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 $ ^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 given domination number $ \gamma $, for $ \alpha\in(-\infty, 0)\cup(1, \infty) $ and $ \alpha\in(0, 1) $, respectively. The corresponding extremal graphs of these bounds are also characterized.