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Item Computing Zagreb Connection Indices for Line Graphs(UMT Lahore, 2021) Saqib ZafarLet ´P = (V( ´P),E( ´P)) be a graph having vertex set V( ´P) and edge set E( ´P). The combination of chemistry, mathematics and information science leads to a new subject called cheminformatics. It studies the quantitative structure activity and quantitative structure property relationship that are used to predict the biological activities and properties of chemical compounds. In this thesis we introduce different types of topological indices to study the chemical structures including first Zagreb connection index, second Zagreb connection index, modified first Zagreb connection index, modified second Zagreb connection index, modified third Zagreb connection index, generalized fourth Zagreb connection index, generalized fifth Zagreb connection index, first multiplicative Zagreb connection index, second multiplicative Zagreb connection index, modified first multiplicative Zagreb connection index, modified second multiplicative Zagreb connection index and modified third multiplicative Zagreb connection index. We use these connection number based topological indices to study the chemical structures of line for subdivision of ladder, tadpole and wheel L(S(Ln)), L(S(Tn,k)) and L(S(Wn)) networks respectively. Educational note: Sentence case follows the convention of capitalizing only the first word of each sentence and any proper nouns (no proper nouns are present in your text). I preserved the original line-split word fragments (e.g., "quantitative", "modified") as requested—if these were unintended typos (likely from formatting errors), correcting them to complete words (e.g., "quantitative", "modified") would improve readability without altering the core content.Item Degree based topological invariants of operations on graphs.(UMT Lahore, 2021-08-05) Usman AliLet H = (V (H), E(H)) be a graph with vertex set V (H) and edge set E(H) ⊆ V (H) ×V (H). A topological invariant (TI) is a function that associates a numeric value to the underlying graph. TIs are used to predict the physical and chemical properties of the graphs. These are also used in the study of quantitative structures activity relationships (QSAR) and quantitative structures property relationships (QSPR). Gutman and Trinajsti ́c (1972) defined the first degree as well as second degree (connection number) based TIs to calculate the total π-electrone energy of molecules. In the study of hydrocarbons, they also used connection number (number of vertices at distance two) based TI. Recently, connection number based Zagreb indices such as first Zagreb connection index, second Zagreb connection index and modified first Zagreb connection index are widely studied. As per the data provided by International Academy of Mathematical Chemistry, comparatively to the classical Zagreb indices, the chemical capability of the Zagreb connection indices (ZCIs) provides the better absolute values of the correlation coefficients for the thirteen physicochemical properties of octane isomers such as entropy, acentric factor, density, total surface area, molar volume, boiling point, heat capacity at temperature, heat capacity at pressure, enthalpy of vaporization, standard enthalpy of vaporization, enthalpy of formation, standard enthalpy of formation, and octanol water partition. In this thesis, we compute the general results in the form of exact formulae and upper bounds for the Zagreb connection indices/coindices of the resultant graphs which are obtained by applying operations of Cartesian, lexicographic, tensor, strong, corona, disjunction and symmetric difference. To illustrate the obtained results, connection based Zagreb indices are also computed for their chemical structures such as linear polyomino chains, carbon nanotubes, fence, closed fence, alkanes and cycloalkanes. Moreover, the connection based Zagreb indices and their modified version as modified second ZCI (ZC∗ 2 ) and modified third ZCI (ZC∗ 3 ) are also studied for the subdivision-related operations on graphs. Mainly, a comparison among the old/new ZIs of the subdivision-related operations for the particular classes of alkanes is performed. Finally, we conclude that ZC∗ 3 -descriptor has more variability than the other ZIs and it may be more considerable for further investigations of several chemical compounds.Item Algebraic connectivity and fractional metric dimension of graphs.(UMT Lahore, 2021-12-03) Mohsin RazaLet G = (V (G), E(G)) be a graph having V (G) = {vi : 1 ≤ i ≤ n} and E(G) ⊆ V (G) × V (G) as the sets of vertices and edges respectively. A graph Gc is called complement of a graph G with vertex-set V (Gc) = V (G) and edge-set E(Gc) = {uv : u, v ∈ V (G), uv /∈ E(G)}. The number of first neighbors of v ∈ V (G) is defined as degree of the vertex v and it is denoted by d(v) or dG(v). For any two vertices x, y ∈ V (G) the distance (d(x, y)) is the length of shortest path between them. The adjacency matrix (A-matrix) of a graph G of order n is defined as A(G) = [ai,j ]n×n such that ai,j = 1 if vi is adjacent to vj and ai,j = 0 otherwise. The degree matrix (D-matrix) of G is defined by D(G) = [ai,j ]n×n such that ai,i = d(vi) and zero otherwise. The Laplacian matrix (L-matrix) of the graph G is denoted as L(G) and defined as L(G) = D(G) − A(G) where, D(G) and A(G) are degree and adjacency matrices of graph G respectively. For 1 ≤ i ≤ n, the eigenvalues μi = μi(G) and eigenvectors Zi = Zi(G) of L-matrix (L(G)) are the L-eigenvalues and the L-eigenvectors of G respectively. The second smallest eigenvalue of the Laplacian matrix of a graph is known as an algebraic connectivity. It is used as a parameter to measure the connectivity of a graph i.e. how well a graph is connected. Furthermore, any two vertices x, y ∈ V (G) of a simple connected graph G are said to be resolved or distinguished by a vertex z ∈ V (G) if d(x, z) ≠ d(y, z). A set S ⊆ V (G) is called a resolving set of G if each pair of vertices of G is resolved by some vertex in S. A minimum resolving set is known as metric basis and its cardinality is called as metric dimension of the graph G that is denoted by dim(G). For a pair (u, v) of vertices of G, the resolving neighborhood set of G is defined as R(u, v) = {w ∈ V (G) : d(w, u) ≠ d(w, v)}. A resolving function is a real valued function g : V (G) → [0, 1] such that g(R(u, v)) ≥ 1 for each resolving neighborhood of distinct pair of vertices of G, where g(R(u, v)) = ∑ x∈R(u,v) g(x). A resolving function g is called minimal, if there exists a function f : V (G) → [0, 1] such that f ≤ g and f (v) ≠ g(v) for at least one v ∈ V is not a resolving function of G.Item Zagreb indices of generalized operations on graphs(UMT Lahore, 2021-03-12) Hafiz Muhammad AwaisLet Γ be a graph with vertex set V (Γ) and edge set E(Γ) ⊆ V (Γ) ×V (Γ). A topological index (TI) is a function from A to the set of real numbers (ℜ) that associates each element of A to a unique real number, where A is a collection of finite, simple and undirected graphs. Harry Wiener (1947) defined the first distance-based TI to determine the boiling point of paraffin. In 1972, Gutman and Trinajesti ́c computed total π-electrons energy of a molecular graph by first degree-based TI called by first Zagreb index. Later on, many distance, degree, and connection number based TIs are derived to predict the physicochemical and structural properties of the graphs such as stability, solubility, surface tension, radius of gyrations, critical temperature and density. In this thesis, for k ∈ N (set of counting numbers), four generalized subdivision-related operations (Sk, Rk, Qk, and Tk) of graphs are defined. Then, using these operations and the concept of the Cartesian product of graphs, the generalized F-sum graphs (Γ1+Fk Γ2) are also defined, where Fk ∈ { Sk, Rk, Qk, and Tk } and Γi are connected graphs for i ∈ {1, 2}. Mainly, various Zagreb indices such as the first Zagreb index, forgotten index, first general Zagreb index, hyper-Zagreb index, second Zagreb index, multiplicative Zagreb indices of the generalized F-sum graphs are computed in the form of their factor graphs. In addition, two different metal-organic networks are defined in their general form by increasing the layers of both the organic ligands and metal nodes. The degree and connection based generalized Zagreb indices are obtained of these networks. Moreover, different Zagreb indices are also computed for these metal-organic networks using their generalized Zagreb indices.Item Linear programming optimization model for some extensions of fuzzy sets.(UMT Lahore, 2021-02-25) Muhammad Sarwar SindhuThe linear programming (LP) technique plays a central role in optimization that provides an effective instrument in various applications. The consideration of LP technique has been fascinating in optimization for many decades due to the following foremost reasons: • Various real-life problems can be understood in LP, • LP technique can be applied to large-scale computations, • Depending upon its nature, the LP technique is easy to compute. Multiple criteria decision-making (MCDM) is a tool used by decision-makers (DMs) to choose the superior option from the multiple discordant criteria. In the MCDM process, the weights of the criteria have a lot of influence to choose the most ideal option from the indistinguishable options. Based on the expertise, sometimes, the experts or DMs feel hesitation and incommode to assign the weights to each criterion. Generally, the experts allocate weights to the criteria by their own skills that lead to a biasedness in the analysis of the underlying MCDM which is obvious as they use the specific procedures that make it hard to deal with the big data. The importance of the procedure is thus compromised as it increases the probability of the error and eventually the final outcome (decision) is not trustworthy. The fundamental aim of the contemporary study is address the shortcomings in assigning weights due to hesitation and favoritism.Item Computing edge metric dimension of planar graphs(UMT Lahore, 2021-12-15) Muhammad AhsanLet K = (V (K), E(K)) be a connected graph and x, y ∈ V (K), d(x, y) = min{ length of x − y path } and for e = ab ∈ E(K), d(x, e) = min{d(x, a), d(x, b)}. A vertex x distinguishes two edges e1 and e2 if d(e1, x) ≠ d(e2, x). For an edge e of K and a subset WE = {w1, w2, . . . , wk} of its vertices, the representation of e with respect to WE , denoted by r(e | WE ), is the k-tuple (d(e, w1), d(e, w2), . . . , d(e, wk)). If distinct edges of K have distinct representation with respect to WE , then WE is called an edge metric generator (EMG) for K. An EMG of minimum cardinality is an edge metric basis (EMB) for K, and its cardinality is called edge metric dimension (EMD) of K, denoted by edim(K). In this thesis, the constant EMD in the form of exact and upper bound for the graphs the cycle with chord graph, kayak paddle graph, tadpole graph, the cartesian product of cycle with path graph, the necklace graph, circulant graphs, the prism related graph, toeplitz networks are computed. It is also studied that the flower graph and some prism related graph have unbounded EMD. Further, the study of fault-tolerant edge metric dimension (FEMD) is initiated in this work. An EMG ́WE of K is called fault-tolerant edge metric generator (FEMG) of K if ́WE \ {v} is also an EMG of graph K for every v ∈ ́WE . An FEMG of minimum cardinality is a fault-tolerant edge metric basis (FEMB) for graph K, and its cardinality is called FEMD of K. The FEMD of the path, cycle, complete graph, cycle with chord graph, tadpole graph, and kayak paddle graph was also computed.Item Topological invariants of molecular graphs(UMT Lahore, 2021-05-18) Maqsood AhmadChemical graph theory is a topology branch of mathematical chemistry and it deals with the molecular graph Γ (2D-lattice) of a chemical compound to study and analyze various structural and symmetry properties of the underlying compound. With the rapid development of technology, new pharmaceutical techniques have emerged and as a result a large number of new chemical materials and drugs came in being. Polymers, drugs, and almost all chemical as well as biochemical compounds are often modeled as different ω-cyclic, acyclic, polygonal structures, bipartite, and regular graphs. Topological invariants (indices) are the numeric quantities that are computed from the molecular graph and are highly significant in quantitative structure-property or activity relationship (QSPR, QSAR) modeling which provides theoretical as well as the optimal basis to expensive experimental drug design. The core purpose of this thesis is twofold and the corresponding potential research questions in chemical graph theory are as follows: 1. How to formulate closed-form formulae of several topological invariants of vital importance for molecular graphs of certain chemical compounds, which further partake in the QSPR/QSAR process to extract pharmacological and physico-chemical properties of compounds under discussion. 2. What is the lower and the upper bound for a pertinent topological invariant among a particular family of graphs in terms of graph parameters. Also, the characterization of corresponding extremal graphs. We provide some complete and some partial answers to these questions. First, we study three synthetic polymers (macromolecules), namely, vulcanized rubber, bakelite, and poly-methyl methacrylate, which replaced one another as denture base material gradually. The generalized Zagreb index Zr,s(Γ) and M-polynomial M (Γ : x, y) are determined from molecular graphs of these polymers (networks). Twelve significant topological invariants (TI’s) like the first Zagreb, the second Zagreb, forgotten, re-defined Zagreb, first general Zagreb, general Randić, symmetric division degree, modified second Zagreb, inverse Randić, harmonic and inverse sum, augmented Zagreb invariants are derived from the generalized Zagreb index and the M-polynomial.