M.Phil / MS

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Derivation of New Class of Topological Indices Through M-Polynomial
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Arooj Asghar; CIIT/SP24-RMT-009/LHR; Dr. Maqsood Ahmad; LHR TP 10071
A topological index (TI) is a numeral quantity computed from the molecule’s graphical model that provides insight into the physical, chemical, bioactive, and structural properties of the associated compound. Chemical graph theory (CGT) is the bridge between graph theory and mathematical chemistry, widely employs this tool to design and analyze existing and new compounds through QSPR/QSAR analyses. The M-Polynomial is a general ex- pression possessing abundant facts about myriad degree-related TIs. Recently, the class of K-Banhatti indices and some famous indices have been extracted from the M-Polynomial. The line graphs of the synthetic denture base polymers, namely bakelite BLst , vulcanite VUst , and polymethyl methacrylate PMst , are novel, intriguing and complex structures. In this work, initially, we aim to calculate the M-Polynomials of the above-described graphs. Using M-Polynomials, we extract several TIs such as the first and second simple, hyper, and modified K-Banhatti indices, respectively. Also, we derive harmonic K-Banhatti , GA, and ABC indices from M-Polynomials of graphs under study. Lastly, we examine the relation among the paraline graphs of bakelite BLst , vulcanite VUst , and polymethyl methacrylate PMst by comparing the computed TIs.
Bounds on Second Hyper Zagreb Index for Four Graph Operations
(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Sumaira Rafique; SP22-RMT-037; Dr. Maqsood Ahmad; LHR TP 8736
In the field of chemical graph theory, a molecular descriptor is a numerical value obtained through a mathematical formula involving vertex degrees, distance, spectrum, and their combination. These values are extracted from the molecular graph of a chemical com pound and are instrumental in the QSPR/QSAR analysis. Operations on the graphs enables us to construct new and valuable molecular graphs of future chemical compounds. A fas cinating area of inquiry in chemical graph theory is determining the upper and lower limits of relevant topological indices within a specific family of graphs. Our project focuses on computing the bounds of the second hyper Zagreb index for four graph operations, and we have included various examples to verify our findings.
The Sigma Index of Four Graph Operations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Humaira Ijaz FA23-RMT-049; Dr. Maqsood Ahmad; LHR TP 9792
that offers information about the structural, physical, bioactive, and chemical characteristics of the related chemical compound. It is used in chemical graph theory (CGT), a field of mathematical chemistry. The sigma index being a variant of Zagreb indices is regarded as the model for degree-based indices, highly influential, and thoroughly studied. The square of the difference of all edges of the molecular graph Γ with weights 􀀀 dΓ(x)−dΓ(y) 2 is the sigma index σ(Γ). These TIs are extracted from the structure Γ regarding chemical substance and are instrumental in the QSPR/QSAR analysis. Graph operations are critical to creating novel and sophisticated graphs which are of general or chemical interest. To build a connection with the topological indices of the base graphs, it is relevant to investigate several graph operations. To build a connection with the topological indices of the base graphs, it is relevant to investigate several graph operations. One of the main issues in extremal and CGT is determining the most precise formulations of the lower bound as well as upper bound for the proper TI over a family of graphs. Gutman [MATCH Commun. Math. Comput. Chem, 2018 79] introduced the sigma index, established its properties, and studied the inverse problem. Eliasi and Taeri [Discrete Applied Mathematics, 2009 157] introduced four novel graph operations called F-sum graphs. In this study, we intend to formulate precise formulas and bounds on the sigma index for the class of F-sum graphs. Furthermore, we use our results on a variety of cases to confirm their validity and correctness

M.Phil / MS

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