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

摘要

Let G = (V(G), E(G)) be a graph. A set D ⊆ V(G) is a distance k-dominating set of G if for every vertex u ∈ V(G)∖D, dG(u, v) ≤ k for some vertex v ∈ D, where k is a positive integer. The distance k-domination number γk(G) of G is the minimum cardinality among all distance k-dominating sets of G. The first Zagreb index of G is defined as M1 = ∑u∈V(G)d²(u) and the second Zagreb index of G is M2 = ∑uv∈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, ). What is worth mentioning, for an n-vertex tree T, is that a sharp upper bound on the distance k-domination number γk(T) is determined.