On Hamilton-connectedness and detour concepts of Generalized Sierpin ´ski Graphs

The structure of a chemical compound can be represented by a molecular graph, a labeled graph whose vertex and edge labels specify the atom and bond types, respectively. 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. Of the numerous topological indices, the detour index has proven to be among the most useful at estimating many biochemical properties of molecules. In this paper 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 upon existing results by loosening the require ment of a molecular graph being Hamilton-connected and only requirement certain sub graphs to be Hamilton-connected. In this paper we prove that the generalized Sierpinski graph of any Hamilton-connected graph is Hamilton connected. Hence we find the Detour index of the same.