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Title: Topological Invariants of Molecular Graphs
Authors: Ahmad, Maqsood
Keywords: Physical Sciences
Issue Date: 2021
Publisher: University of Management & Technology, Lahore
Abstract: Chemical graph theory is a topology branch of mathematical chemistry and it deals with the molec ular graph Γ (2D-lattice) of a chemical compound to study and analyze various structural and symmetry properties of the underlying compound. With the rapid development of technology, new pharmaceutical techniques have emerged and as a result a large number of new chemical materials and drugs came in being. Polymers, drugs, and almost all chemical as well as biochemical compounds are often modeled as different ω-cyclic, acyclic, polygonal structures, bipartite, and regular graphs. Topological invariants (indices) are the numeric quantities that are computed from the molecular graph and are highly significant in quantitative structure-property or activity relationship (QSPR, QSAR) modeling which provides theoretical as well as the optimal basis to expensive experimental drug design. The core purpose of this thesis is twofold and the corresponding potential research questions in chemical graph theory are as follows: 1. How to formulate closed-form formulae of several topological invariants of vital importance for molecular graphs of certain chemical compounds, which further partake in the QSPR/QSAR process to extract pharmacological and physico-chemical properties of compounds under dis cussion. 2. What is the lower and the upper bound for a pertinent topological invariant among a particular family of graphs in terms of graph parameters. Also, the characterization of corresponding extremal graphs. We provide some complete and some partial answers to these questions. First, we study three synthetic polymers (macromolecules), namely, vulcanized rubber, bake lite, and poly-methyl methacrylate, which replaced one another as denture base material gradually. The generalized Zagreb index Zr,s(Γ) and M-polynomial M(Γ : x, y) are determined from molec ular graphs of these polymers (networks). Twelve significant topological invariants (TI’s) like the first Zagreb, the second Zagreb, forgotten, re-defined Zagreb, first general Zagreb, general Randi´c, symmetric division degree, modified second Zagreb, inverse Randi´c, harmonic and inverse sum, aug mented Zagreb invariants are derived from the generalized Zagreb index and the M-polynomial. XVIIIXIX Further, atom bond connectivity, its fourth version ABC4, geometric arithmetic, its fifth version GA5, and Sanskruti indices are presented. We extended our study to three closely related (isomeric) natural polymers (polysaccharides) called cellulose, glycogen, and amylopectin. Closed-form formulae of all TIs worked out for synthetic polymers, are also computed for the aforementioned natural polymers. Moreover, a comparative study using computer-based graphs has been made to clarify their nature for these families of networks. TIs such as forgotten index, first generalized Zagreb index, general harmonic index, general sum-connectivity index, Re-defined third Zagreb index (second Gourava index), first Gourava index, multiplicative Zagreb indices, reformulated Zagreb indices, reduced second Zagreb index, and San skruti index of Hex-derived networks of two kinds, namely, HDN1(n) and HDN2(n) are computed. Exact formulae of various significant topological invariants like Atom-bond connectivity index, its fourth version ABC4, geometric arithmetic index, its fifth version GA5 and nine other indices (from M-polynomial) for subdivided hex-derived network SHDN1(n) and SHDN2(n) are computed. Fur thermore, the above-mentioned nine invariants are also recovered from M-polynomial for hexagonal HX(n) and honeycomb HCn networks. The study of the general sum-connectivity index χβ and the general Randi´c index Rα have largely been limited to graph operations. We provide the exact formula to compute the general-sum connectivity index for S-sum graphs. We offered, as far as we aware, tight bounds on χβ in terms of graph parameters and topological invariants of base graphs for F-sum graphs when β ∈ N and F ∈ {R, Q, T}. We computed the lower and the upper bounds on the general Randi´c index Rα for the F-sum graphs where α ∈ N and F ∈ {S, R, Q, T}. We present numerous examples to support and check the reliability, as well as the validity of our bounds. The final aspect of this thesis is concerned with the characterization of the extremal graphs with minimum and maximum forgotten-index in the class of unicyclic graphs with certain pendent vertices. Furthermore, lower and upper bound on forgotten-index in the family of unicyclic graphs is established
Gov't Doc #: 23119
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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