Browsing by Author "Dr. Hani Shaker"

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Ergodic Properties of Full Branch Maps
(Library Information Services, CUI Lahore, 2022) Hadiqa Iqbal; FA20-RMT-023; Dr. Hani Shaker
Dynamical System is the study of a point in state space and its behavior over the time. This system contain functions which describe time dependence of a point in a topological space. There are some examples of dynamical system such as celestial bodies motion, ﬂow of water through a pipe, the movement of clock etc. Markov chain in dynamical system is a stochastic process that describes sequence of possible events whose probabilities relies on the states achieved in the previous event. In simple terms, outcomes of future state is dependent solely on current state. Markov chain has many applications e.g. lines of cus- tomers arriving at the airport or the ques in front of cash counter of shop, Exchange change rates of currency etc. Ergodic theory play a huge role in dynamical systems. Ergodic theory explains the statistical properties of dynamical system which are deterministic. By deter- ministic dynamical system, we mean the equation which determines the dynamics which do not have any random noise, perturbations. In this thesis, we will discuss ergodic prop- erties of full branch map. We will describe more general result about Markov maps which indirectly implies to full branch map. We will discuss Markov Chains, Markov Measure and Conjugacy of Markov Shifts and Interval Maps to reach out to our main result.
Graph Labelings via Abelian Groups
(Library Information Services, CUI Lahore, 2022) Rabie Ashraf; SP21-RMT-038/; Dr. Hani Shaker
2017. For any graph ? = ( ?,? ) o f orde r ? , whic h i s oriente d ther e i s bijectio n ?++⃗: ? ( ? ) → Г ? ∈ ?(?), defines a directed group distance magic labelling. The property that there e x ists ? ∈ Г (referred to as the magic constant) We say that a graph G is orientable group distance magic if there is an orientation in ? for group distance magic labeling G. In this thesis we used direct product of directed graph antiprism and cycles to show that that their triple product has orientable group distance magic labeling by using the different module groups; ?
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 9909
The 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.
Metric Dimension of Honeycomb Rectangular Torus
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Sahar Ali; CIIT/SP20-RMT-028/LHR; Dr. Hani Shaker; LHR TP 7655
Distance between two vertices of a graph is number of edges in a shortest path between them. One vertex having different distance from two distinct vertices is said to be a resolved vertex. Resolving set is a subset of vertex set of a graph and it consists of such vertices which resolve all the vertices of a graph. Metric dimension is an integer that is associated to resolving set and its defined as minimum count of resolving set. Honeycomb rectangular torus is one of the type of honeycomb torus. In this thesis, we will discuss metric dimension of honeycomb rectangular Torus. Metric dimension of honeycomb rectangular torus ( , ) for case (2, ) is 3. Metric dimension of honeycomb rectangular torus ( , ) > 4 for = 4 and is any even integer.
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 9893
The 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.
Metric Dimension of Related Wheel Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Zubair Aslam; SP21-BSM-033; Dr. Hani Shaker; LHR TP 9893
The 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.
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 9894
The 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.
Network Analysis Using Graph Indices
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Shamsa Liaqat (CIIT/FA-BSM-010/LHR), Mahnoor (CIIT/FA20-BSM-043/LHR); Dr. Hani Shaker; LHR TP 9917
In the digital age, social networking platforms like Twitter, LinkedIn, and Facebook have significantly influenced how individuals interact and form communities. These platforms, along with transportation networks, play crucial roles in shaping social dynamics and fa- cilitating physical mobility. This thesis employs graph theory to analyze these networks, focusing on the complexities of their interactions using Graph Indices, a mathematical framework that enhances the precision of network analysis. Graph theory provides a robust foundation for understanding the structure and flow of information within networks. This study specifically utilizes concepts such as betweenness centrality and the beta index to an- alyze network properties. Betweenness centrality identifies key influencers by measuring how often a node lies on paths between other nodes, while the beta index assesses network complexity by calculating the ratio of edges to vertices. The research encompasses a de- tailed examination of subgraphs, which reveal clusters or communities within the network, providing insights into user behavior and network dynamics. This analysis is applied to various social networks and transportation systems, demonstrating how graph indices can be used to optimize and enhance network structures. By integrating Graph Indices into graph theory, this study offers a more versatile framework for capturing social interactions’ complexities. The findings underscore the importance of mathematical tools in developing strategies for managing and leveraging social networks, leading to improved efficiency and robustness of network systems. This research not only advances theoretical understand- ing but also provides practical solutions for network analysis, paving the way for future applications in the field.
Network Analysis Using Graph Indices
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Shamsa Liaqat (FA20-BSM-010) : Mahnoor (FA20-BSM-043); Dr. Hani Shaker; LHR TP 9917
In the digital age, social networking platforms like Twitter, LinkedIn, and Facebook have significantly influenced how individuals interact and form communities. These platforms, along with transportation networks, play crucial roles in shaping social dynamics and fa cilitating physical mobility. This thesis employs graph theory to analyze these networks, focusing on the complexities of their interactions using Graph Indices, a mathematical framework that enhances the precision of network analysis. Graph theory provides a robust foundation for understanding the structure and flow of information within networks. This study specifically utilizes concepts such as betweenness centrality and the beta index to an alyze network properties. Betweenness centrality identifies key influencers by measuring how often a node lies on paths between other nodes, while the beta index assesses network complexity by calculating the ratio of edges to vertices. The research encompasses a de tailed examination of subgraphs, which reveal clusters or communities within the network, providing insights into user behavior and network dynamics. This analysis is applied to various social networks and transportation systems, demonstrating how graph indices can be used to optimize and enhance network structures. By integrating Graph Indices into graph theory, this study offers a more versatile framework for capturing social interactions’ complexities. The findings underscore the importance of mathematical tools in developing strategies for managing and leveraging social networks, leading to improved efficiency and robustness of network systems. This research not only advances theoretical understand ing but also provides practical solutions for network analysis, paving the way for future applications in the field.
On Cubic Pythagorean Fuzzy Topological Spaces and their Properties
(Library Information Services, CUI Lahore, 2023) Farwa Raza; CIIT/FA21-RMT-035/LHR; Dr. Hani Shaker
In the domain of set theory, fuzzy mathematics is an extension of traditional mathematics. From the inspection of literature on fuzzy mathematics, Simple observation reveals that fuzzy set theory hold a broad scope of mathematical contents that were previously known. It has a comprehensive applications, containing automobiles, transport systems, communi- cation, and other activities. The main idea of a FS, provides an appropriate framework for both constructing new region of fuzzy mathematics. The most important region of research is to expand the concepts of PFTSs, which are basically extensions of FTSs and IFTSs. Here we study, to develop the concept of CPFTSs and CPF continuity of a mapping among them. Also, we will discuss some useful features of these concepts. Furthermore, we will design a CPFT on an accessible nonempty reference set using continuity ideas. We will begin by going over some definitions and fundamental concepts, including fuzzy sets, and fuzzy topological spaces. These ideas will be extremely useful to us as we do this research. After that, we’ll explore PFTSs. By using this idea, our major main goal to expand and used to distinguish between FTS and IFTS, also aim develop the concept in cubic form and continuity map between them. Furthermore, we will design it on a nonempty reference set using continuity ideas.
On Fermatean Fuzzy Bipolar Soft Topological Space
(Library Information Services, CUI Lahore, 2023) Zainab Zaka; CIIT/FA21-RMT-056/LHR; Dr. Hani Shaker
In the current thesis, on fermatean fuzzy bipolar soft topological space is investigated. A fermatean fuzzy set is converted by using bipolar soft set in topological space. The soft sets are a family of parameterized sets. The concepts of a soft neighborhood of a point, a soft open set, and a soft closed set are introduced. Soft topological space offers variety of topological spaces that are parameterized.The BSS is made with the two SS. One gives negative information while the other one gives us positive information. Two mappings are used to explain the FBSS. In FBS, one mapping is used to approximate fuzziness relative to the degree of positivity while another mapping is used to approximate fuzziness relative to the degree of negativity in the initial universal set objects. To cope with uncertainty in various real-world condition, a reducing mathematical technique called fermatean fuzzy set is being developed. FFS is more ﬂexible than intuitionistic and PFS.The result are evaluated in on FFBSTS and FFBS. We work on some features such as neighborhood, continuity, and others.
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 7640
Assume 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 !
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 Shaker
A 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.
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 9577
Consider 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 .
On Spanning Simplicial Complexes Associated to Ladder Graph
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) MUHAMMAD AZIZ; CIIT/FA19-RMT-011/LHR; Dr. Hani Shaker; LHR TP 7427
In this thesis, we are going to discuss the algebraic and combinatorial aspects of span- ning simplicial complex ∆s(G) associated with the simple connected graph G, namely Ladder graph Ln. We mainly emphasis the characterization of s(Ln), all spanning tree, of Ln. We intend to provide the algebraic and combinatorial characterization of SSC ∆s(Ln) associated with the ladder graph Ln. In particular, we compute the formula for the f -vector of the SSC associated with the ladder graph Ln.
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 7385
This 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.
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 Shaker
Graph 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.
Polynomial Invariants Of Knots
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) SAAD AHMAD; FA17-RMT-002; Dr. Hani Shaker; LHR TP 5745
Knot theory is the branch of algebraic topology. A knot is considered as the embed ding of a unit circle in R3. Polynomial knot invariants play a vital role to understand underlying aspects of knotted structures. Tutte polynomial provides many combina torial properties for knots. In this thesis we study the Tutte polynomial by associating directed planar multi graphs with knotted structures. We compute Tutte polynomial for (2,n) torus knots and Dn k family of knots. For (2,n) torus knots and Dn k family of knots we also compute number of spanning trees and chromatic polynomial which are the combinatorial properties of Tutte polynomial.
Spanning Simplicial Complex of Certain Family of Graphs
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Zain Ul Abideen (FA23-RMT-053); Dr. Hani Shaker; LHR TP9796
The spanning simplicial complex Ds(Gw) related to a family of graphs known as w-wing graphs is thoroughly examined in this thesis. The combinatorial essence of cycle-free edge subsets is captured by Ds(Gw), where each facet corresponds to a spanning tree of the graph Gw. We examine this complex’s combinatorial invariants, which together carry significant structural information. These invariants include the f -vector, Euler characteristic, and Hilbert series. In addition, we investigate the algebraic features of the Stanley–Reisner ring of Ds(Gw) and define and analyze the facet ideal. In addition to generalizing existing concepts, the results provide a tangible link between commutative algebra and combinatorial topology.
Structural Analysis and Comparisons of Identity Graph of Groups
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Muhammad Sufyan (FA23-RMT-029); Dr. Hani Shaker; LHR TP 9775
The identity graphs of three groups are examined in this thesis: the quaternion group Q8, the symmetric group S5, and the alternating group A4. Every group element in an identity graph is a point (vertex) that is connected to both the identity and its inverse. We compare these graphs by examining fundamental characteristics such as the number of connections and points, triangles, self-inverse elements, and special groupings within the graph. Our research demonstrates how these graphs provide a clear and visual understanding of the group structure. This work proposes new techniques to use graphs to study groups by tying together concepts from group theory and graph theory. Due to vast usage of networks, numerous networks are considered and widely practiced in several branches of science, i.e in engineering, chemistry, biology, and computer networking. Networks can be describable in the aspect of graphs, where a vertex is affiliated with a node, and the connection between them as edges