The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=∑v∈V(Γ)f(d(v)) for a f(x) on non-negative real numbers. In this paper, we determine the extremal graphs with maximum (minimum) vertex degree in set of all n-vertex chemical trees, and connected graphs. We also present Nordhaus–Gaddum-type results Hf(Γ)+Hf(Γ¯) Hf(Γ)·Hf(Γ¯).