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Item Dominant and fault-tolerant metric dimensions of graphs(UMT Lahore, 2025-07-04) Imtiaz AliA graph represented by G = (V(G), E(G)) is a collection of vertices V(G) and edges E(G). The distance between any two vertices u, v of a graph G is represented as d(u, v) and is equal to the number of edges of the shortest path connecting u and v. For an ordered set U= {u1 , u2, u3, ⋯⋯ ,uk} ⊆ V(G) and a vertex v ∈ V(G), the representations of v with respect to U are the ordered k-tuple r(v/U) = {d(v , u1) , d(v , u2), d(v , u3), ⋯⋯ ,d(v , uk)}. The set U ⊆ V(G) of a graph G is a resolving set if for all u, v ∈ V(G), then r(u/U) ≠ r(v/U). A metric basis is a resolving set with minimum vertices. The cardinality of a metric basis is called the metric dimension. A set U ⊆ V(G) is a fault-tolerant resolving set if U\v is again a resolving set for each v ∈ U. A fault-tolerant resolving set is called a fault tolerant metric basis if it carries minimum vertices. The cardinality of a fault tolerant metric basis is called fault-tolerant metric dimension. The notation t ~ u implies tu ∈ E(G). The set U ⊆ V(G) is a dominant set if for every vertex t ∈ V(G)\U, there exists a vertex u ∈ U such that there is an edge between t and u means t ~ u. In 2020, the authors combined the dominant set with the resolving set and developed the idea of the dominant metric dimension in graph theory. A resolving set U ⊆ V(G) which is also a dominant set is called the dominant resolving set. A dominant metric basis is a dominant resolving set that contains the minimum vertices. The cardinality of a dominant metric basis is called dominant metric dimension of G. In graph theory metric dimension is a key research area because of its various applications like sensor networking, operation research problems, linear optimization problems, drug discoveries, robot navigation, classification of chemical compounds, source localization, comparing the interconnected networks, detecting network motifs, embedding biological sequence data and in image processing. Dominant and fault tolerant metric dimension problem is just like an optimization problem in which we have to obtain the best solution from all the possible solutions. Now a days, whole industry is changing into automation industry. The need of automatic machines is increasing day by day. As in a restaurant robots are working on the places of waiters. In order to minimize our cost and energy, we need a locating set such that minimum number of automatic machines can move or communicate with each other in a network. Dominant and fault tolerant metric dimension of a graph provide us locating sets, which are helpful for automatic machines. The contradiction method, also known as proof by contradiction, indirect proof, is a common mathematical proof technique that establishes the truth of a statement by assuming the opposite of the statement and showing that the assumption leads to a contradiction. Mathematical induction is a method for proving that a mathematical statement is true for all natural numbers. In order to compute the dominant and fault-tolerant metric dimension of graphs, authors use contradiction and mathematical induction methods. In this thesis, we also use these methods to compute dominant and fault-tolerant metric dimension of graphs. In this research work, we obtain minimum dominant resolving sets of few connected graphs such as wheel, anti-web wheel, gear graphs, generalized anti-web wheel and generalized anti-web gear graphs. We also study the dominant metric dimension of aforesaid graphs. Furthermore, we obtain the fault-tolerant metric dimension of generalized anti-web wheel and generalized anti-web gear graphs. We also study the boundedness of aforementioned connected graphs. Moreover, we classify non bipartite graphs whose metric and dominant metric dimensions are equal. Upon introducing the metric dimension a distance-based parameter, the concept of fault-tolerant metric dimension appeared in the literature. For the dominant metric dimension of graphs introduced by the authors, a fault-tolerant parameter is needed. In this research, we introduce the concept of fault-tolerant dominant metric dimension of graphs and initiate research related to its mathematical properties. We compare fault-tolerant resolving sets with fault-tolerant dominant resolving sets. We present a method to obtain the aforesaid dimension of graphs and expressions for a family of wheel-related graphs.Item Study on fractional metric dimensions of connected graphs(UMT Lahore, 2025-08-07) AROOBA FATIMAConsider a graph N = (V (N), E(N)), where V (N) represents the set of vertices and E(N) ⊆ V (N) × V (N) represents the set of edges. A walk in N is defined as a sequence of alternating vertices and edges. A path is a specific type of walk in which no vertices are repeated, except possibly the first and last vertices. For any graph N = (V (N), E(N)), the distance d(u, v) between two vertices u and v is the length of the shortest path connecting them. A graph N is classified as connected if a path exists between every pair of vertices. For a vertex u ∈ V (N), the degree of u, denoted as d(u), refers to the number of vertices directly adjacent to u. The neighborhood of u is defined as N(u) = {v ∈ V (N) : uv ∈ E(N)}, and the second neighborhood set of u, denoted as N2(u), is given by N2(u) = {v ∈ V (N) : d(u, v) = 2}. The degree sequence of N is the ordered list of vertex degrees in non-increasing order. The independence number α(N) of N is the size of the largest set of vertices in N such that no two vertices in the set are adjacent. Let U = {u1, u2, u3, . . . , um} ⊆ V (N), then a m-tuple metric form of u ∈ V (N) with respect to U is given by r(u|U ) = (d(u, u1), d(u, u2), . . . , d(u, um)). The set U is called a resolving set if every pair of distinct vertices in N has unique metric representation. The smallest resolving set, in term of cardinality, is named as metric basis of N and its cardinality is referred as the metric dimension of N. A vertex w is said to resolve a pair of vertices {u, v} if d(w, u) ̸ = d(w, v). A resolving neighborhood set for a pair of vertices {u, v} ⊆ V (N), denoted by R(u, v), is the set of all vertices that resolve the pair {u, v}, i.e., R(u, v) = {w ∈ V (N) | d(w, u) ̸= d(w, v)}. A function µ : V (N) → [0, 1] is called a resolving function if, for every resolving neighborhood set R(u, v), µ(R(u, v)) = ∑{w∈R(u,v)} µ(w) ≥ 1. A resolving function µ is termed a minimal resolving function if, for any other function φ : V (N) → [0, 1] such that φ ≤ µ and φ (w) ̸ = µ(w) for at least one w ∈ V (N), φ is not the resolving function of N. The fractional metric dimension of a graph N is defined as dimFm(N) = η, where η= min{|µ| : µ is minimal resolving function of N}.Item Study of cosmic acceleration and relativistic stellar(UMT Lahore, 2025-07-01) Muhammad SaleemThis thesis investigates suitable candidates for dark energy by exploring phenomena like cosmic acceleration through cosmological parameters and cosmic coincidence problems by thermodynamical analysis. We consider the fractal universe model with a timelike fractal profile in the flat FRW universe model and check the thermodynamic stability of various dark energy models. These dark energy models include the family of Chaplygin gas models (such as generalized Chaplygin gas, modified Chaplygin gas, generalized cosmic Chaplygin gas, variable modified Chaplygin gas) and parameterized equation of state (Chevallier-Polarski-Linder) model. To check the stability of these models in a fractal framework thermodynamically, we construct a total equation of state parameters for each case. The conditions to check the stability of models depending on some thermodynamic quantities yield three conditions on this parameter. We graphically check the behavior of the equation of state parameter along with the stability of each model. It is concluded that the dark energy models are thermodynamically stable under appropriate choices of model parameters. In the framework of f(P) gravity, we examine the nature of the cosmological parameters by choosing the different models of f(P) gravity at past, present, and future epochs. It is found that the equation of state parameter leads to quintessence behavior and also its ranges lie within Planck data constraints. The square speed of sound leads to instability in the linear f(P) model while giving stable behavior in the non-linear f(P) model. We study the validity of the generalized second law of thermodynamics and observe that it holds while failing for the non-linear f(P) model, respectively. However, the thermal equilibrium condition holds for both f(P) models. Moreover, we use deflection angle formalism to study the thermal stability and phase transitions of 4D charged Einstein-Gauss-Bonnet-AdS black hole in the presence of exponential entropy. We examine the phase structure of the black hole through optical aspects by using the elliptic function analysis. We observe that thermal variation of the deflection angle can be used to obtain stable and unstable phases. We also study the Hawking-Page phase transition from the Gibbs free energy optical dependence using exponential corrected entropy. The particular points of the deflection angle yield transition for large black holes and small black holes. Our Gibbs free energy versus deflection angle behavior confirmed that to obtain the critical behavior of AdS black holes the deflection angle can be utilized as a relevant quantity. Thermodynamic geometry of 4D charged Einstein-Gauss-Bonnet-AdS black hole is also discussed in the presence of corrected exponential entropy.Item Systematic numerical algorithms and analysis of fractional order nonlinear delay epidemic systems.(UMT Lahore, 2025-04-24) Mudassar RafiqueThis research work is particularly concerned with the analysis of the behavior of infectious diseases and measures for their prevention. Diseases play a significant role in endangering human life, and mathematical modelling has emerged to be a very useful tool in managing disease outbreaks. In this context, we have transformed the integer-order models into a time-delayed fractional-order model by using the Caputo fractional differential operator and a delay factor. It offers a general solution at any time t and guarantees that all solutions are positive and bounded as expected in the real-world concerning disease transmission. The research investigates two key equilibrium states, disease-free equilibrium, which refers to the situation when the infection is not present in the population, and the endemic equilibrium, which implies the constant presence of the disease in the community. The quantity R0 is to assess the characteristics of outbreak, it is called the basic reproductive number. This is because the value of R0 is distinguishing in analyzing the stability of equilibria and outcome of the threat or control of the infection. A local and global stability analysis of dynamic model at both equilibrium states are carried out. For solving these delayed fractional-order model, a hybridized finite difference numerical method is proposed to get desired numerical solutions. This method makes it possible to simulate the behavior of the model with a high level of accuracy to identify the dynamics of the disease. The stability and structure of the numerical approach are discussed, and simulated figures are given to explain the biological characteristics of the model and the efficiency of the numerical technique. The results from these simulations are very useful in helping understand how disease control measures can be implemented and the effects of varying parameters based on the spread of the infection. This research can be very valuable in the study of infectious disease modeling as it gives a fresh perspective on how models can be made to be more satisfactory, useful, and helpful in the fight against diseases. Hence, the study goes ahead to establish the importance of R0 in disease control and establishes that fractional-order model analysis is a significant way of capturing the contagious disease transmission complexity. The conclusions made in this study provide a basis for further research on creating highly effective mathematical models for public health and epidemiology. National significance/ linked sdgs This research on "systematic numerical algorithms and fractional-order nonlinear delay epidemic systems" is of immense significance for Pakistan. The fundamental advantage of epidemic modeling with delay and fractional dynamics is that they provide efficient methods for modeling and predicting the spread of various infectious diseases, including dengue, hepatitis, and COVID-19, among other diseases. These diseases present multifaceted risks to the health of the Pakistani population, the country's economy, and the maintenance of society. It brings a new level of realism to policymaking and healthcare planning by employing more advanced numerical methods, allowing for more effective interventions, more efficient use of resources, and less strain on current healthcare infrastructure. This research is relevant to "SDG 3: good health and well-being" since it relates to disease controlling and the improvement of healthcare systems with the help of modeling. It also contributes to "SDG 9: industry, innovation, and infrastructure" since proposals and subsequent advancements in computational and scientific investigations enhance Pakistan's capability in responding to multifaceted issues in the health sector. Additionally, it contributes to "SDG 4: quality education" by stimulating local scholars and scientists to investigate innovative technological fields.Item Dynamics of stellar evolution in modified Gauss-Bonnet gravity(UMT Lahore, 2025-05-14) MUHAMMAD AWAIS SADIQThis thesis investigates the dynamics of stellar evolution by focusing on the implications of modified Gauss-Bonnet gravity as an exotic energy candidate. In this setting, we explore the evolution of axial and spherical stellar systems in context of f(G, T) gravity. First of all, we examine the evolution of dissipative, axially symmetric collapsing fluid in the presence of dark sources. We use the modified Gauss-Bonnet gravity in order to formulate the dynamical variables and investigate the impact of dark sources on heat dissipation and pressure anisotropy. We derive scalar functions through the orthogonal decomposition of the Riemann tensor and assess the physical behavior of these scalars in both matter and dark source configurations. The evolution equations related to dynamical variables, heat transport, the Weyl tensor and the super-Poynting vector are developed, highlighting features such as thermodynamics, density inhomogeneity and gravitational radiations in the presence of exotic terms. These findings reveal that dark source terms influence the system's thermodynamics, kinematical variables' evolution and density inhomogeneity, potentially generating repulsive radiations that induce cosmic expansion. Further we explore the effects of dark sources on the evolution of anisotropic, dissipative and shear-free fluid with axial symmetry within the modified Gauss-Bonnet gravity framework. In this configuration, governing equations and heat transfer equations are derived to investigate shear-free evolution, exploring different fluid models, including dissipative non-geodesic and geodesic fluids. The non-geodesic model requires rotational distribution for a radiating scenario, while the geodesic model is non-radiating and irrotational, resembling the FRW model for positive expansion parameters. In this thesis, we also analyze the dynamical impacts of the f(G, T) gravity model on star clusters by considering spherically symmetric interior geometry with anisotropic fluid. We express the modified field equations by using a specific f(G, T) model, with the observational data of the compact star 4U 1820-30 which we utilize to explore the evolutionary behavior of the stellar. We use scalar functions to find factors causing density irregularities. We also calculate the evolution parameters and investigate structure scalars for dust balls, finding that the Gauss-Bonnet parameter significantly governs the dynamics of star clusters. The expansion-free model of star clusters in modified Gauss-Bonnet gravity is investigated, considering dissipative anisotropic viscous models. We examine field equations, junction conditions and dynamical equations to explain the physical meaning of expansion and shear effects through radial and relative radial velocities. It is deduced that cavities appear in the expansion-free evolution of star clusters, with shear-free and expansion-free collapse determined by the relative velocity between neighboring fluid layers. The Skripin model, corresponding to a non-dissipative, expansion-free isotropic cluster, exhibits homologous evolution in the shear-free case. Lastly, we analyze the evolution of cavities in star clusters within the framework of modified Gauss-Bonnet gravity. For this purpose, we consider the spherically symmetric geometry with locally anisotropic fluid distribution, assuming that the proper radial distance among neighboring stellar components remains unchanged during the purely areal evolution stage. We provide the analytical solutions by using general formalism in f(G, T) gravitation theory, with thick-shell cavities at boundary surfaces satisfying the Darmois conditions. It is concluded that dark matter significantly impacts the evolution of cavities in star clusters.Item Impact of modified theories of gravity on the propagation of gravitational waves.(UMT Lahore, 2025-07-02) ARSAL KAMALThis thesis investigates the fluctuations of GWs in FLRW and Bianchi Type-I universe models within framework of f(R, Tφ) theory. To incorporate GWs into these backgrounds, we employ the axial and polar perturbation techniques established by Regge and Wheeler, simultaneously perturbing the corresponding scalar field as well as its potential in each case. We aim to examine the effect of scalar field on GWs through the modified gravity model. In each scenario, we mainly formulate field equations for unperturbed and perturbed cases. By solving these two sets of field equations simultaneously, along with the perturbed scalar field and its potential, we obtain a set of differential equations involving unknown perturbation parameters. The resulting equations are solved using the separation of variables method, yielding solutions for both the temporal and spatial components. Furthermore, we analyze the perturbation parameters graphically by varying the model parameter λ. The graphical analysis in FRW case as well as Bianchi Type-I scenario, revealed that the scalar field influences propagation of axial and polar GWs. Our results highlight influence of scalar field on GWs propagation, particularly through variations in model parameter. Graphical analysis observed that how scalar field influences the temporal components of the perturbations. Meanwhile, the spatial part remains unaffected by the scalar field. These findings offer insights into the interaction between scalar fields and GWs, with implications for early universe dynamics and modified gravity theories.