Phd

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Dynamical analysis of compact objects in f(R) and f(R, T) theories
(UMT Lahore, 2016-10) Ifra Noureen
The astrophysics and astronomical theories are invigorated largely by the gravitational evolution and instability range explorations of gravitating sources. Gravitational collapse is the fundamental phenomenon to account evolution within galaxies and assemble supergiant structures. This dissertation is based on the explorations regarding dynamical instability of gravitating sources in f(R) and f(RT) theories of gravity. The considered modified gravitational theories provide dark energy substitutes constituting large negative pressure and thought to be responsible for the cosmic speed-up. The dynamical systems are studied by considering spherically and axially symmetric backgrounds with anisotropic matter distribution. The modified eld equations and conservation equations for spherically symmetric dynamical system are constructed in f(RT) gravity. The variations in gravitating system are estimated by implementation of rest order perturbations on dynamical equations. Insertion of perturbed physical quantities derived from perturbed eld equations in perturbed Bianchi identities leads to the evolution equation. The expression for adiabatic index is constructed from evolution equation to investigate the variation in pressure stresses with the given energy density. Moreover, terms lying in Newtonian and post Newtonian eras are identified to establish the corrections to weak eld limit. We have also studied the dynamics of spherically symmetric anisotropic stars under the influence of shear-free condition. The modified eld equations accompanying vanishing shear scalar are obtained. On establishment of evolution equation, it is observed that the ow variables are less constrained in shear-free case and so leads to a wider range of stability. The vi corrections to Newtonian and post Newtonian approximations are estimated as well. The dynamics of spherical stars evolving under expansion-free condition in f(RT) gravity is explored by taking anisotropic matter configuration. The collapse equation is acquired from linearly perturbed dynamical equations. It is concluded that in zero-expansion case, the unequal stresses and density pro le defines instability rage rather than the adiabatic index. However, the physical quantities are constrained to maintain positivity of energy density and stable stellar configuration. Motivating from the incidental deviations from spherical symmetry of gravitating objects, we study the dynamical instability of axially symmetric sources (avoiding reflection and rotation terms about symmetry axis). Furthermore, the evolution equation is settled for both the considered modified theories leading to the instability range of axially symmetric dynamical system in Newtonian and post Newtonian regimes.
Existence and approximation of solutions of differential equations
(UMT, Lahore, 2017) Imran Talib
Existence and approximation of solutions of differential equations
(UMT Lahore, 2017-09-20) Imran Talib
The dynamical study of compact objects in General Relativity
(UMT Lahore, 2018-03-16) Syed Ali Mardan Azmi
In this thesis, we discuss the dynamical stability of charged compact objects with the help of some mathematical models. For this purpose, we have selected three different models of charged compact objects to discuss the possible occurrence of cracking under different conditions. In first selected model, we discuss charged anisotropic compact objects with a linear equation of state. In second model, we study anisotropic charged compact object PSR J1614-2230 in quadratic regime, while in third model, we study charged compact stars corresponding to embedded class one metric with perfect inner fluid distribution. We investigate the impact of electromagnetic field on the stability regions of charged self-gravitating compact objects by using the concept of cracking. For this, we have applied local density perturbation scheme to the hydrostatic equilibrium equation as well as on physical parameters involved in the model. In particular, we have examined the cracking of charged compact objects (a) PSR J1614-2230, PSR J1903+327, Vela X-1, SMC X-1 and Cen X-3 with linear equation of state (b) PSR J1614-2230 with quadratic equation of state (c) Her X-1, PSR 1937+21, PSR J1614-2230, PSR J0348+0432 and RX J1856-37 corresponding to embedded class one metric. We conclude that these objects exhibit cracking and stability regions decreases with the increase of charge. We also extend two conventional polytropic equations of state to generalized polytropic equations of state for spherical and cylindrical symmetries in the context of general relativity. For this purpose, we formulate the general framework to discuss the physical properties of spherical and cylindrical polytropes with charged anisotropic inner fluid distribution under conformally flat condition. We investigate the stability of generalized polytropic models through Tolman-mass and Whittaker formula for spherical and cylindrical symmetries respectively. We also discuss the possible occurrence of cracking in charged anisotropic polytropes developed under the assumption of generalized polytropic equation of state in two different ways (i) by carrying out local density perturbation under conformally flat condition (ii) by parametric perturbations. We conclude that one of the generalized polytropic equations of state results into a physically viable model and cracking appears for a specific range of density and model parameters.
Aggregation operators based on some extension of fuzzy sets.
(UMT Lahore, 2018-03-22) Raja Noshad Jamil
Bonferroni mean (BM) and heronian mean (HM) operators are useful tools for group decision making problems, when arguments are interrelated to each other. In this thesis, we developed some BM and HM based aggregation operators. We defined some aggregation operators for dual hesitant fuzzy (DHF) sets, for instance, we defined dual hesitant fuzzy geometric bonferroni mean (DHFGBM) and different properties of DHFGBM are discussed. Some special cases are also studied in detail for DHFGBM. In addition, dual hesitant fuzzy weighted geometric bonferroni mean (DHFWGBM) and dual hesitant fuzzy chouqet geometric bonferroni mean (DHFCGBM) proposed. We also model a system of fuzzy soft differential equations (FSDEs) to analyze the behavior over the time of an individual depending on their companion’s actions under any particular situation against some decision by the help of BM. Using the ability of BM to capture the interrelationship of arguments, we defined bonferroni fuzzy soft matrix (BFSM) and weighted bonferroni fuzzy soft matrix (WBFSM) for data representation. WBFSM is a decision matrix and provide optimum fuzzy soft constant (OFSC), which is the key element of FSDEs. By utilizing the OFSC, we developed a system of FSDEs to study a dynamical process with nonlinear uncertain and vague data. We presented a novel efficient technique for analyzing the future attitude of people due to their present decisions. To illustrate the practicality and feasibility of proposed technique, an example is also discussed with the help of phase portrait and line graphs. With respect to multiple attribute group decision making problems, in which the value of the attributes are taken in the form of hesitant 2-tuple (H2T) or intuitionistic 2-tuple (I2T) linguistic information are called CW. H2T linguistic arguments are used to evaluate the group decision making problems which have inter-dependent or interactive attributes. Some operational laws are developed for H2T linguistic elements and based on these operational laws hesitant 2-tuple weighted averaging (H2TWA) operator and generalize hesitant 2-tuple averaging (GH2TA) operator are proposed. Combining choquet integral (CI) with H2T linguistic information, defined hesitant 2-tuple correlated averaging (H2TCA) and generalize hesitant 2-tuple correlated averaging (GH2TCA) operators. In the existing literature review, we observed that during aggregation procedure for H2T, more hesitation produces in the resultant element. We targeted this issue and developed a diminishing hesitant 2-tuple averaging operator (DH2TA) operator for H2T linguistic arguments. DH2TA operator work in the way that it’s aggregate all H2T linguistic elements and during the aggregation process it also controls the hesitation in the translation of the resulting aggregated xv xvi linguistic term. We developed a scalar product for H2T linguistic elements and based on the scalar product, a diminishing weighted hesitant 2-tuple averaging operator (DWH2TA) is introduced. Moreover, combining CI with H2T linguistic information, the diminishing choquet hesitant 2-tuple average operator (DCH2TA) operator is defined. Most of existing operational laws in literature for handling the process for CW are not bounded and hence a logical problem comes. We targeted this issue and developed closed operational laws based on Archimedean t-norm and t-conorm. Some aggregation operators intuitionistic 2-tuple linguistic heronian mean (I2THM) and intuitionistic 2-tuple linguistic chouqet heronian mean (I2TCHM) based on these closed operational laws developed and discussed desired properties of the proposed operators. Linkages between industry and university are the significant parts in the entire advancement of any country. To assess university’s reputation for industry, we proposed a fusion approach by using heronian intuitionistic fuzzy analytic hierarchy process (HIF-AHP), fuzzy geometric bonferroni mean (FGHM) operator and 2-tuple fuzzy linguistic elements. In each chapter, we developed some techniques based on proposed operators and demonstrated the validity and feasibility of these techniques by some examples. Educational note: Sentence case formatting prioritizes readability by limiting capitalization to only the first word of each sentence, proper nouns/adjectives (e.g., "Archimedean," derived from the mathematician Archimedes), and acronyms (e.g., BM, DHF). This consistency avoids visual clutter, which is particularly useful for academic texts like the thesis excerpt provided—readers can quickly parse sentence boundaries without distraction from overcapitalized terms. Acronyms are retained in uppercase to preserve their symbolic meaning and avoid confusion with generic terms (e.g., "BM" remains distinct from the common noun "bm").
Multi-criteria decision making techniques based on some extensions of fuzzy set
(UMT Lahore, 2018-07) Shahzad Faizi
Multi-criteria decision making (MCDM) is a common activity in everyday life and the objective is to select the most feasible alternative from a set of given alternatives with the highest level of satisfaction in the presence of multiple, usually conflicting, criteria. Similarly, there are many real-life complex problems, where we need involve wide domain of knowledge which are beyond a single expert. Therefore, it is usually necessary to allocate more than one expert to decision process from different fields, including the education backgrounds, work experience and knowledge structure. Consequently, the multi-criteria group decision making (MCGDM) is also an important tool to deal with human activities and their problems of daily lives. In this thesis, different MCDM/MCGDM techniques are discussed based on some important extensions of fuzzy set. This thesis is comprised of three stages. In the first stage, the MCDM method called the Characteristic Objects Method (COMET) is extended to solve problems for MCGDM with hesitant fuzzy sets (HFSs) and intuitionistic fuzzy sets (IFSs). It is a completely new idea for solving problems of group decision making (GDM) under uncertainty. In this approach, we use L-R-type generalized fuzzy numbers (GFNs) and triangular intuitionistic fuzzy numbers (TIFNs) to get the degree of hesitancy in the form of hesitant fuzzy elements (HFEs) and the degree of membership and non-membership in the form of intuitionistic fuzzy numbers (IFNs) respectively for an alternative under a certain criterion. Therefore, the classical COMET method was adapted to work with GFNs and TIFNs in GDM problems. The proposed extensions are presented in detail, along with the necessary background information. The second stage of the thesis is comprised of three parts. In the first part, an outranking method is constructed using hesitant intuitionitic fuzzy linguistic term sets (HIFLTSs) for ranking alternatives in MCGDM problems based on intuitionistic fuzzy support function (IFSF), intuitionistic fuzzy risk function (IFRF), intuitionistic fuzzy credibility function (IFCF) and the net outranking flow index (NOFI). In the second part, the notion of directional Hausdorff distance between two HIFLTSs has been proposed and used it to formulate ELECTRE-based outranking method in hesitant intuitionistic fuzzy linguistic (HIFL) environment.
Generalization of fractional calculus operators with applications
(UMT Lahore, 2018-07-31) Muhammad Khurshid Azam
Generalized forms of fractional calculus operators (integrals and derivatives) are introduced. Caputo k-fractional derivative and Hadamard Caputo type k-fractional derivative are discussed and their results with some applications are presented. Extensions of Weyl k-fractional integral and Hadamard k-fractional integral are also introduced. Boundedness of the extended Hadamard k-fractional integral in spaces is determined. The generalized k-fractional derivative and generalized Caputo type k-fractional derivative are introduced and their properties and results are found. Finally, the generalized type k-fractional integral (unifying eleven existing fractional integrals) is introduced and its boundedness in spaces is also determined. Further, integral transforms of k-fractional and extended k-fractional operators are found. Proofs of properties including semigroup, commutative and some other results for k-Weyl fractional integral are given. Moreover, some inequalities for k-Weyl fractional integral are discussed and examples are also given to illustrate the results. Relationship between these new generalized forms of fractional calculus operators with the existing fractional operators are discussed by substituting the different values of involved parameters. Integral transforms of new fractional operators and their applications are also given.
Analytical solutions for different motions of differential and rate type fluids with fractional derivatives.
(UMT Lahore, 2018-10-24) Muhammad Bilal Riaz
In this dissertation, we present the analytical studies of some fluid flow models. We analyze the fractional models for the flow of non-Newtonian fluids via classical computational techniques to obtain analytical solutions. This study includes the investigation of the unsteady natural convection flow of Maxwell fluid with fractional derivative over an exponentially accelerated infinite vertical plate. Slip condition, chemical reaction, transverse magnetic field and Newtonian heating effects are also considered using a modern definition of fractional derivative. Moreover, the unsteady flow of Maxwell fluid with non-integer order derivatives through a circular cylinder of infinite length in a rotating frame is studied. The motion of Maxwell fluid is generated by a time dependent torsion applied to the surface of the cylinder. As novelty, the dimensionless governing equation related to the non-trivial shear stress is used and the first exact solution analogous to a ramped shear stress on the surface is obtained. The rotational flow of an Oldroyd-B fluid with fractional derivative induced by an infinite circular cylinder that applies a constant couple stress to the fluid is investigated. It is worth mentioning that the considered problem of Oldroyd-B fluid in the settings of fractional derivatives has not been found in the literature. Some unsteady Couette flows of an Oldroyd-B fluid with non-integer derivative in an annular region of two infinite co-axial circular cylinders are investigated. Flows are due to the motion of the outer cylinder, that rotates about its axis with an arbitrary time dependent velocity while the inner cylinder is held fixed. Finally, the analysis of the second grade fluid with fractional derivative is made. The fluid fills the annulus region between two coaxial cylinders in which one cylinder is at rest while the other experiences time dependent shear stress. In all the flow models, we obtained the exact or semi analytical solutions for the motions with technical relevance. These solutions correspond to some flows in which either velocity or the shear stress is given on the boundary are established for different kinds of rate and differential type fluids. The obtained solutions presented in all the fluid flow models satisfy the imposed initial and boundary conditions. Further, the flow properties and comparison of models with respect to derivative (fractional or ordinary) are highlighted by graphical illustrations.
Development of hybrid metaheuristic for global optimization
(UMT Lahore, 2019-06-13) Javaid Ali
Metaheuristics is a research area that delivers general purpose high quality optimization algorithms, proved effectual in dealing with complex global optimization problems. Success of metaheuristics greatly depends on their aptitude to establish equilibrium between their essential characters: exploration and exploitation. But the advent of No Free Lunch theorems by Wolpert and Macready established a general opinion that all algorithms perform equally when averaged over the whole function space and hence none of them can be claimed to be the best over the entire function space. For this reason, the basic algorithms require essential refinements and enhancements. The main goal of this thesis is twofold: to develop new effective hybrid metaheuristic strategies for solving selected global optimization problems and to analyze the performances of developed hybrid metaheuristics on mathematical benchmark functions and complex real world problems that can be modeled as global optimization problems. Generally, hybridization is carried out by integrating powerful components of different algorithms. The first hybrid metaheuristic proposed in this work is Controlled Showering Optimization (CSO) algorithm which is a combination of Artificial Showering Algorithm and frame based search mechanism. The second proposed hybrid algorithm is Cooperative Multi-Simplex algorithm (CMSA) that is based on collaborative search of multiple simplexes working under the iterations of a Non- Stagnated Nelder-Mead Simplex algorithm (NS-NMSA). The evolvement of the provably convergent variant NS-NMSA is also carried out in this work by identifying and coping the failures and stagnations of standard Nelder-Mead simplex algorithm. Multi-Simplex Imperialist Competitive Algorithm (MS-ICA) is the third hybrid metaheuristic which is designed by embedding NS-NMSA iterations in Imperialist Competitive Algorithm. The fourth hybrid metaheuristic designed in this continuation is obtained by integrating CMSA and Differential Evolution (DE) algorithm. In a specifically constructed computational framework, this hybrid algorithm in collaboration with Padé approximation is named as hybrid Evolutionary Padé Approximation (EPA) scheme.
Hybrid nature-inspired algorithms for engineering design optimization problems
(UMT Lahore, 2019-11) Muhammad Luqman
The focus of this dissertation is on the development of hybrid nature inspired metaheuristics for engineering design optimization problems. In this study, three nature inspired metaheuristics naming Artificial Showering Algorithm (ASHA), Artificial Bee Colony (ABC) algorithm and Differential Evolution (DE) have been considered for improvement and hybridization. We propose several improved as well as novel mixtures of the Nature Inspired Computational (NIC) methods, such as Targeted Showering Optimization (TSO), Radial ABC (RABC), hybrid of ABC and a modified ASHA (ABC-MASH) and Differential Targeted ABC (DTABC) algorithms. The structures and working principles of the proposed algorithms are discussed and analyzed in details. The performance of our proposed hybrid NIC algorithms has been investigated by statistical analysis of their results on nonlinear, unimodal, multi-modal, multi-objective, nonlinear systems in engineering and engineering design optimization problems. The analysis reveals that the proposed hybrid NIC algorithms overcome the deficiencies of individual algorithms and outperform several past hybrid methods on engineering design optimization problems. It has been established through computer simulations and non-parametric analysis of the results that our designed hybrid NIC algorithms are consistent in producing superior optimization results over the standard individual NIC algorithms as well as the past hybrid methods with respect to the exploration efficiency, speed of convergence and quality and quantity of the best and mean optimal solutions attained.
The study of topological invariants of graphs
(UMT Lahore, 2020-02) Abaid ur Rehman Virk
Topological indices (TIs) catch symmetry of molecular structures and give it a scientific dialect to foresee properties, for example, boiling points, viscosity, the radius of gyrations and so forth. Drugs and other chemical compounds are often modeled as various polygonal shapes, trees, graphs, etc. In this thesis, three molecular structures namely, Silicon Carbides, Bismuth Tri-Iodie and Dendrimers are discussed. This thesis consist of three parts. In the first part of the thesis, M-polynomials and Zagreb polynomials for aforementioned molecular structures are computed. M-polynomial is rich in information about several degree based TIs, like Zagreb index (ZI), Randić index (RI), etc. By applying basic rules of calculus on M-polynomial, first and second ZIs, modified second ZI, general RI, inverse RI, Symmetric division index, Harmonic index, Inverse sum index and Augmented ZI are recovered. In the second part, redefined first, second and third ZIs, generalized version of first and second ZIs and Geometric arithmetic (GA) index are computed. From these generalized versions, multiple first and second ZIs, multiple first and second hyper ZIs, multiple sum and product connectivity indices and multiple GA index are recovered. Multiple Atomic-bond connectivity (ABC) index, Shigehalli and Kanabur indices, Gourava indices and irregularity indices are also computed. In the third part, some new indices based on the reversed degree of edges are introduced. First and second reversed ZIs, first and second reversed hyper ZIs, reversed Zagreb polynomials are computed while reversed ABC index and inverse GA index are introduced keeping in view the “smoothness property” that helps us to understand the effectiveness of a TI. “Smoothness property” have two axioms, structure sensitivity and abruptness. According to this property a TI is superior if its structure sensitivity is high as possible and abruptness is low as possible. While checking their effectiveness some of the highly investigated TIs have reverse reaction, hence, to overcome this deficiency, reversed indices play an important rule.
Linear programming optimization model for some extensions of fuzzy sets
(UMT Lahore, 2020-02-25) Muhammad Sarwar Sindhu
The 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.
Spatio-temporal numerical modeling of continuous dynamical system
(UMT Lahore, 2020-08-27) Nauman Ahmed
The main objective of this study is to design some novel and efficient structure preserving numerical schemes for spatio-temporal continuous dynamical systems. The designed numerical schemes along with well-known classical finite difference methods are applied to four different reaction-diffusion models. The reaction diffusion models considered for this work are extended in one, two and three space dimensions. This work is divided into two types of reaction diffusion models: autocatalytic chemical reaction models and infectious disease epidemic models. For chemical reaction models, numerical solution is investigated with the help of proposed and classical finite difference schemes. For epidemic models, the numerical stability, numerical bifurcation value of infection parameter and numerical solution is examined. The state variables involved in the reaction diffusion systems under investigation are the concentrations of chemical substances and population sizes, so it is a basic need for the solution to be positive. Also, the continuous systems have stable equilibrium points. Therefore, the proposed numerical schemes are developed in such a way that they preserve all important structures of the continuous systems. Von Neumann stability method and Taylor series expansion are used to analyze the stability and consistency of the proposed schemes respectively. M matrix theory is used for the verification of positivity of proposed implicit schemes. Routh-Hurwitz criteria is used for numerical stability and bifurcation analysis of epidemic models. Several numerical examples are given in this study. The simulations of all finite difference methods under consideration are also presented for the validation of all the attributes.
Enhancement of heat and mass transfer flow of Newtonian and non Newtonian fluids with fractional derivatives
(UMT Lahore, 2020-10) Maryam Aleem
The main aspire of this desertion is to represent the heat and mass transfer mechanism in Newtonian and non-Newtonian fluids flowing through different geometries under different boundary conditions. Analytical solutions of the proposed physical models are obtained by applying integral transform method i.e., Laplace transform method. Enhancement is made with fractional derivatives by including the memory effect i.e., influence of the past on the behavior of the system at present time. Caputo and Caputo-Fabrizio fractional approaches are adapted to analyze the flow behavior of proposed physical problems. Furthermore, additional impacts on flow like MHD, radiation, chemical reaction, heat sink/source, slip effects etc., are considered for different kinds of motions. We also have drawn the comparison between the obtained solutions for temperature, concentration and velocity fields and analyze their flow behavior for fractional and physical parameters like Prandtl number Pr, mass Grashof number Gm, thermal Grashof number Gr, Schmidt number Sc, Renold number Re, magnetic field parameter M and fractional parameters α, β, γ on temperature, concentration and velocity fields have been discussed graphically by using mathematical softwares i.e., Mathematica and Mathcad. Rate of heat and mass transfer in the form of Nusselt and Sherwood numbers respectively and skin friction are also obtained for proposed physical models.
Stochastic algorithms for practical optimization
(UMT Lahore, 2020-12-18) Muhammad Farhan Tabassum
Stochastic is a research area that delivers general purpose high quality optimization algorithms, proved effectual in dealing with complex practical optimization problems. Success of stochastic greatly depends on their aptitude to establish equilibrium between their essential characters like diversification and intensification. But in 1995, no free lunch theorems by Wolpert and Macready established a general opinion that all algorithms perform equally when averaged over the whole function space and hence none of them can be claimed to be the best over the entire function space. For this reason, the basic algorithms require essential refinements and further developments. The main goal of this thesis is to develop new effective hybrid stochastics strategies and then to apply the developed hybrid stochastic algorithms to complex practical problems. Generally, hybridization is carried out by integrating powerful components of different algorithms, possibly of different natures. The first hybrid stochastic algorithm proposed in this work is Evolutionary Simplex Adaptive Hooke-Jeeves (ESAHJ) Algorithm which is a combination of Genetic Algorithm and modified Hooke-Jeeves method. The second proposed hybrid optimization algorithm is based on Differential Evolution (DE), Gradient Evolution (GE) and Jumping Technique named as Differential Gradient Evolution Plus (DGE+). The proposed algorithm hybridizes the above mentioned algorithms with the help of an improvised probability distribution, additionally provides a new shake off method to avoid premature convergence towards local optima. The third hybrid technique is implemented with the collaboration of Differential Evolution with Padé Approximation and named as Evolutionary Padé Approximation (EPA) scheme. The last approach has been developed based on modified TOPSIS named Rank Based TOPSIS (RB-TOPSIS) for the multi-criteria decision making for Congress on Evolutionary Computation 2017 competition.
Computing Zagreb Connection Indices for Line Graphs
(UMT Lahore, 2021) Saqib Zafar
Let ´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.
Linear programming optimization model for some extensions of fuzzy sets.
(UMT Lahore, 2021-02-25) Muhammad Sarwar Sindhu
The 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.
Zagreb indices of generalized operations on graphs
(UMT Lahore, 2021-03-12) Hafiz Muhammad Awais
Let 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.
Topological invariants of molecular graphs
(UMT Lahore, 2021-05-18) Maqsood Ahmad
Chemical 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.
Degree based topological invariants of operations on graphs.
(UMT Lahore, 2021-08-05) Usman Ali
Let 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.

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