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.
Explicit Formulas and Bounds on General Randic Index for Complex Graph Models
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Sajjad Asghar; SP23-RMT-021; Dr. Maqsood Ahmad; LHR TP 9586
In chemical graph theory, a topological index is a numeric value extracted from the graph ical model of the molecule that provides insight into the structural, physical, bio-active, and chemical properties of the corresponding chemical compound. Randi´c index is very influential, well investigated, and considered to be the prototype of degree-based indices. General Randi´c index Rγ(Γ) is defined as the sum over all edges of the molecular graph Γ with weights dΓ(x)×dΓ(y) γ when γ is a positive integer. Graph operations play a phenomenal role in obtaining new and complex graphs, either of general or of chemical in terest. It is pertinent to explore various graph operations to establish a relationship with the topological indices of the base graphs. In this article, we intend to derive some explicit ex pressions and bounds on the general Randi´c index when γ ∈ Z+ for various complex graph operations like the cartesian product, tensor product, lexicographic product, strong prod uct, co-normal product, corona product, splices, links, and Haj´ os construction of graphs. Besides, we compute the general Randi´c index by applying the formulas obtained for few small cases to check the validity and reliability of our results.
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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