A Study of Vertex-Degree Function Indices via Branching Operations on Trees

Document Type : Research Paper

Authors

Instituto de Matem\'aticas‎, ‎Universidad de Antioquia‎, ‎Medell\'{\i}n‎, ‎Colombia

10.22052/ijmc.2024.254896.1865

Abstract

‎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: ‎$H_{f}\left(G \right) =\sum_{u\in V\left(G \right)}f\left(d_{u} \right)‎,$
‎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.

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References

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Iranian Journal of Mathematical Chemistry
Volume 16, Issue 1
In progress
March 2025
Pages 1-12

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