Trees of given order and independence number with extremal general zeroth-order randi\’{c} index

The general zeroth-order Randi\’c index ${^0}R_{\beta}(G)=\sum_{u\in V(G)}(d_{G}(v))^{\beta}$ is a vertex-degree-based graph invariants, where $d_{G}(v)$ denotes the degree of a vertex $v$ in $G$ and $\beta$ is an arith real number, which was defined by Li and Zheng in 2005, under the name “the first general Zagreb index”. In this paper, we explore the general zeroth-order Randi\’c index in terms of independence number. First, we give the upper and lower bounds for the general zeroth-order Randi\’{c} index of trees with given order $n$ and the independence number $\alpha$. Moreover, the corresponding extremal graphs are characterized. For unicyclic graphs, bounds for $^0R_{\beta}(G)$ are also be determined.