Some Topological Invariants of Graphs Associated with the Group of Symmetries

A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reflects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randic index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randic index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.