Computation of three topological indices on some molecular graphs and families of nanostar dendrimers

Chemical graph theory is a branch of mathematical chemistry which applies graph theory in mathematical modeling of chemical phenomena. One of the most active fields of research in chemical graph theory is the study of topological indices that can be used for describing and predicting physicochemical and pharmacological properties of organic compounds. A topological index is a single unique number characteristic of the molecular graph and is mathematically known as the graph invariant. Eccentric connectivity Index, Zagreb-eccentricity indices and Wiener index are three of the most popular topological indices and used in wide spectrum of applications in chemical graph theory. Motivated by the works done on characterization of mathematical properties for some nanostructures (dendrimers, nanotubes, nanotori, fullerenes etc.), we continue to investigate and obtain novelty formulas of the eccentric connectivity index for unicyclic chemical graph, chemical trees and some families of nanostar dendrimers. Also, we consider novelty formulas of the Zagreb-eccentricity indices for some families of nanostar dendrimers. Finally, novelty formulas for Wiener index of a new class of nanostar dendrimers are considered and new formulas associated with it are determined. In this thesis, we study and analyses the molecular structures and structural properties of chemical compounds with the objective to represent them graphically and construct new classes of graphs. We use mathematical methods of mathematical induction and mathematical logic to arrive at our theorems. In particular, the Eccentric Connectivity Indices ξ (G) are obtained for certain special graphs constructed by joining some special graphs to path graph. Through those graphs constructed are found ξ (G) for graphs associated with some of molecular graphs such as chemical trees, chemical unicyclic graphs and some infinite families of nanostar dendrimers. Also, the Zagreb-eccentricity indices Z(G) are found for some families of chemical trees, chemical unicyclic graphs and some infinite families of nanostar dendrimers. Finally, novel formulas for Wiener index of some dendrimers such as Polyphenelene dendrimers are established. Based on these investigations and graphical analysis novel formulas for the topological indices of these chemical compounds and nanotechnology are then obtained.

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