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

Let G be a graph with vertex set V (G). The vertex-degree function index Hf (G) is defined on G as: X Hf(G)= uEV (G) f (du) , where f (x) is a function defined on positive real numbers. Our main concern in this paper is to study Hf over the set Tn of trees with n vertices, over the set Tn,k of trees with n vertices and k branching vertices, and over the set Tnp 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 Hf, when f (x + 1) - f (x) is a strictly increasing function. (c) 2025 University of Kashan Press. All rights reserved.