a study of vertex-degree function indices via branching operations on trees

‎let $g$ be a graph with vertex set $vleft(g right)$‎. ‎the vertex-degree function index $h_{f}left(g right) $ is defined on $g$ as‎: ‎$h_{f}left(g right) =sum_{uin vleft(g right)}fleft(d_{u} right)‎,$‎where $fleft(x right) $ is a function defined on positive real numbers‎. ‎our main concern in this paper is to study $h_{f}$ over the set $mathcal{t}_{n}$ of trees with $n$ vertices‎, ‎over the set $mathcal{t}_{n,k}$ of trees with $n$ vertices and $k$ branching vertices‎, ‎and over the set $mathcal{t}^{p}_{n}$ of trees with $n$ vertices and $p$ pendant vertices‎. ‎namely‎, ‎we will show in each of these sets of trees that it is possible via branching operations to construct a strictly monotone sequence of trees that reaches the extremal values of $h_{f}$‎, ‎when $fleft( x+1right)-fleft( xright) $ is a strictly increasing function‎.