Asian Research Thesis Index

Abstract

Molecular topology is an important branch of graph theory which has many appli- cations in di erentelds of life like Pharmacology, Chemistry, Bio and many more. Its objective isnd the topological characterization of molecules in terms of numerical invari- ents named as topological indices. A topological index is very important in application of QSPR/QSAR study. Organic chemistry, nanotechnology and biotechnology also have various applications of these descriptors because of their unique properties. The purpose of this thesis is to compute the characterisation of degrees based on topo- logical indices in certain graphs. The application of Atom bond connectivity index are found in rationalizing the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. Basically, the ABC index will be a ected when edges are inserted and deleted. Therefore, motivated by the works done on this, we continue investigate the inserting and deleting of two and three edges of graph. Then, we obtain the complete solution of derivation of ABC index for inserting and deleting of two and three edges. For the latter part, we also investigate inserting of two and three edges for geometric- arithmetic index and obtain the results, respectively. The result in this thesis also covers computation and study on the the degrees based topological indices for some types of nanostructure such as dendrimers, nanotubes nanotori, nanocones and networks. In the  rst part, we investigate and obtain the novelty formulas for edge version of ABC(e) , GA(e) and second TUC4C6C8[p; q] nanotube, TUC4C8(s)[m; n] nanotube, H-Naphtalenic NPHX[m; n] nanotube,TUC4C6C8[p; q] nanotori, TC4C8(s)[p; q] nanotori and Nanocones CNCk[n] for K3and n0. Then, we calculate analytical closed results for counting related polynomials like Zagreb polynomial for some families of dendrimers, namelyrst, second and third kind of nanostars, (NS1[n], NS2[n] and NS3[n]). We also continue to investigate and obtain novelty formulas for Boron, Boron and line graphs of TiO2 neighbourhood polynomial, fourth atom-bond connectivity andfth geo- metric arithmetic indices. In the next phase, we study and obtain novelty formulas for the augmented Zagreb,rst reformulated Zagreb, connectivity and sum connectivity indices of certain networks like silicate networks, chain silicate networks, hexagonal networks, oxide networks and honeycomb networks. Many new results for degrees based on topo- logical indices in certain graphs we obtained. The thesis culminates by including some open problems for further investigations.

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