On extremal multiplicative Zagreb indices of trees with given domination number

TL;DR

This paper investigates the extremal values of multiplicative Zagreb indices in trees with a fixed domination number, providing bounds and characterizations of extremal graphs.

Contribution

It establishes sharp bounds and characterizes extremal trees for multiplicative Zagreb indices given a specific domination number.

Findings

Sharp upper and lower bounds for $ ext{prod}_1$ and $ ext{prod}_2$

Characterization of extremal trees achieving these bounds

Extension of Zagreb index analysis to trees with fixed domination number

Abstract

For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajsti\'c in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$ and $\prod_2(G)$ are given. In addition, the corresponding extreme graphs are charaterized.