Explicit Formulas and Bounds on General Randic Index for Complex Graph Models

Abstract

In chemical graph theory, a topological index is a numeric value extracted from the graphical model of the molecule that provides insight into the structural, physical, bio-active, and chemical properties of the corresponding chemical compound. Randic index is very ´ influential, well investigated, and considered to be the prototype of degree-based indices. General Randic index ´ R γ(Γ) is defined as the sum over all edges of the molecular graph Γ with weights dΓ(x) × dΓ(y) γ when γ is a positive integer. Graph operations play a phenomenal role in obtaining new and complex graphs, either of general or of chemical interest. It is pertinent to explore various graph operations to establish a relationship with the topological indices of the base graphs. In this article, we intend to derive some explicit expressions and bounds on the general Randic index when ´ γ ∈ Z+ for various complex graph operations like the cartesian product, tensor product, lexicographic product, strong product, co-normal product, corona product, splices, links, and Hajos construction of graphs. ´ Besides, we compute the general Randic index by applying the formulas obtained for few ´ small cases to check the validity and reliability of our results.