University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 16 1 2025 03 01 A Study of Vertex-Degree Function Indices via Branching Operations on Trees 1 12 114704 10.22052/ijmc.2024.254896.1865 EN Roberto Cruz Instituto de Matem\'aticas‎, ‎Universidad de Antioquia‎, ‎Medell\'{\i}n‎, ‎Colombia ‎Carlos Espinal Instituto de Matem\'aticas‎, ‎Universidad de Antioquia‎, ‎Medell\'{\i}n‎, ‎Colombia Juan Rada Instituto de Matem\'aticas‎, ‎Universidad de Antioquia‎, ‎Medell\'{\i}n‎, ‎Colombia Journal Article 2024 05 06 ‎Let $G$ be a graph with vertex set $V\left(G \right)$‎. ‎The vertex-degree function index $H_{f}\left(G \right) $ is defined on $G$ as‎: ‎$H_{f}\left(G \right) =\sum_{u\in V\left(G \right)}f\left(d_{u} \right)‎,$<br />‎where $f\left(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 $f\left( x+1\right)-f\left( x\right) $ is a strictly increasing function‎. https://ijmc.kashanu.ac.ir/article_114704_676d009fc053b149539538ad94fdc02c.pdf