Some Bounds on Zeroth-Order General Randić$ Index

Abstract

For a graph $G$ without isolated vertices, the inverse degree of a graph $G$ is defined as $ID(G)={\sum }_{u\in V(G)}d(u{)}^{-1}$ where $d(u)$ is the number of vertices adjacent to the vertex $u$ in $G$. By replacing $-1$ by any non-zero real number we obtain zeroth-order general Randić index, i.e. ${}^0{R}_{\gamma }(G)={\sum }_{u\in V(G)}d(u{)}^{\gamma }$ where $\gamma$ is any non-zero real number. In \cite{xd}, Xu et. al. determined some upper and lower bounds on the inverse degree for a connected graph $G$ in terms of chromatic number, clique number, connectivity, number of cut edges. In this paper, we extend their results and investigate if the same results hold for $\gamma <0$. The corresponding extremal graphs have been also characterized.

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