Department of Mathematics

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Author: search.filters.author.Dr. Imran Zulfiqar CheemaSubject: search.filters.subject.Department of Mathematics

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    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 7649
    When 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.
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    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 7414
    A 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 graphs
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    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 Cheema
    Topological 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.
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    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 Cheema
    Fuzzy 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.
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    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 Cheema
    The 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.