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AAAG MONTHLY MEETING & MOCK FIRST ASSESSMENT PRESENTATIONS SEMESTER II, 2017/2018

Day/Date : Monday, 14 May 2018 Time : 2.30 PM – 4.55 PM Venue : Main Meeting Room, C22-310, Faculty of Science

TENTATIVE SCHEDULE

Time (PM) Speakers 2.30 – 2.35 Welcoming speech by AAAG research group leader,

• Assoc. Prof. Dr. Nor Muhainiah Mohd Ali

2.35 – 3.10 Nur Idayu Alimon “THE TOPOLOGICAL INDICES OF SOME GRAPHS OF SOME FINITE GROUPS” Supervisors: Prof. Dr. Nor Haniza Sarmin (Main), Prof. Dr. Ahmad Erfanian (Co) 3.10 – 3.45 Nurul Izzaty Ismail “MATHEMATICAL MODELLING OF DNA SPLICING SYSTEMS WITH PALINDROMIC AND NON- PALINDROMIC RESTRICTION ENZYMES” Supervisor: Dr. Fong Wan Heng (Main), Prof. Dr. Nor Haniza Sarmin (Co) 3.45 – 4.20 Nurhazirah Mohamad Yunos “CHARACTERIZING THE SPHEROIDS BASED ON THE POLARIZATION TENSOR” Supervisor: Dr. Taufiq Khairi Ahmad Khairuddin 4.20 – 4.55 Amira Fadina Ahmad Fadzil “THE CAYLEY GRAPHS FOR SOME FINITE GROUPS AND THEIR ENERGY” Supervisors: Prof. Dr. Nor Haniza Sarmin (Main), Prof. Dr. Ahmad Erfanian (Co)

Organised by Applied Algebra and Analysis Group (AAAG), Frontier Materials Research Alliance Universiti Teknologi Malaysia, Johor Bahru, Johor www.science.utm.my/AAAG

ABSTRACT _____________________________________________

THE TOPOLOGICAL INDICES OF SOME GRAPHS OF SOME FINITE GROUPS

Nur Idayu Alimon Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor nuridayualimon@yahoo.com Supervisors:

• Prof. Dr. Nor Haniza Sarmin

Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor nhs@utm.my

• Prof. Dr. Ahmad Erfanian

Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran erfanian@um.ac.ir Abstract Topological indices are the numerical values that can be analyzed to predict the chemical properties of the molecular

• structure. There are many types of topological indices that have been studied by other researchers such as Wiener

index, Zagreb index, Szeged index, Degree-distance index and Harary index. The Wiener index is the sum of half of the distances between every pair of vertices of a graph. Meanwhile, there are two types of Zagreb index which called first Zagreb index and second Zagreb index. First Zagreb index is defined as the sum of squares of the degrees of the vertices of a graph, while the second Zagreb index is the sum of the products of the degrees for pair of adjacent vertices of a graph. The Szeged index involves the number of vertices of a graph that lying closer to endpoint to endpoint of the edge and vice versa. Then, their products are sum up for each edge. Harary index is a half-sum of the elements in the reciprocal distance matrix. Next, both degree of the vertices and the shortest distance between them are involved in computing the Degree-distance index. These topological indices can be obtained for various connected graphs associated to groups. The graphs included in this research are non-commuting graph, conjugacy class graph, generalized conjugacy class graph and coprime graph. In addition, this research will focus only on the dihedral groups, symmetric groups, quaternion groups and quasidihedral gorups. The aim of this research is to find the generalization of the Wiener index, the Zagreb index, the Szeged index, the Harary index and the Degree-distance index of non-commuting graph, conjugacy class graph, generalized conjugacy class graph, and coprime graph for the dihedral groups, symmetric groups, quaternion groups, and quasidihedral groups. In this first assessment presentation, this research proposal will be presented. Keywords: Topological index, graphs of groups, finite groups

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MATHEMATICAL MODELLING OF DNA SPLICING SYSTEMS WITH PALINDROMIC AND NON- PALINDROMIC RESTRICTION ENZYMES Nurul Izzaty binti Ismail Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor

iamnurulizzaty1112@gmail.com

Supervisors:

• Dr. Fong Wan Heng and Prof. Dr. Nor Haniza Sarmin

Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor

fwh@utm.my, nhs@utm.my

Abstract

The mathematical modelling of DNA splicing system is one of the models in DNA computing that generates languages by using formal language theory. In DNA splicing system, the existence of restriction enzymes allows DNA molecules to be cleaved and recombined to generate new molecules. The molecules resulting from the splicing system are depicted as a splicing language which is analysed using formal language theory. DNA molecules generally read in two directions; forward and backward. A sequence of string that reads the same forwards and backwards is called a palindrome. The palindromic and non-palindromic sequences can also be recognised in the restriction enzymes. Research on DNA splicing systems with different restriction enzymes has been done previously. In this research, splicing languages in DNA splicing system with palindromic and non- palindromic restriction enzymes will be generalised and given as mathematical theorems. Next, the generalisations of splicing languages from DNA splicing systems will be illustrated as graphical representations via automata and de Bruijn graphs. Besides, a graphical user interface (GUI) for DNA splicing systems with palindromic and non-palindromic restriction enzymes will be developed using C++ visual programming. The GUI will simplify the manual computations of splicing languages from the respective DNA splicing system. The mathematical model of DNA splicing system with palindromic and non-palindromic restriction enzymes will be verified through in vitro experiments to compare the molecules resulting from the experiments and these

• generalisations. In this first assessment presentation, the proposal will be presented.

Keywords: DNA, Splicing system, Splicing language, Palindrome, Restriction Enzymes

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CHARACTERIZING THE SPHEROIDS BASED ON THE POLARIZATION TENSOR Nurhazirah Mohamad Yunos Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor

nurhazirah12394@yahoo.com

Supervisor:

• Dr. Taufiq Khairi Ahmad Khairuddin

Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor taufiq@utm.my Abstract

Polarization Tensor (PT) is an old terminology in sciences and engineering where, it has recently appeared in many applications of electric or electromagnetics such as electrical imaging for medical purposes or industrial problems, material science and metal detection. In these applications, the PT can be used for example to describe conducting objects presented in electric or electromagnetic fields. In this study, the main purpose is to investigate specifically the first order PT when the conducting object is a spheroid. When the conducting object is spheroid, the formula of the first order PT can be reformulated by adapting the depolarization factors. In

• rder to describe the spheroid, it is essential to investigate the properties of the formula itself. Since the formula

consist of the depolarization factors, some new properties of depolarization factors for ellipsoid are revealed. In this study, it is found that the value of depolarization factors are always positive. Furthermore, each depolarization factors must lies between 0 and 1. The depolarization factors can be characterized based on the values of the semi principal axes of the ellipsoid. Reversely, the semi principal axes of the ellipsoid can be classified based on the values of the depolarization factors. Next, the properties of the first order PT for spheroid are explained by using the properties of depolarization factors for ellipsoid. In this thesis, it is shown that when the first order PT for the spheroid is a positive definite matrix, then the conductivity of the spheroid is greater than 1, whereas, the first order PT is a negative definite matrix when the conductivity is between 0 and 1. Besides, the existence condition for both prolate and oblate spheroid are explained in details which can also describe the conductivity according to the elements of the first order PT. In addition, some numerical examples are also provided in this thesis to support our proposed theorems. Keywords: depolarization factors, conductivity, integrals

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THE CAYLEY GRAPHS FOR SOME FINITE GROUPS AND THEIR ENERGY

Amira Fadina Ahmad Fadzil Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor amirafadinaahmadfadzil@yahoo.com Supervisor:

• Prof. Dr. Nor Haniza Sarmin

Department of Mathematical Sciences, Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru, Johor nhs@utm.my Co-supervisor:

• Prof. Dr. Ahmad Erfanian

Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran erfanian@um.ac.ir

Abstract

A Cayley graph of a finite group G with respect to a Cayley subset S of G is a graph where its vertices are the elements of G and two vertices a and b in G are connected if ab-1 is in the set S. The Cayley subset S does not consist of the identity of G and is inverse-closed. This research will focus to find the Cayley graphs of some finite groups, and their energy. The defined energy of a graph is the sum of all absolute values of the eigenvalues of the adjacency matrix of the graph where the adjacency matrix of a graph is a square matrix with entries 0 and 1. The ij-th entries becomes 1 if there an edge between vertices i and j and becomes 0 otherwise. From the adjacency matrix of the graph, the eigenvalues can be found by finding its characteristic polynomial. Therefore, the target of this research is to compute the energy of the Cayley graphs associated to some finite groups which include symmetric groups, alternating groups and dihedral groups. The procedures will consist of finding the elements, vertices and edges for the Cayley graphs of the groups being studied, finding the isomorphism for the graphs, building the adjacency matrix for each Cayley graph, finding the eigenvalues of the adjacency matrix of the graphs, and lastly calculating the energy of the Cayley graphs related to the groups. Besides, the computations especially for the groups of higher order will be assisted by Maple software. Some properties and generalizations for the energy of the Cayley graphs will also be determined at the end of the study. In this first assessment presentation, the research proposal will be presented. Keywords: Energy of graph, Adjacency matrix, Eigenvalues, Cayley graph, Finite groups