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

Abstract

Abstract Let G = ( V ( G ) , E ( G ) ) $G=(V(G),E(G))$ be a graph. A set D ⊆ V ( G ) $D\subseteq V(G)$ is a distance k-dominating set of G if for every vertex u ∈ V ( G ) ∖ D $u\in V(G)\setminus D$ , d G ( u , v ) ≤ k $d_{G}(u,v)\leq k$ for some vertex v ∈ D $v\in D$ , where k is a positive integer. The distance k-domination number γ k ( G ) $\gamma_{k}(G)$ 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 = ∑ u ∈ V ( G ) d 2 ( u ) $M_{1}=\sum_{u\in V(G)}d^{2}(u)$ and the second Zagreb index of G is M 2 = ∑ u v ∈ E ( G ) d ( u ) d ( v ) $M_{2}=\sum_{uv\in E(G)}d(u)d(v)$ . 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 γ k ( T ) $\gamma _{k}(T)$ is determined.

Keywords

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