Department of Mathematics
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Item On Topological Aspects of Molecular Graph of Copper Sulfide(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Maryam Shahid; CIIT/SP24-RMT-011/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 10073The use of graph theory to many fields of study has increased significantly, especially in the field of chemical graph theory. Many scholars have been able to explore a range of new directions in the last few years. A chemical graph is a named graph in which the atoms of a compound are represented at the vertices, and the edges represent chemical bonds between atoms. Determining various physical and chemical properties of the molecular structures under study requires an examination of topological indices. Firstly, we focused on degree based topological indices, namely Randic index, atom bond connectivity index, geometric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices, of Copper Sulfide Network CuS.Item Degree Based Topological Co-Indices of Chemical Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Nazir Hussain; CIIT/SP20-RMT-021/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7631Graph theory is playing a key role in applied and pure mathematics. The scope of graph theory is expanding day by day. In this thesis, we have evaluated some degree based topological co-indices, namely, the general Randic co-index for ∝= 1, −1, 1 2 , − 1 2 , the atom bond connectivity co- index, the geometric arithmetic co-index, the first and second Zagreb co-indices, the first and second Zagreb co-indices, the forgotten co-index, the hyper Zagreb co-index, the Balaban co-index, the first and second multiple Zagreb co-indices, the redefined first, second and third Zagreb co-indices for some chemical and lattices graphs like as line graph of Metal Organic Networks, Triangular Benzenoids Gn ⱯϵN, Zigzag edge cornoid fused with starphene zcs(k,l,m), Hony comb network (Hcn), Hydro chloroquine HCQ Conjugated Hydroxyethyl. My thesis comprising of eight chapters. In the first chapter we will define some fundamental definitions. In the 2nd chapter, we will find degree based topological Co- indices for line graph of Metal Organic Networks. In the 3rd chapter, we will find degree based topological Co- indices for Triangular Benzenoids Gn ⱯϵN. In the 4th chapter, we will find degree based topological Co- indices for Zigzag edge cornoid fused with starphene zcs(k,l,m). In the 5th chapter, we will find degree based topological Co- indices for Hony comb network (Hcn). In the 6th chapter, we will find degree based topological Co- indices for Hydro chloroquine HCQ Conjugated Hydroxyethyl. In the 7th chapter, we will present the conclusion of our thesis. In the last, we will present the references of our thesisItem Reverse Degree Based Topological Indices of Graph Products(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Inzimam Sajid; CIIT/SP20-RMT-035/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7638Graph theory is playing a key role in applied and pure mathematics. The scope of graph theory is expanding day by day. In this thesis, we have evaluated some reverse degree based topological indices, namely, the reverse general Randic index for 𝛼 = 1, −1, 1 2 ,− 1 2 the reverse atom bond connectivity index, the reverse geometric arithmetic index, the reverse first and second Zagreb indices, the reverse first and second Zagreb co-indices, the reverse forgotten index, the reverse hyper Zagreb index, the reverse first and second multiple Zagreb indices, the reverse redefined first, second and third Zagreb indices for some reverse degree based topological indices for line graphs, Reverse Indices of various Products of graph. My thesis comprising of six chapters. In the first chapter we will define some fundamental definitions. In the 2nd chapter, we will find reverse degree based topological indices for various products of graphs. In the 3rd chapter, we will find reverse degree based topological indices for 𝐿(𝑀𝑂𝑁1(𝜒)). In the 4th chapter, reverse degree based topological indices for 𝐿(𝑀𝑂𝑁2(𝜒)). In the 5th chapter, we will present the conclusion of our thesis. In the 6th chapter, we will present the references of our thesis.Item Computing Degree Based Topological Indices of Calcium Hydroxide(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Hammad Ali; CIIT/FA20-RMT-036LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7938‘Graph Theory is playing a key role in applied and pure Mathematics. The scope of Graph Theory is expanding day by day. ‘ In this thesis, we computed some topological indices, Co-indices and reverse degree- based indices such as ffirst‘ and ‘second Zagreb‘co-index, first and‘second multiple ‘Zagreb co-index‘forgotton‘topological co-index, atom bond‘connectivity co- index,‘geometric arithmetic co-index , Balaban index and general‘Randic index ‘for‘∝ = 1, −1, 1 2 , − 1 2 ‘for Calcium Hydroxide. My thesis comprising of five chapters. In the first chapter, we will define some basic definitions. In the 2nd chapter, we will find topological indices of Calcium Hydroxide structure. In the 3rd chapter, we will find topological Co-Indices of Calcium Hydroxide structure. In the 4th chapter, we will find reverse degree based topological indices of Calcium Hydroxide structure. In the 5th chapter, we will present the conclusion of our thesis.Item Reverse Degree Based Topological Indices of Chemical Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Tayyab Younas; CIIT/FA19-RMT-064/LHR; Dr. Muhammad Kamran Siddiqui; LHR TP 7428Graph theory is playing a key role in applied and pure mathematics. The scope of graph theory is expanding day by day. In this thesis, we have evaluated some reverse degree based topological indices, namely, the reverse general Randic index for ∝= 1, −1, 1 2 , − 1 2 , the reverse atom bond connectivity index, the reverse geometric arithmetic index, the reverse first and second Zagreb indices, the reverse first and second Zagreb co-indices, the reverse forgotten index, the reverse hyper Zagreb index, the reverse Balaban index, the reverse first and second multiple Zagreb indices, the reverse redefined first, second and third Zagreb indices for some chemical and lattices graphs like as Remdesivir, transition metal tetra cyano benzene organic network (𝑇𝑀 − 𝑇𝐶𝑁𝐵),metal organic superlattice and trans-Pd-(𝑁𝐻2)S lattice. My thesis comprising of five chapters. In the first chapter we will define some fundamental definitions. In the 2nd chapter, we will find reverse degree based topological indices for Remdesivir and transition metal tetra cyano benzene organic network (𝑇𝑀 − 𝑇𝐶𝑁𝐵). In the 3rd chapter, we will find reverse degree based topological indices for metal organic superlattice and trans-Pd-(𝑁𝐻2)S lattice. In the 4th chapter, we will present the conclusion of our thesis. In the 5th chapter, we will present the references of our thesis.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 Physical Analysis of Topological Indices for Palladium Selenide Crystals(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Muhammad Aleem; FA22-RMT-023; Dr. Muhammad Kamran Siddiqui; LHR TP 9336The use of graph theory in many fields of study has increased significantly, especially in the field of chemical graph theory. Many scholars have been able to explore a range of new directions in the last few years. A chemical graph is a named graph in which the atoms of a compound are represented at the vertices, and the edges represent chemical bonds between atoms. Determining various physical and chemical properties of the molecular structures under study requires an examination of topological indices. Firstly, we focused on degree-based topological indices, namely the Randic index, atom bond connectivity in dex, geometric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, for gotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices, of Palladium Selenide Crystals PdSe2. Further, we gave graphical representations of these in dices. These mathematical frameworks might provide a way to study the thermo dynamical properties of the chemical structure of Palladium Selenide Crystals PdSe2.Item On Topological Aspects of Molecular Graph of Iron Telluride(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Hira Ahmed; FA22-RMT-025; Dr. Muhammad Kamran Siddiqui; LHR TP 9337The use of graph theory to many fields of study has increased significantly, especially in the field of chemical graph theory. Many scholars have been able to explore a range of new directions in the last few years. A chemical graph is a named graph in which the atoms of a compound are represented at the vertices, and the edges represent chemical bonds between atoms. Determining various physical and chemical properties of the molecular structures under study requires an examination of topological indices. Firstly, we focused on degree based topological indices, namely Randic index, atom bond connectivity index, geomet ric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices, of Iron Telluride Network FeTe2. Afterwards, we found the physical measures like heat of formation of Iron Telluride Network FeTe2. Then, we fitted curves between different indices and corre sponding heat of formation. Curve fitting was done in MATLAB through different methods based on linearity and non-linearity. The performance of the method was tested using mean squared error (MSE), the sum of squared errors (SSE) or R 2 . Further, we gave graphical representations of these indices. These mathematical frameworks might provide a way to study the thermo dynamical properties of the chemical structure Iron Telluride Network FeTe2.Item On Some Regular Properties of Fuzzy Graph(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Mohsan Majeed; FA19-RMT-027; LHR TP 7388; Dr. Muhammad Kamran SiddiquiGraph theory is playing a key role in applied and pure mathematics. The scope of graph theory is expanding day by day. In this thesis, we study the definition of fuzzy graph, complete fuzzy graph, strong fuzzy graph, regular fuzzy graph, totally regular fuzzy graph with illustrative example and properties of regular and totally regular fuzzy graph. We discuss the concept of uniform vertex and uniform edge fuzzy graph with examples also we check the results related to uniform vertex and uniform edge fuzzy graph. We see that the concept of uniform vertex and uniform edge fuzzy graph is useful in the study of regular and totally regular fuzzy graph. We also see the definition of complement of fuzzy graph and properties of self-complement fuzzy graph and we study some other operations on fuzzy graph such that union, join, composition, direct product, semi strong product and strong product. We read the notion of balanced fuzzy graph. At last we see the fuzzy soft graph, complete fuzzy soft graph, regular fuzzy soft graph, totally regular fuzzy soft graph with examples and some of their important properties. My thesis comprising of seven chapters. In the first chapter we will define of fuzzy graph regular fuzzy graph and totally regular fuzzy graph. In the 2th chapter, we will define uniform vertex fuzzy graph and uniform edge fuzzy graph. In the 3th chapter, we will discuss complement of fuzzy graph. We will check union and join of two fuzzy graphs. In the 4th chapter, we will discuss three more operations on fuzzy graph. In the 5th chapter, we will see the concept of the fuzzy soft graph and various types of fuzzy soft graph with examples. In the 6th chapter, we will present the conclusion of our thesis. In the 7th chapter, we will present the references of our thesis.Item A Comparative Study of Topological Descriptors for Benzyl Sulfamoyl Network(Library Information Services COMSATS University Lahore Campus, 2024-03-14) Ayesha Maqbool; FA22-RMT-043; Dr. Muhammad Kamran Siddiqui; LHR TP 9358The use of graph theory to many fields of study has increased significantly, especially in the field of chemical graph theory. Many scholars have been able to explore a range of new directions in the last few years. A chemical graph is a named graph in which the atoms of a compound are represented at the vertices, and the edges represent chemical bonds between atoms. Determining various physical and chemical properties of the molecular structures under study requires an examination of topological indices. Firstly, we focused on degree based topological indices, namely Randic index, atom bond connectivity index, geomet ric arithmetic index, the first and the second Zagreb indices and co-indices, the first and the second multiple Zagreb indices and co-indices, hyper Zagreb index, forgotten index, augmented Zagreb index, Balaban index, redefined Zagreb type indices, of Benzyl Sul famoyl Network BSb. Afterwards, we found the physical measures like heat of formation of Benzyl Sulfamoyl Network BSb. Then, we fitted curves between different indices and corresponding heat of formation. Curve fitting was done in MATLAB through different methods based on linearity and non-linearity. The performance of the method was tested using mean squared error (MSE), the sum of squared errors (SSE) or R 2 . Further, we gave graphical representations of these indices. These mathematical frameworks might provide a way to study the thermo dynamical properties of the chemical structure Benzyl Sulfamoyl Network BSb.