Sharp bounds for the general Randić index of transformation graphs

Abstract

Inequalities are a useful method to investigate and compare topological indices of graphs relatively. A large collection of graph associated numerical descriptors have been used to examine the whole structure of networks. In these analysis, degree related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree related indices is one of the successful topic of research. Given a molecular graph $\mathcal{H}$ , the general Randić connectivity index is interpreted as ${\mathcal{R}}_{\alpha }\left(\mathcal{H}\right)={\sum }_{ℛ\in \mathsf{E}\left(\mathcal{H}\right)}\left({deg}_{\mathcal{H}}\right(\mathsf{a}\left){deg}_{\mathcal{H}}\right(\mathsf{b}){)}^{\alpha },$ with α is a real quantity. Also a graph transformation of $\mathcal{H}$ provides a comparatively simpler isomorphic structure with an ease to work on different chemical properties. In this article, we determine the sharp bounds of general Randić index of numerous graph transformations, such that semi-total-point, semi-total-line, total and eight individual transformations ${\mathcal{H}}^{\mathit{fgh}}$ , where f, g, h ∈ {+ , -} of graphs by using combinatorial inequalities.