Department of Mathematics

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Author: search.filters.author.Dr. Hani Shaker

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    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 !
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    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.
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    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.
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    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 7439
    Algebraic 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”.
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    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 7438
    The 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.
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    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.
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    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.
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    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.
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    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; ?
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    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 .