On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree

TL;DR

This paper investigates extremal values of two multiplicative Zagreb indices in trees with a fixed number of maximum degree vertices, identifying the extremal structures and their properties.

Contribution

It provides the maximum and minimum values of the indices for such trees and characterizes the extremal graphs, extending understanding of Zagreb indices in tree structures.

Findings

Derived extremal values of multiplicative Zagreb indices for trees.

Characterized the structure of extremal trees with given maximum degree.

Extended existing results to trees with specified maximum degree vertices.

Abstract

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.