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

Abstract

Let G=(V(G),E(G)) be a graph. A set Dsubseteq V(G) is a distance k-dominating set of G if for every vertex uin V(G)setminus D, d_{G}(u,v)leq k for some vertex vin D, where k is a positive integer. The distance k-domination number gamma_{k}(G) of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as M_{1}=sum_{uin V(G)}d^{2}(u) and the second Zagreb index of G is M_{2}=sum_{uvin E(G)}d(u)d(v). 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 gamma _{k}(T) is determined.