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

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.