Sharp bounds on the zeroth-order general Randić index of trees in terms of domination number

The zeroth-order general Randić index of graph $G=(V_G,E_G)$, denoted by ${}^0{R}_{\alpha }(G)$, is the sum of items $(d_v{)}^{\alpha }$ over all vertices $v\in V_G$, where $\alpha$ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on ${}^0{R}_{\alpha }$ of trees with a given domination number $\gamma$, for $\alpha \in (-\mathrm{\infty },0)\cup (1,\mathrm{\infty })$ and $\alpha \in (0,1)$, respectively. The corresponding extremal graphs of these bounds are also characterized.