On the Zeroth-Order General Randić Index of Unicycle Graphs with $k$ Pendant Vertices

Abstract

Let $G$ be a graph. The zeroth-order general Randić index of a graph is defined as $R_{\alpha }^0(G)=\sum _{v\in V(G)}d(v{)}^{\alpha }(v)$, where $\alpha$ is an arbitrary real number and $d(v)$ is the degree of the vertex $v$ in $G$. In this paper, we give sharp lower and upper bounds for the zeroth-order general Randić index $R_{\alpha }^0(G)$ among all unicycle graphs $G$ with $n$ vertices and $k$ pendant vertices.