| Summary: | <span>The zeroth-order general Randić index of a graph </span><span class="math inline"><em>G</em></span><span> is defined as </span><span class="math inline"><em>R</em><sub><em>a</em></sub>(<em>G</em>)=∑<sub><em>v</em> ∈ <em>V</em>(<em>G</em>)</sub><em>d</em><sub><em>G</em></sub><sup><em>a</em></sup>(<em>v</em>)</span><span>, where </span><span class="math inline"><em>a</em> ∈ ℝ</span><span>, </span><span class="math inline"><em>V</em>(<em>G</em>)</span><span> is the vertex set of </span><span class="math inline"><em>G</em></span><span> and </span><span class="math inline"><em>d</em><sub><em>G</em></sub>(<em>v</em>)</span><span> is the degree of a vertex </span><span class="math inline"><em>v</em></span><span> in </span><span class="math inline"><em>G</em></span><span>. We obtain bounds on the zeroth-order general Randić index for trees of given order and distance </span><span class="math inline"><em>k</em></span><span>-domination number, where </span><span class="math inline"><em>k</em> ≥ 1</span><span>. Lower bounds are given for </span><span class="math inline">0 < <em>a</em> < 1</span><span> and upper bounds are given for </span><span class="math inline"><em>a</em> < 0</span><span> and </span><span class="math inline"><em>a</em> > 1</span><span>. All the extremal graphs are presented which means that our bounds are the best possible.</span>
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