Department of Mathematics
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Item Discussions on Solutions of Fredholm Integral Equations(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muhammad Adeel Khan; CIIT/SP21-BSM-022/LHR; Dr. Kashif Nazar; LHR TP 9888This thesis investigates the analytical and numerical solutions of Fredholm integral equa- tions of the second kind, focusing on both homogeneous and non-homogeneous cases. In the analytical framework, we explore solutions with degenerate kernels and methods to ap- proximate complex kernels by degenerate forms, providing efficient pathways for problem simplification. The study covers both theoretical and applied aspects, emphasizing cases where the kernel exhibits specific structures that facilitate closed-form solutions. Numer- ical methods are also developed and analyzed for solving these integral equations in both homogeneous and non-homogeneous settings. The numerical approaches are evaluated for their accuracy and computational efficiency, providing insights into practical implemen- tations and approximations for real-world applications. Through comprehensive analysis and comparison of methods, this research contributes to the effective solution of Fredholm integral equations, offering new perspectives for both theoretical exploration and applied problem-solving.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 Neutrosophic Fuzzy Number Optimized Path in a Network(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Shahzaib Sajid; CIIT/SP21-BSM-029/LHR; Dr. Madiha Khalid; LHR TP 9889Fuzzy and neutrosophic numbers are mathematical tools designed to handle un- certainty and imprecision in real-world problems. Fuzzy numbers, grounded in fuzzy set theory, represent values with gradual transitions between membership and non-membership, characterized by a membership function. These numbers effectively model uncertainty when precise values are unavailable, as seen in applications like decision-making, optimization, and engineering systems. Neutrosophic numbers extend this concept by incorporating indeterminacy into the model, a key feature of neutrosophic set theory. They consist of three components: truth (T), indeterminacy (I), and falsity (F), each of which is independently assessed. This structure enables a more nuanced representation of uncertainty, capturing scenarios where incomplete, contradictory, or vague information coexists. Both approaches play crucial roles in addressing the complexity of real-world problems where classical mathematics may fall short. While fuzzy numbers are suitable for problems with well-defined boundaries of uncertainty, neutro- sophic numbers offer superior flexibility in environments with higher levels of indeterminacy. Together, these tools have broadened the scope of uncertainty modeling across disciplines such as artificial intelligence, economics, and risk managementItem Topological Indices of Graphs and Some Related Aspects(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Ayesha Mehmood; CIIT/SP21-BSM-018/LHR; Dr. Sana Javed; LHR TP 9886This project explores the application of graph theory in chemistry, emphasizing topological indices such as the Leap Eccentric Connectivity Index (LECI) and Leap Zagreb Indices. These indices serve as mathematical descriptors to model molecular structures and predict chemical properties, contributing to quantitative structure- property (QSPR) and structure-activity relationships (QSAR). The study includes graph theory concepts, including vertices, edges, and connectivity. Detailed computations of the Leap Eccentric Connectivity Index (LECI) and Leap Zagreb Indices for various types of graphs, including complete, bipartite, wheel, and star graphs. Extensions to specialized graph structures like subdivision graphs, windmill graphs, and flower graphs are also included. This project is divided into five chapters. Chapter 1 provides basic terminologies, definitions and notions from graph theory and Leap Eccentric connectivity indices. Chapter 2 gives a literature review. Chapter 3 is about main results of the project while a discussion is provided in chapter 4. The references used in the project are given in chapter 5.Item Shortest Path Solution of an Intuitionistic Fuzzy Network(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Aleesha; CIIT/SP21-BSM-031/LHR; Dr. Madiha Khalid; LHR TP 9891This study focuses on solving the shortest path problem in networks where edge weights are represented as Intuitionistic Fuzzy Numbers (IFNs). Intuitionistic fuzzy sets allow the incorporation of both membership and non-membership degrees, providing a more com- prehensive representation of uncertainty compared to traditional fuzzy sets. The Floyd- Warshall algorithm, a well-established method for finding the shortest paths in weighted graphs, is adapted to work with IFNs. The algorithm operates iteratively, leveraging the properties of IFNs and appropriate fuzzy arithmetic to handle uncertainty in edge weights effectively. The proposed approach ensures that the shortest path calculations consider both the degree of certainty and uncertainty, providing a more robust solution for real- world problems involving imprecise or uncertain data. Numerical examples are presented to demonstrate the effectiveness and accuracy of the adapted algorithm, highlighting its po- tential applications in fields such as transportation, logistics, and communication networks.Item Application of Bilinear Neural Network Method to Sawada-Kotera Model(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Syeda Maryam Zahra; CIIT/SP21-BSM-013/LHR; Dr. Syed Tahir Raza Rizvi; LHR TP 9882In this work, we use the bilinear neural network method (BNNM) for solving the (2+1)- dimensional bidirectional Sawada-Kotera (bSK) equation. This model is the extension of the famous Korteweg-de Vries (KdV) equation which represents the shallow water waves on surfaces. Firstly, we use generalized Hirota bilinear approach (gHBA) and then by us- ing BNNM, we obtain exact analytical solution. BNNM is an important step toward the integration of neural networks with classic methodologies for nonlinear partial differential equations (NLPDEs), presenting both the solution in exact form and offering powerful vi- sualization of complex phenomena. This may be a method applied to a wide range of fields such as plasma, optics, acoustics, fluid dynamics, and so on. BNNM models the neural network with a tensor formula and derives the exact solutions by symbolic computation. This neural network consists of layers, with their contribution in finding the solutions using activation functions and weight matrices. This method generates new test functions and in this thesis, the test functions consist of ”3-2-1” and ”3-3-1” neural network models. Here ”3-2-1” means 3 neurons in input layer, 3 neurons in first hidden layer and a single neuron in second hidden layer. Lump, lump with one kink, rogue wave, and breather lump soliton and periodic solution for the governing model will be obtained by BNNM with the men- tioned layer. SevItem Soliton Solutions to Electromagnetic Model via Two Integration Schemes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Syed Zaheer Abbas; CIIT/SP21-BSM-005/LHR; Prof. Dr. Kashif Ali; LHR TP 9879Solitons are the special types of solitary waves who does not change their shape and speed during propagation. When a light pulse in space or time travels in some host medium, it does not dispersive or diffract is called optical Solitons. Optical solitons are produced due to balance between group velocity dispersion and self-phase modulation. In this thesis, we have studied the electromagnetic model with two integration schemes namely the enhanced algebraic method and Riccati equation method. Our purpose is to attain bell-type, kink and anti kink type, singular solitons solutions for various forms of NLEEs. We’ll also obtain Weierstrass elliptic functions, Jacobi elliptic functions in terms of hyperbolic, trigonometric and rational functions. We have present the solutions graphically in various dimensions like 3D, 2D and contour.Item Metric Dimension of Related Wheel Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muhammad Zubair Aslam; CIIT/SP21-BSM-033/LHR; Dr. Hani Shaker; LHR TP 9893The metric dimension of a graph is a central concept in graph theory, which deals with de- termining the minimum number of vertices required to uniquely identify every other vertex in the graph through their distances to the selected set of vertices, known as a resolving set. This thesis explores the metric dimension of related wheel graphs, a family of graphs ob- tained by attaching a cycle graph to a central vertex, thus forming a wheel like structure. We examine the metric dimension in both classical and related wheel graphs, where modifica- tions such as additional connections or altered structures are introduced. Through a detailed analysis, we aim to provide bounds on the metric dimension for different classes of related wheel graphs, propose efficient algorithms for finding the resolving set, and compare the metric dimension of these graphs with other well-studied graph families. The results of this study contribute to a deeper understanding of the interplay between graph structure and metric dimension, with potential applications in network design, sensor placement and graph labeling.Item On Topological Indices of Molecular Graph of Icosahedral Nanosheet(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Seerat; CIIT/SP21-BSM-032/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 9892The utilization of graph theory in multiple branches of research has been vastly enhanced, particularly in the domain of chemical graph theory. Over the past few years, numerous re- searchers have been able to pursue a variety of novel avenues. A chemical graph is a named graph where the edges reflect chemical bonds between atoms and the vertices reflect the atoms of a compound. The investigation of topological indices is crucial for determining numerous physical and chemical characteristics of the under-investigation molecular struc- tures. Firstly, we focused on degree-based topological indices, namely the Randic index, atom bond connectivity index, geometric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augm,ented Zagreb index, Balaban index, redefined Zagreb type indices, of Molecular ,Graph of Icosahedral Nanosheet.Item On Some Degree Based Topological Indices of Tetracyano Benzene Metal Organic Framework(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Aqib Javaid; CIIT/SP21-BSM-028/LHR; Dr. M. Faisal Nadeem; LHR TP 9888Metal–organic frameworks (MOFs) play a pivotal role in modern materials science due to their highly porous and customizable structures, which are ideal for applications such as gas storage, catalysis, and drug delivery. To model these complex structures, chem- ical graph theory offers a powerful mathematical framework that captures the molecular architecture of MOFs. Within this framework, topological indices—also known as molec- ular descriptors—serve as mathematical formulations derived from the molecular models. These descriptors allow researchers to analyze the physicochemical properties of MOFs without resorting to expensive laboratory experiments, thus streamlining the study of struc- ture–property and structure–activity relationships in mathematical chemistry. In this project, we focus on the tetracyanobenzene-based metal–organic framework, systematically computing and examining its various molecular descriptors. A numerical comparison of these descriptors is provided, offering insights into the framework’s charac- teristics and potential applications.