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J Inequal Appl. 2018;2018(1):16. doi: 10.1186/s13660-017-1597-3. Epub 2018 Jan 10.

# Extremal values on Zagreb indices of trees with given distance k-domination number.

### Author information

1
School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601 China.

### Abstract

Let [Formula: see text] be a graph. A set [Formula: see text] is a distance k-dominating set of G if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where k is a positive integer. The distance k-domination number [Formula: see text] of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as [Formula: see text] and the second Zagreb index of G is [Formula: see text]. In this paper, we obtain the upper bounds for the Zagreb indices of n-vertex trees with given distance k-domination number and characterize the extremal trees, which generalize the results of Borovićanin and Furtula (Appl. Math. Comput. 276:208-218, 2016). What is worth mentioning, for an n-vertex tree T, is that a sharp upper bound on the distance k-domination number [Formula: see text] is determined.

#### KEYWORDS:

distance k-domination number; first Zagreb index; second Zagreb index; trees

### Conflict of interest statement

The authors declare that they have no competing interests.

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