Extremal vertex-degree function index of trees with some given parameters

Abstract

For a graph $G$, the vertex-degree function index of $G$ is defined as $H_{f}(G)=\sum_{u\in V(G)}f(deg_{G}(u))$,
where $deg_{G}(u)$ stands for the degree of vertex $u$ in $G$ and $f(x)$ is a function defined on positive real numbers. In this article, we determine the extremal values of the vertex-degree function index of trees with given number of pendent vertices/segments/branching vertices/maximum degree vertices and
with a perfect matching when $f(x)$ is strictly convex (resp. concave). Moreover, we use the results directly to some famous topological indices which belong to the type of vertex-degree function index, such as the zeroth-order general Randic index, sum lordeg index, variable sum exdeg index, first and second multiplicative Zagreb indices.