The study of topological invariants of graphs
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Date
2020-02
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UMT Lahore
Abstract
Topological indices (TIs) catch symmetry of molecular structures and give it a scientific dialect to foresee properties, for example, boiling points, viscosity, the radius of gyrations and so forth. Drugs and other chemical compounds are often modeled as various polygonal shapes, trees, graphs, etc. In this thesis, three molecular structures namely, Silicon Carbides, Bismuth Tri-Iodie and Dendrimers are discussed. This thesis consist of three parts. In the first part of the thesis, M-polynomials and Zagreb polynomials for aforementioned molecular structures are computed. M-polynomial is rich in information about several degree based TIs, like Zagreb index (ZI), Randić index (RI), etc. By applying basic rules of calculus on M-polynomial, first and second ZIs, modified second ZI, general RI, inverse RI, Symmetric division index, Harmonic index, Inverse sum index and Augmented ZI are recovered. In the second part, redefined first, second and third ZIs, generalized version of first and second ZIs and Geometric arithmetic (GA) index are computed. From these generalized versions, multiple first and second ZIs, multiple first and second hyper ZIs, multiple sum and product connectivity indices and multiple GA index are recovered. Multiple Atomic-bond connectivity (ABC) index, Shigehalli and Kanabur indices, Gourava indices and irregularity indices are also computed. In the third part, some new indices based on the reversed degree of edges are introduced. First and second reversed ZIs, first and second reversed hyper ZIs, reversed Zagreb polynomials are computed while reversed ABC index and inverse GA index are introduced keeping in view the “smoothness property” that helps us to understand the effectiveness of a TI. “Smoothness property” have two axioms, structure sensitivity and abruptness. According to this property a TI is superior if its structure sensitivity is high as possible and abruptness is low as possible. While checking their effectiveness some of the highly investigated TIs have reverse reaction, hence, to overcome this deficiency, reversed indices play an important rule.