M.Phil / MS
Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/52
This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.
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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 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 7655Distance 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.Item 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 7427In 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.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 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 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 9775The 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 edgesItem 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 TP9796The 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.