# On extremal results of multiplicative Zagreb indices of trees with given distance $k$-domination number

## TL;DR

This paper establishes extremal bounds for the first and second multiplicative Zagreb indices of trees with a specified distance k-domination number, and characterizes the trees that attain these bounds.

## Contribution

It provides sharp bounds for Zagreb indices of trees with a given distance k-domination number and characterizes the extremal trees.

## Key findings

- Sharp lower bound for $_1$ of trees with given distance k-domination number.
- Sharp upper bound for $_2$ of trees with given distance k-domination number.
- Characterization of trees attaining the bounds.

## Abstract

The first multiplicative Zagreb index $\Pi_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $\Pi_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $\Pi_1$ and upper bound for $\Pi_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.