Bounds and power means for the general Randic index

Abstract

We review bounds for the general Randić index, $R_{\alpha }={\sum }_{ij\in E}(d_i{d}_j{)}^{\alpha }$, and use the power mean inequality to prove, for example, that $R_{\alpha }\ge m{\lambda }^{2\alpha }$ for $\alpha <0$, where $\lambda$ is the spectral radius of a graph. This enables us to strengthen various known lower and upper bounds for $R_{\alpha }$ and to generalise a non-spectral bound due to Bollobás \emph{et al}. We also prove that the zeroth-order general Randić index, $Q_{\alpha }={\sum }_{i\in V}{d}_i^{\alpha }\ge n{\lambda }^{\alpha }$ for $\alpha <0$.

References in corpus (1)

• On comparing Zagreb indices

Cited by in corpus (1)

• Randic index, radius, and diameter for cactus graphs