Extremal Graphs to Vertex Degree Function Index for Convex Functions

The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=∑v∈V(Γ)f(d(v)) for a function f(x) 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 n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for Hf(Γ)+Hf(Γ¯) and Hf(Γ)·Hf(Γ¯).