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The <i>vertex-degree function index</i> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all <i>n</i>-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mi mathvariant="sans-serif">Γ</mi><mo>)</mo></mrow><mo>·</mo><msub><mi>H</mi><mi>f</mi></msub><mrow><mo>(</mo><mover><mi mathvariant="sans-serif">Γ</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.