Department of Mathematics
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Item γ-Edge Irregular Intuitionistic Fuzzy Graph(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Habiba Samreen; CIIT/SP20-RMT-018/LHR; Dr. Imran Zulfiqar Cheema; LHR TP 7649When available knowledge is unable to convey ambiguity using sets, intuitionistic fuzzy sets can be considered as a different technique. In a fuzzy set, just the element membership degree is considered, while in intuitionistic fuzzy Edge irregular intuitionistic fuzzy graph, edge totally irregular intuitionistic fuzzy graph, sets, both membership and non-membership degrees are included. Intuitionistic fuzzy graph is a more advanced fuzzy graph idea. In this thesis degree of an edge 𝜅𝐺(𝑦𝑖𝑦𝑗) = (𝜅1𝐺(𝑦𝑖𝑦𝑗), 𝜅2𝐺(𝑦𝑖𝑦𝑗) of an intuitionistic fuzzy graph, total degree of an edge 𝑡𝜅𝐺(𝑦𝑖𝑦𝑗) = (𝑡𝜅1𝐺(𝑦𝑖𝑦𝑗), 𝑡𝜅2𝐺(𝑦𝑖𝑦𝑗) and operations (complement, union and join) on an intuitionistic fuzzy graph are define. Edge irregular intuitionistic fuzzy graph, edge totally irregular intuitionistic fuzzy graph, highly edge irregular intuitionistic fuzzy graph, highly edge totally irregular intuitionistic fuzzy graph, strongly edge irregular intuitionistic fuzzy graph and strongly edge totally irregular intuitionistic fuzzy graph are also defined with suitable illustration. On an intuitionistic fuzzy graph, such as a γ-edge irregular intuitionistic fuzzy graph, we examine and discuss properties. Moreover, we are establishing results and providing various examples to help understand these concepts.Item Topological Indices on Line Graph of Multi Web and Multi Jahangir Graph(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Awais Amanat; CIIT/FA19-RMT-082/LHR; Dr. Imran Zulfiqar Cheema; LHR TP 7414A topological index is a numeric measure from the molecular graph. Usage of topological indices in chemistry is remarkable. In a structure related graph, a real number that always remains constant is called a topological index. A great number of topological indices have been presented till date, several of them are being applied in different industries as means like pharmaceutical industry modal chemical industry etc. In this thesis, we will calculate the degree based topological indices so-called Randic index, atom bond connectivity (ABC) index, geometric arithmetic (GA) index, first and second Zagreb, third Zagreb, hyper Zagreb, Harmonic index, F-index, Redefined first, second and third Zagreb indices, Sum connectivity index for line graph of multi- Web graph (𝑊𝑛,𝑚) (Tn × Cm) where m ≥ 5 and n ≥ 2 and as well as line graph of multi- Jahangir graph (𝐽𝑛,𝑚) where n ≥ 4 m ≥ 3. We will also compute degree sum based topological indices ABC₄ index known as the fourth version of atom bond connectivity (ABC) index and GA₅ index which is the fifth class of geometric arithmetic (GA) index for these graphsItem Topological Indices on Line Graphs of Titania Nanotube ð ‘‡ð ‘ ð ‘‡3 and Connection Number of Titania Nanotube ð ‘‡ð ‘ ð ‘‡3 Graph(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Kiran Ilyas; FA22-RMT-014; Dr. Imran Zulfiqar Cheema; LHR TP 9344This research attempts to apply topological indices to Titania Nan otube line graphs based on degree to drive the chemical structures of titania nanitube. In recognition of the variety of their applications, the exploration of graphs through chemicals has captivated the at tention of multiple scholars worldwide. For these chemical graphs, we calculate a number of neighborhood degree based topological in dices, including first zagreb index, second zagreb index, hyper zagreb index, redefined third zagreb index, general sum-connectivity in dex, general randic-connectivity index and their polynomials. These topological indices involved with the drug’s molecular structure and related biological, medicinal, pharmacological, and chemical attributes. We are able to develop a mathematical model utilising these neigh bourhood degree-based topological indices, and then we can calcu late physical characteristics from the structure of the modItem Fuzzy Graph Based on Einstein Operator(Library Information Services, CUI Lahore, 2022) Hafiza Sobia Ishfaq; FA20-RMT-049; Dr. Imran Zulfiqar CheemaA fuzzy graph is useful in representing structures of relationships between items where the existence of a concrete item (vertex) and relationship between two items (edge) are matters of degree. In this thesis, the fuzzy graph based on Einstein operator is introduced. Using the Einstein sum and product operator, we will introduce fuzzy variants of many basic graph-theoretical notions. Furthermore, we recognise such results in the context of Intuitionistic fuzzy graphs based on Einstein operators that preserve strong property.Item Topological Indices on Line Graphs of Multi Wheel Graphs and Multi Prism Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Zeeshan Anwar; FA19-RMT-078; LHR TP 7413; Dr. Imran Zulfiqar CheemaTopological index is a numerical quantity which characterizes the whole structure of a graph. These indices are invariants under the graph isomorphism which characterizes its topology and are usually hold mathematical properties of graph invariant. Related to chemistry, actually it is a numerical quantity linked with chemical structure express for the relationship of chemical network with various physical-chemical characteristics like temperature, boiling points and path number etc. Actually these topological indices are formulated by transforming a chemical structure into a numeric number. In the area of research graph theory has been found to be very beneficial. In this thesis, we will calculate the degree based topological indices so-called Randic index, atom bond connectivity (ABC) index, geometric arithmetic(GA) index, first and second Zagreb, second modified Zagreb, third Zagreb, hyper Zagreb, Harmonic index, F-index, Shigehalli and Kanubar index for line graph of multi-wheel graph 𝑊𝑚 = (𝐶𝑛 × 𝑃𝑚) where 𝑚 ≥ 2 as well as line graph of multi-prism graph 𝑌𝑚,𝑛 = (𝐶𝑚 × 𝑃𝑛) where 𝑛 ≥ 4. We will also calculate degree sum based topological indices ABC₄ index known as the forth version of atom bond connectivity (ABC) index and GA₅ index which is the fifth class of geometric arithmetic (GA) index for these graphs. This study will give us a very interesting and different results by using new topological index that is first Zagreb and second Zagreb, second modified Zagreb, third Zagreb, hyper and augmented Zagreb topological indices.Item Weighted Mean Product of Certain Fuzzy Graph Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Tariq Mehmood; FA19-RMT-044; LHR TP 7353; Dr. Imran Zulfiqar CheemaFuzzy set theory and fuzzy logics are examples of fuzzy mathematics. It began in 1965, when Lotfi Asker Zadeh's important work Fuzzy sets was issued. Fuzzy Graph theory is becoming very popular these days, so it is used in many major fields. In this thesis, by means of weighted mean fuzzy transitivity here also I want to define for understanding (weighted mean transitivity) Weighted mean transitivity is defined as: μ(u,v) ≤ λ(σ(u)∨σ(v))+(1-λ)(σ(u)∧σ(v)) u₁,u₂∈ Ri′ where λ is optimized parameter and λ∈[0,1]. Now here we check λ at different values of interval. We discuss the different graph structure at λ=0, 0.5, 1. But here only we discuss λ= 0.5, 1. We calculate the Semi Strong min-product on different structures, degree of a vertex and total degree of a vertex. At the end of thesis we write the application as well as Algorithm.Item Fractional Metric Dimension of Different Chemical Structures(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Muhammad Junaid Rashid; FA18-RMT-055; LHR TP 6459; Dr. Imran Zulfiqar CheemaThe Metric Dimension of a graph G is the minimum number of basis element in a resolving set. Let Q= {q1, q2, q3,…, qk} is the set of vertices of G. The representation r(u/Q) of u with respect to Q is the k-tuple {d(u,q1 ), d(u,q2 ), d(u,q3 ), …, d(u,qk )}, where Q is called a resolving set [4], if every vertex of G is uniquely identified by its distances from the vertices of Q, or equivalently, if distinct vertices of G have distinct representations with respect to Q. Minimum number of vertices in a resolving set is called the basis for G and the number of basis elements is known as the metric dimension of G denoted by dim (G) [10]. In recent years some mathematics work on the concept of resolving sets and metric basis [1, 7, 13]. We have an ordered set of vertices Q ={q1, q2,…, qk } of a graph G, the d(u,qi ) is zero iff u= qi. If r(s/Q) ≠ r(t/Q) for each pair of distinct vertices s,t belongs to V(G)\Q then Q is called a resolving set. The Fractional Metric Dimension of G is defined as dim f(G) = {|g|: g is minimal Resolving function of G} Where |g| = ∑ V∈Vg (v). In the graph theory the most important research study in integer-valued graph theoretic concept of fractionalization. The research is about different chemical structure and convert these structure into planar network graphs. Consequently to calculate the fractional metric dimension of these network graphs. In this thesis, we investigated metric dimension and fractional metric dimension of Line Graph of Polythiophene Network and Backbone Network.Item Topological Indices on k-Subdivision Graphs of Cubic Zirconia Network (CZ)(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Mudassarah Tasneem; FA23-RMT-024; Dr. Imran Zulfiqar Cheema; LHR TP 9770Graph theory has gained extensive application across various dis ciplines, with chemical graph theory emerging as one of the most vibrant areas of research. Graph theoretical approaches have been used in several studies in recent years to study various chemical net works and structures. These representations use atoms as vertices and chemical bonds as edges. In order to understand the physical and chemical characteristics of molecules, this field of study focuses on investigating topological indices. In this work, we examined a number of degree-based topo logical indices, such as the Augmented Zagreb index, the First and Second Zagreb indices, the Atom Bond Connectivity (ABC) index, the Geometric Arithmetic (GA) index, and Hyper Zagreb index. For the cubic zirconia (ZrO2) network in particular, we also took into account revised Zagreb-type indices. These mathematical models could provide useful resources for investigating the cubic zirconia structure’s thermodynamic features, improving our comprehension of its physics and chemical activity.Item Computation of Some Fundamental Topological Indices for the Line Graph of M-Subdivision of Triangular Oxide Network (TOX)(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Majid Siddique; FA23-RMT-020; Dr. Imran Zulfiqar Cheema; LHR TP 9767Afascinating field of study in graph theory is the investigation of topological invariants for the line graph of m-subdivision of Triangular Oxide Network (TOX). This abstract seeks to identify basic traits that can be utilized to differentiate and categorize this structure by examining the intrinsic qualities of this line graph. To understand the complexity and con nectivity of this line graph, topological invariants such the chromatic number, girth, and connectedness are investigated. In order to reveal further structural details, algebraic invari ants like the spectrum and eigenvalues are also investigated. The findings of this study not only advance our knowledge of the Triangular Oxide Network but also have the potential to influence a number of disciplines where accurate characterization of complex systems is crucial, such as computational chemistry, materials science, and network analysis.Item Analysing Some Topological Indices of the Line Graph Derived from the K-subdivision of the Wheel Graph(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Rabi Ijaz Cheema; SP21-BSM-030; Dr. Imran Zulfiqar Cheema; LHR TP 9890Topological indices play a crucial role in the mathematical characterization of chemical structures and complex networks. In this paper, we analyze several topological indices of the line graph derived from the k-subdivision of the wheel graph Wn. The k-subdivision of a graph is obtained by replacing each edge with a path of k edges, transforming the original structure into a more intricate graph. The line graph of this k-subdivision further encodes adjacency relationships between edges, enabling a deeper structural analysis. 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 in dices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices for Analysing some topological indices of the line graph derived from the K-subdivision of the Wheel graph . These indices provide insights into the connectivity and complexity of the graph. Additionally, we analyze the behavior of these indices with respect to the parameter k and the number of vertices n in the original wheel graph. Our f indings reveal trends and relationships between k, n, and the computed indices, shedding light on the impact of k-subdivision on graph-theoretic properties. Theresults presented herein contribute to the broader study of graph invariants and their applications in mathematical chemistry, structural biology, and network theory