Department of Mathematics
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Item On Group Labeling of Antiprism Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Mazhar Hussain; CIIT/SP20-RMT-038/LHR; Dr. Hani Shaker; LHR TP 7640Assume a graph G with n vertices and Λ be any abelian group having n elements. Group distance magic labeling of G is a bijective map f: V → Λsuch that ∃μ∈Λ for that ∑ ∈ ( ) ( ) = ∀ v ∈ V, where are the vertices adjacent to v. In this thesis , we find the × × × , × × × , × × × , × × × , × × × , × × × , group distance magic labeling for antiprism graph !Item The Fundamental Groups and Classification of Covering Spaces(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Shoaib Khan; CIIT/FA19-RMT-050/LHR; Dr. Hani Shaker; LHR TP 7439Algebraic topology is a very rich subject and its beauty lies in the way chosen to study space. Based on the global properties of spaces, the constructions and developments in this subject give an abstract and general notion. Defining algebraic topology as the study of topological spaces by using the algebraic invariants is the very basic interpretation of algebraic topology. However, the fundamental group is one of these topological invariants used to study topological spaces. The algebraic variants such as fundamental group are used for the classification of topological spaces up to the homeomorphism. However, most of the invariants classify topological spaces up to homotopy equivalence. The general idea of algebraic topology and fundamental group has been given in the 1st chapter. In this thesis work, to explain fundamental group, homeomorphism, and homotopy, the 2nd chapter includes the gathered ideas and concepts of homotopy of loops and maps, fundamental group, and induced homeomorphism. This chapter forms a basis for covering spaces which has a direct link with the fundamental group. After setting the basis for covering spaces, the 3rd chapter explains all the aspects related to covering spaces such as the lifting of maps to the covering space, and the applications of the fundamental group are explained through the major algebraic topology theorems; “Brouwer Fixed Point Theorem” and “Borsuk Ulam Theorem”. A main result named as “Van-Kampan Theorem” has also been proved which helps a lot to calculate the fundamental groups of different spaces. In chapter 4 all the possible covering spaces of the given topological space have been discussed leading to the classification of covering spaces. Moreover, this section discusses the classification of covering spaces by starting from “The Covering Transformation”, “Galois Covering Space”, “Universal Cover” and “Galois Correspondence”.Item Topological Indices of Boron Nano Tubes and Some Chemical Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) HAFIZ MUHAMMAD ASIF MEHMOOD; CIIT/FA19-RMT-114/LHR; Dr. Hani Shaker; LHR TP 7438The goal of this research study is to discuss some degree based topological descriptors of molecular graphs of (Boron Nano Tubes) by the assistance of fundamental ideas of graph theory. Using different terminologies we will calculate degree-based topological indices like “Gourava Indices”, “Shigehalli and Kanabur Indices”, “Sanskruti Index” of Boron Nano Tubes and also find some multiplicative version of above discussed topological indices of different molecular graphs.Item On Topological Study Of Face Cubic Lattice FCC(N)(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Anum Shahzadi; FA19-RMT-062; Dr. Hani Shaker; LHR TP 7385This study explores the topological properties of the Face-Centered Cubic (FCC) lattice, denoted as FCC(N), which is widely used to model crystalline structures in solid-state physics and materials science. The research focuses on analyzing the structural arrangement, connectivity, and symmetry of the FCC lattice using concepts from graph theory and topology. Key properties such as vertex coordination, edge relationships, and lattice transformations are examined to understand how the topology influences physical characteristics like stability, density, and atomic interactions. The findings highlight the mathematical significance of FCC(N) and its applications in modeling complex three-dimensional networks.Item On Secure Resolving Number of Networks(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Shahid Ikram; SP23-RMT-039; Dr. Hani Shaker; LHR TP 9577Consider a graph G(V,E) that is simple, connected, and finite.The resolving set for G is defined as the subset of V(G) for which each vertex of G has a distinct representation and the minimal cardinality of such a subset is said to be the metric dimension of G.If for any m ∈V \T, there exists n ∈ T such that T −{n}∪{m} is a resolving set, then the resolving set T is called a secure resolving set. The secure metric dimension of a graph G is defined as the cardinality of the minimum secure resolving set. Establishing the secure metric dimension of a specific graph presents an NP complete challenge.The SMD of various graph networks is already defined and determined.In this thesis we have examined the exact value of secure resolving number of Web Graph W∗ n .Item On Magic Type Labeling of Some Families of Subdivided Stars(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) HINA BIBI; FA19-RMT-049; LHR TP 7415; Dr. Hani ShakerA graph G is simple means it is undirected,does not contain any multiple edges and loop. In a directed, the direction of edges are given. The graph is con- nected if all the edges and vertices are given. A graph is said to be loop when starting and ending vertex is same. If we assigns positive integers to the set of nodes(vertices) and set of lines(edges) by a mapping that is bijection then it is known as labeling. If we assigns positive integers to set of nodes(vertices) only then the labeling become vertex(node) labeling. If we assigns positive in- tegers to set of lines(edges) then the labeling is known as edge(line) labeling. If we label the vertex(nodes) set and edge(lines) set both the resulting labeling is known as total labeling. There are many types of labeling but in this we will discuss about the magic type labeling. The terms edge label, edge sum and edge weights will help us to label the graph. In research magic and anti-magic labeling is growing fast. We will introduce new results on the basis of a concept labeling that is SEMTL of the families of subdivided star.Item Orientable Group Distance Magic Labeling of Regular Graphs and Their Direct Product(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Sana Ali; FA22-RMT-048; LHR TP 9362; Dr. Hani ShakerGraph labeling provides connectivity operation in networks used in computer networking, chemical structures, circuit design, and database administration.Group Distance Magic La beling (GDML) combines graph theory with group theory by using Abelian groups. A graph G has a GDML if we use elements of group for the labeling of graph’s components in such a way that the weight of each vertex in its neighborhood is contants in that group. The main focus of this work is on digraph orientable group distance magic labeling(OGDML). If an group H exits a digraph G, and if there is a injective map φ from G vertex set to the group members, then for every x ∈V, there exists a set of values such that ∑y∈N + G(x) φ→(x)− ∑y∈N − G(x) φ→(x). We study oriented graphs in this work. In particular, special labeling (OGDML) on directed graphs is the main emphasis of this study on oriented graphs.In this study, we prove that the directed direct product of Prism graphs Pn and CycleCn is OGDML under these non-isomorphic modulo groups Z2nm, Z2×Zn, Zn×Z2m, Z2×Zn/2×Z2m, and Z2 ×Z2 ×Zn/4 ×Z2m.Item Topological Description of Mobius Strip via Degree Based Indices(Library Information Services COMSATS University Lahore Campus, 2023-03-13) MAIRA RIAZ; SP22-RMT-016; Dr. Hani Shaker; LHR TP 8723Scientists study tiny building blocks called molecules. These molecules have shapes and patterns, and scientists use something called molecular graph theory to understand them. They also use special math formulas called topological indices. In our project we looked at a special shape called Hexagonal Mobius strip. It’s like a twisted cylinder with loop on one side, and it can be big or small. It comes in variety of sizes and shapes and while some Mobius strip are simple to picture in regular space, few are more difficult. We wanted to know two things: how molecules act (heat of formation/entropy) and the structure of molecules (topological indices and co indices). To figure this out, we used a method called Curve Fitting on the Hexagonal Mobius strip.Item Metric Dimension of Wheel Based Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muzammal Ashraf; CIIT/SP21-BSM-037/LHR; Dr. Hani Shaker; LHR TP 9894The concept of metric dimension is essential for solving a variety of structural and opti- mization issues. The smallest size of a set of vertices that allows the original vertex to be uniquely determined by the shortest distances from any vertex to these selected vertices is known as the graph’s metric dimension. This idea finds use in the construction of effective routing algorithms, network navigation, and locating issues. In this study, we investigate the metric dimension of different classes of graphs, such as trees, pathways, cycles, and more intricate structures like grids and hypercubes. We explore methods for computing the metric dimension and present novel approaches to obtain accurate values or constraints for particular graph families. The study sheds light on the behavior of the metric dimen- sion under various graph settings, emphasizing how it varies with respect to graph factors including degree, vertex connectivity, and diameter. Furthermore, we examine the compu- tational difficulty of figuring out the metric dimension and suggest effective strategies for real-world uses. In order to provide more efficient solutions for graph-based problems in practical settings, we seek to optimize the determination of metric dimensions by utilizing recent developments in computational techniques.Item House Price Prediction by using Machine Learning(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Laiba Faisal (CIIT/FA20-BSM-057/LHR), Ameema Umar (CIIT/FA20-BSM-053/LHR); Dr. Hani Shaker; LHR TP 9909The goal of this project is to create a machine learning model that can predict home selling prices by taking into account variables like location, square footage, number of bedrooms and bathrooms, and other relevant characteristics. The model prioritizes optimization for managing missing data in order to produce trustworthy estimates for the real estate market. This study advances real estate predictive modelling by using rigorous analysis and experimentation. It provides practitioners and stakeholders with useful information for making precise cost projections.