| Summary: | In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs with a fixed maximum degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Δ</mi></semantics></math></inline-formula> and the maximum and the second maximum general Randić index for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≤</mo><mi>η</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> among all <i>n</i>-vertex unicyclic graphs, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≈</mo><mo>−</mo><mn>0.21</mn></mrow></semantics></math></inline-formula>. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes.
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