A study on detour concepts in graphs and their applications

Abstract

Distance is a concept that runs through all of graph theory, and it s employed in isomorphism tests, graph operations, hamiltonicity difficulties, extremal connectivity problems, diameter, and convexity in graphs. The length of the longest u v path in G is the detour distance l(u,v) between two vertices u and v in a connected graph G. A molecular graph, a labelled graph whose vertex and edge labels describe the atom and bond types, can be used to represent the structure of a chemical compound. Topological indices are graph invariants defined to analyze the drug molecular structures and pharmacy characteristics, such as melting point, boiling point, and other biochemical activities. Computational chemistry is a discipline of chemistry that deals with utilizing mathematical methods to compute and model molecular behaviour and properties. The detour index is one of the topological index in the collection of computational chemistry works. newlineIn this thesis we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-Connected (HC) graphs. This improves on previous results by requiring only particular subgraphs of a molecular network to be Hamilton-connected, rather than the entire molecular graph. We have also calculated the detour index of various graph s joins. newlineFinding topological indices of subdivision graph line graphs is a recent research topic of interest. The problem of computing the Detour index of a generic graph is NP-complete. newline