Browsing by Author "Dr. M. Faisal Nadeem"

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Degree and Distance Based Parameters of Certain Graph Structures Zeshan Saleem Mufti
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) Zeshan Saleem Mufti; FA12-PMATH-006; LHR TP 5950; Dr. M. Faisal Nadeem
Topological indices are a rapidly developing area of research in graph theory. There are various topological indices such as degree-based topological indices, spectrum based topological indices and distance related topological indices. These topological indices correlate certain physico-chemical properties such as boiling point, stability of chemical compounds and many more. In this dissertation, we focus on degree based and spectrum based topological indices. Under the degree based topological indices, we study topological indices such Randic index, general Randic index, Zagreb index, general sum connectivity index, first and second multiple Zagreb indices, geometric index, atom bond connectivity index for para-line graphs of some chemical structures such as Phenylene, Anthracene, Benzenoid, Pentacene. Similarly, we study the spectrum based topological indices such as Estrada index for certain chemical graph structures such as Phenylene, Anthracene, Bi-smuth Tri-iodide, benzene ring embedded in P-type-surface in 2D network (BRE). Also we find the energy of above mentioned structures. We also discuss the notion of distance related parameters such as metric dimension and edge metric dimension. We compute the edge metric dimension for the barcycentric Subdivision of Cayley Graphs. Also we have studied the M-polynomial and entropy of generalized Sierpinski graphs.
M-Polynomias and Degree Base Topological Indices of Strontium Structure
(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Maryam Javed; SP22-RMT-018; Dr. M. Faisal Nadeem; LHR TP 8725
M-Polynomias and Degree Base Topological Indices of Strontium Structure
On Connection Based Topological Descriptors of Magnesium Silicide
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Eisha Afzal; FA22-RMT-009; Dr. M. Faisal Nadeem; LHR TP 9341
Chemical graph theory focuses on the physical features of molecular structures using topo logical descriptors, where topological descriptors are mathematical formulas that repre sents a molecular structure and can be applied to any graph. Chemical graph theory depend extensively on topological descriptors, particularly in relation with chemical features of molecules. According to the structural characteristics of the graphs implemented in their computation, they can be categorised. This thesis focuses on several types of topologi cal descriptors for Magnesium Silicide’s molecular structure. The structure denoted by (Mg2Si), comprises of eight Magnesium Mg and four Silicon Si atoms. The structure and shape of the molecular graph are commonly described by the topological indices. The significance of these indices in simulating the chemical characteristics of the Magnesium Silicide’s network is emphasised in the thesis. Chapter 1 of the thesis includes the introduction, which involves the history of graph the ory, applications, basic definitions and representation of chemical graphs. Chapter 2 includes literature review, some well-known topological descriptors and known results of topological descriptors on different types of graphs. Chapter 3 and 4 includes the construction of chemical structure of Magnesium Silicide, results of degree and connection number-based topological descriptors which are also em phasized using figures and table
On Entropy Measure of Different Families of Fullerenes
(Library Information Services, CUI Lahore, 2023) Hafsa Iqbal; FA21-RMT-006; Dr. M. Faisal Nadeem
A topological index is a numerical value assigned to a graph, which represents speciﬁc topological properties or characteristics of the graph. It serves as a quantitative measure to analyze and compare the structural features of different graphs. Topological indices have extensive applications in the ﬁeld of graph theory, enabling the study and understanding of various graph structures and their properties.Topological indices can be categorized based on three main criteria: degree, distance, and spectrum. These categories help organize the different types of indices and provide a framework for analyzing graph structures.Hosoya entropy is a metric in graph theory that quantiﬁes the complexity of a graph’s struc- ture and the connectivity patterns it exhibits. It is computed using the Hosoya polynomial, which counts the number of distinct walks between any pair of vertices in the graph. The resulting Hosoya entropy value provides a numerical representation of the graph’s topolog- ical characteristics and symmetry properties. In this thesis we compute the Hosoya entropy of different fullerene graphs by understanding the structures of fullerenes and compute dis- tance to partition them in Hosoya classes.
On Equitable Coloring of Certain Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Ahmed Muhammad (CIIT/FA20-BSM-022/LHR), Sohail Abu Haraira (CIIT/SP20-BSM-013/LHR); Dr. M. Faisal Nadeem; LHR TP 9912
In this project, we explore several fundamental concepts in graph theory, focusing primarily equitable edge coloring. An equitable edge coloring of a graph is a proper edge coloring for which the difference between any two color classes is at most one. The minimum cardinality of G for such coloring is called equitable edge chromatic number.
On Radio Labeling of Some Wheel-Related Graphs
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Rimsha Nawaz; FA22-RMT-029; Dr. M. Faisal Nadeem; LHR TP 9388
Graph theory is an important branch of mathematics which has wide range of applications in different areas such as computer science,engineering and communications networks etc. There are different types of labeling defined in graph theory. Our work is at the radio labeling of some wheel-related graphs. First chapter includes the introduction and basic definitions in graph theory which includes study of different types of graphs including wheel graph. Second chapter of our thesis based on types of labeling introduced in the literature. We have discussed the radio labeling and its conditions. We also focused on the known results of some radio labeling of different types of graphs. In chapter three, we have discussed some results of radio labeling of wheel related graphs, we see different bounds on different types of Jahangir graph and prove the radio condition on pair of distinct vertices and illustrate the results with examples.
On Some Degree Based Topological Indices of Tetracyano Benzene Metal Organic Framework
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Aqib Javaid; CIIT/SP21-BSM-028/LHR; Dr. M. Faisal Nadeem; LHR TP 9888
Metal–organic frameworks (MOFs) play a pivotal role in modern materials science due to their highly porous and customizable structures, which are ideal for applications such as gas storage, catalysis, and drug delivery. To model these complex structures, chem- ical graph theory offers a powerful mathematical framework that captures the molecular architecture of MOFs. Within this framework, topological indices—also known as molec- ular descriptors—serve as mathematical formulations derived from the molecular models. These descriptors allow researchers to analyze the physicochemical properties of MOFs without resorting to expensive laboratory experiments, thus streamlining the study of struc- ture–property and structure–activity relationships in mathematical chemistry. In this project, we focus on the tetracyanobenzene-based metal–organic framework, systematically computing and examining its various molecular descriptors. A numerical comparison of these descriptors is provided, offering insights into the framework’s charac- teristics and potential applications.
On the Bounds of the Second Fuzzy Zagreb Index in Fuzzy Graphs
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Talha Ishaq (FA20-BSM-034) : Mehrab Saeed (FA20-BSM-046); Dr. M. Faisal Nadeem; LHR TP 9918
This project explores the fuzzy Zagreb topological indices within the framework of fuzzy graph theory, focusing on their applications and significance in various real-world scenar ios. Fuzzy graph theory, an extension of classical graph theory, accommodates the concepts of uncertainty and imprecision, which are inherent in many practical situations. This study begins by introducing the fundamental concepts of fuzzy sets and fuzzy graphs, as pio neered by Zadeh and Rosenfeld. In the first chapter, we discuss the basic definitions of graph theory. In the second chapter, we go through a history or literature review. In the third chapter, we have discussed the second Zagreb indices and their main results. In the fourth chapter, we have mentioned the conclusion of our project. The research then delves into the specific domain of fuzzy Zagreb indices, including both the first and second indices, and their roles in analyzing and predicting properties of fuzzy graphs. These indices provide a robust mathematical approach for handling complex systems characterized by fuzziness. The project presents detailed definitions, theorems, and proofs related to the second Zagreb index for various fuzzy graph structures, includ ing fuzzy path graphs, cycle graphs, wheel graphs, star graphs, complete bipartite graphs, complete graphs, and Petersen graphs. Asignificant portion of the study is dedicated to calculating the second Zagreb index for different fuzzy graph models, supported by illustrative examples. These examples demon strate the practical application of theoretical concepts, providing a clear understanding of how fuzzy Zagreb indices can be utilized to solve problems in fields such as chemistry, ix pharmaceutical engineering, and network analysis. Overall, this project contributes to the field of fuzzy graph theory by offering com prehensive insights into fuzzy Zagreb indices, their theoretical underpinnings, and their practical applications. The findings highlight the importance of these indices in modeling and analyzing systems where traditional crisp graph approaches are inadequate, thus paving the way for further research and development in fuzzy systems and their applications.
On the Study of Fuzzy Z-Graphs with Structure, Properties, and Potential Applications
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Samrin Akbar; CIIT/SP24-RMT-016/LHR; Dr. M. Faisal Nadeem; LHR TP 10077
Real world networks are frequently described by information that is not only imprecise but also of varying reliability. Classical graphs represent relations crisp, while fuzzy graphs incorporate graded memberships; however, standard fuzzy graph models typically do not distinguish between the imprecision of a statement and the credibility of that statement. Motivated by Zadeh’s Z numbers, this thesis investigates a reliability aware network model called the fuzzy Z-graph. In this framework, vertices and/or edges are labeled by Z-numbers of the form Z = (A,B), where A is a fuzzy restriction describing the primary uncertain value (e.g., strength, cost, risk, or affinity) and B ∈ [0, 1] represents the reliability (confidence) associated with that restriction. We develop a rigorous mathematical foundation for fuzzy Z-graphs by providing pre- cise definitions, consistent notation, and canonical constructions for vertex labeled, edge labeled, and mixed Z-labeling schemes. Suitable aggregation operators and orderings on Z-numbers are introduced to enable comparison and optimization in Z-labeled networks. Fundamental structural properties are then established, including closure behavior under basic graph operations, well-defined Z-degree concepts under admissible aggregation, and path-based calculus with reliability compounding. Algorithmic aspects are also addressed through Z-weight evaluation and scalarization strategies that connect fuzzy Z-graphs to classical combinatorial optimization methods. Finally, the thesis outlines potential applica- tion settings such as risk reliability modeling, decision support, and uncertain infrastructure networks demonstrating how fuzzy Z-graphs offer a principled way to encode both “what is believed” and “how reliably it is believed” within a single graph-theoretic object.
Roman Domination Number of Split Graphs of Certain Graphs
(Library Information Services, CUI Lahore, 2022) Ghulam Mustafa; SP23-RMT-009; Dr. M. Faisal Nadeem
A dominating set D in a graph G = (V; E) is a subset of vertices such that every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of such a set is called the dominating number of G. A Roman dominating function on G is a function f : V ! f0; 1; 2g with the property that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function f is deﬁned as and the minimum possible weight over all Roman dominating functions on G is called the Roman dominating number, denoted by gadjacent to every neighbor of v. In this work, we explore how different graphs relate to their corresponding split graphs S
Secure and Efficient Image Encryption via LSTM-Enhanced Chaotic Systems
(Library Information Services COMSATS Unoiversity Islamabad Lahore Campus, 2025) Shehrooz FA23-RMT-051; Dr. M. Faisal Nadeem; LHR TP 9794
This study aims to utilize a chaos-based color image cryptosystem by applying deep learning. In particular, our method uses a four-dimensional chaotic system, whose increased dimensionality enables greater complexity and randomness, both of which are crucial considerations in image encryption. It employs several analytical techniques to ensure the system is chaotic, including the largest Lyapunov exponent and the 0-1 test for chaos. Later, sequences generated from the four-dimensional chaotic system will be used to train a Long Short-Term Memory (LSTM) network. The LSTM Network will predict four new sequences, which need to be tested for chaos verification and randomness to confirm their applicability in encryption. Secure ciphering of the image is performed by applying two proposed scrambling methods and one diffusion method to a chosen color image using the processed chaotic sequences. Due to the inherent complexity of deep learning and improved properties of chaotic signals, decrypting an image without prior knowledge of system parameters is quite difficult for an attacker. Comprehensive security analysis shows that the proposed encryption method achieves optimal results, such as high entropy ( 7.9978), a large key space analysis of 110840, and great differential attack shown by NPCR ( 99.61%) and UACI ( 33.44%). By this method, pixel correlation is successfully broken, offering a strong defence against various cryptographic threats. These results demonstrate that the proposed algorithm, incorporating deep learning and chaos theory, provides a high level of security to meet the significant requirements of robust color image encryption.
Zero Forcing Number of Optical Transpose Interconnection System Swapped Network
(Library Information Services, CUI Lahore, 2023) Shamza Jabeen; SP20-RMT-040; Dr. M. Faisal Nadeem
nodes is a two-level swapped architecture built by 𝑛 copies of an n-node basis network that constitutes its clusters. We explore the relations between basis network and optical transpose interconnection system (OTIS) swapped networks. We assume that the processor/nodes of the basis network are labeled [𝑛] = 1, . . . , 𝑛, and the processor or node labeled by 〈g, p〉 in OTIS network Ω identifies the node g in cluster p, and this corresponds to node 〈g, p〉 ∈ V(Ω). Subsequently, the cluster address of node 〈g, p〉 will be referred to as g, while the processor address will be referred to as 𝑝. In this thesis, our focus is on finding the zero-forcing number of OTIS swapped networks. Here, we show that the swapped connectivity actually introduces a desirable property that may not exist in the basis network. We construct OTIS swapped networks from the basis networks path, cycle, star, complete, wheel, fan and friendship and then find the zero forcing number of these OTIS swapped networks. We find new auxiliary for zero-forcing number of these graphs. We described that the zero-forcing number will be different for different graphs.