Sharp bounds on the zeroth-order general Randi\'c index of trees in terms of domination number

Abstract

The zeroth-order general Randi\'c index of graph $G=(V_G,E_G)$, denoted by $^0R_{\alpha}(G)$, 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 $^0R_{\alpha}$ of trees with a domination number $\gamma$, in intervals $\alpha\in(-\infty,0)\cup(1,\infty)$ and $\alpha\in(0,1)$, respectively. The corresponding extremal graphs of these bounds are also characterized.