graphs, respectively. The results show that the Seidel energy of Cayley graphs with respect to subsets of order one associated to the groups are equal to their energy. It is also found that the Seidel energy of Cayley graphs with respect to some subsets of order two and the generating sets associated to the groups are larger than their energy.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Ahmad Fadzil, Amira Fadina
author_facet	Ahmad Fadzil, Amira Fadina
author_sort	Ahmad Fadzil, Amira Fadina
title	The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
title_short	The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
title_full	The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
title_fullStr	The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
title_full_unstemmed	The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
title_sort	energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2021
url	http://eprints.utm.my/id/eprint/102102/1/AmiraFadinaAhmadPFS2021.pdf
_version_	1776100846234238976

The energy and seidel energy of cayley graphs associated to dihedral, alternating and symmetric groups

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