Department of Mathematics
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Item Analyzing Topological Indices for Phenylacetone Monooxygenase Network Using Curve Fitting Model(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Rimsha Noreen; SP23-RMT-025; Dr. Muhammad Kamran Siddiqui; LHR TP 9568Graph theory has been widely utilized across various fields, with a significant rise in its application in molecular graph theory. In recent years, researchers have explored numerous new directions in this domain. A chemical graph is a labeled graph in which vertices represent atoms in a compound, and edges denote chemical bonds between these atoms. To determine the physical and chemical properties of molecular structures, the study of topological indices is crucial. This work focuses on topological indices, co-indices, and reverse degree-based indices of the Phenylacetone Monooxygenase Network (Pa3Mo). Subsequently, physical proper ties, such as the heat of formation for Pa3Mo are analyzed. Curve fitting techniques were employed to establish relationships between various indices and the corresponding heat of formation. These analyses were conducted using MATLAB, utilizing both linear and non-linear methods. Metrics such as Mean Squared Error (MSE), Sum of Squared Errors (SSE), and the coefficient of determination (R 2 ) were used to evaluate the performance of these methods. Graphical representations of these indices were also provided to aid inter pretation. These mathematical frameworks enable a detailed study of the thermodynamic characteristics of the chemical structure Pa3Mo. Additionally, machine learning techniques, particularly regression models, were ap plied to investigate the relationship between the indices and the corresponding volume of Pa3Mo. Models were developed using data from ten iterations of Pa3Mo, and their perfor mance was evaluated using metrics such as correlation coefficient (R), (R 2 ), and Standard Error (SE).Item On Some Types of Pythagorean Fuzzy Labelings(Library Information Services, COMSATS University, Lahore Campus, 2025) Husnain Ali; Fa23-rmt-013; Dr. Madiha Khalid; LHR TP 9761With the extensive application of networks in every field of science such as engineering, chemistry, biology, and computer networking, the modeling of complex uncertain systems has become more prominent. Graph theory is one of the efficient mathematical methods of representing such systems in which issues of real life are described with the assistance of vertices and edges. Classical graphs are not satisfactory in models involving vagueness and indecision. Fuzzy graphs and their extensions have been developed to address these issues for their handling. Intuitionistic fuzzy graphs (IFGs) on the basis of both membership and non-membership grades provided a more versatile structure to handle uncertainty. However, the condition that the sum of membership and non-membership cannot be more than one puts a limit to their expressiveness. To solve this, Pythagorean fuzzy graphs (PFGs) have been introduced so that the sum of squares of such degrees can be less than or equal to one, thus giving a broader and more precise representation of uncertainty. In This thesis we solves the issue of converting IFGs to PFGsbydevelopingamathematicalframeworkthat preserves the graph structure but enhances its fuzzy representation capability. By using super edge magic labeling we assign unique la bels to the vertices and edges and sum should be constant. The transformation from IFG to PFGis carried out systematically. Additionally, the research explores the consequences of this change in super edge magic labeling, introducing new information on the labeling structures that can be employed on PFGs. The findings prove that PFGs not only generalize IFGs but also are more flexible and effective in graph-based modeling, particularly in scenarios where greater uncertainty and intricate relationships exist.