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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G=(V(G),E(G))$\end{document}G=(V(G),E(G)) be a graph. A set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D\subseteq V(G)$\end{document}D⊆V(G) is a distance k-dominating set of G if for every vertex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\in V(G)\setminus D$\end{document}u∈V(G)∖D, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{G}(u,v)\leq k$\end{document}dG(u,v)≤k for some vertex \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v\in D$\end{document}v∈D, where k is a positive integer. The distance k-domination number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma_{k}(G)$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{1}=\sum_{u\in V(G)}d^{2}(u)$\end{document}M1=∑u∈V(G)d2(u) and the second Zagreb index of G is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{2}=\sum_{uv\in E(G)}d(u)d(v)$\end{document}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, 2016). What is worth mentioning, for an n-vertex tree T, is that a sharp upper bound on the distance k-domination number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{k}(T)$\end{document}γk(T) is determined.


Introduction
Throughout this paper, all graphs considered are simple, undirected and connected. Let G = (V , E) be a simple and connected graph, where V = V (G) is the vertex set and E = E(G) is the edge set of G. The eccentricity of v is defined as ε G (v) = max{d G (u, v) | u ∈ V (G)}. The diameter of G is diam(G) = max{ε G (v) | v ∈ V (G)}. A path P is called a diameter path of G if the length of P is diam(G). Denote by N i G (v) the set of vertices with distance i from v in G, that is, The pendent vertex is the vertex of degree 1.
A chemical molecule can be viewed as a graph. In a molecular graph, the vertices represent the atoms of the molecule and the edges are chemical bonds. A topological index of a molecular graph is a mathematical parameter which is used for studying various properties of this molecule. The distance-based topological indices, such as the Wiener index [2,3] and the Balaban index [4], have been extensively researched for many decades. Meanwhile the spectrum-based indices developed rapidly, such as the Estrada index [5], the Kirchhoff index [6] and matching energy [7]. The eccentricity-based topological indices, such as the eccentric distance sum [8], the connective eccentricity index [9] and the adjacent eccentric distance sum [10], were proposed and studied recently. The degree-based topological indices, such as the Randić index [11][12][13], the general sum-connectivity index [14,15], the Zagreb indices [16], the multiplicative Zagreb indices [17,18] and the augmented Zagreb index [19], where the Zagreb indices include the first Zagreb index M 1 = u∈V (G) d 2 (u) and the second Zagreb index M 2 = uv∈E(G) d(u)d (v), represent one kind of the most famous topological indices. In this paper, we continue the work on Zagreb indices. Further study about the Zagreb indices can be found in [20][21][22][23][24][25]. Many researchers are interested in establishing the bounds for the Zagreb indices of graphs and characterizing the extremal graphs [1,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40].
A set D ⊆ V (G) is a dominating set of G if, for any vertex u ∈ V (G) \ D, N G (u) ∩ D = ∅. The domination number γ (G) of G is the minimum cardinality of dominating sets of G.
The distance k-domination number γ k (G) of G is the minimum cardinality among all distance k-dominating sets of G [41,42]. Every vertex in a minimum distance k-dominating set has a private k-neighbor. The domination number is the special case of the distance k-domination number for k = 1. Two famous books [43,44] written by Haynes et al. show us a comprehensive study of domination. The topological indices of graphs with given domination number or domination variations have attracted much attention of researchers [1,[45][46][47].
Borovićanin [1] showed the sharp upper bounds on the Zagreb indices of n-vertex trees with domination number γ and characterized the extremal trees. Motivated by [1], we describe the upper bounds for the Zagreb indices of n-vertex trees with given distance k-domination number and find the extremal trees. Furthermore, a sharp upper bound, in terms of n, k and , on the distance k-domination number γ k (T) for an n-vertex tree T is obtained in this paper.

Lemmas
In this section, we give some lemmas which are helpful to our results. In what follows, we present two graph transformations that increase the Zagreb indices.
Transformation I ( [49]) Let T be an n-vertex tree (n > 3) and e = uv ∈ E(T) be a nonpendent edge. Assume that Tuv = T 1 ∪ T 2 with vertex u ∈ V (T 1 ) and v ∈ V (T 2 ). Let T be the tree obtained by identifying the vertex u of T 1 with vertex v of T 2 and attaching a pendent vertex w to the u (= v) (see Figure 1). For the sake of convenience, we denote T = τ (T, uv).  Proof It is obvious that d T (u) = d T (u) + d T (v) -1 and Let x ∈ V (T) be a vertex different from u and v. Then This completes the proof.  Let G be a connected graph of order n. If γ k (G) ≥ 2, then n ≥ k + 1. Otherwise, γ k (G) = 1, a contradiction. Hence, by Lemma 2.4, we have γ k (G) ≤ n k+1 and n ≥ (k + 1)γ k for any connected graph G of order n if γ k (G) ≥ 2.

Lemma 2.5 Let T be an n-vertex tree with distance k-domination number
Proof Suppose that ≥ nkγ k + 1. 1, . . . , . Let D be a minimum distance k-dominating set of T, k+1 . Since γ k ≥ 2, γ k ≤ n k+1 by Lemma 2.4, a contradiction. Thus, |S 1 | ≥ 1. Let i 1 ∈ S 1 and Hence, all the vertices in i∈S 1 \{i 1 } V (T i ) can be dominated by y ∈ D . Therefore, D is a distance k-dominating set of T, so the claim is true.
In view of a contradiction as desired.
Determining the bound on the distance k-domination number of a connected graph is an attractive problem. In Lemma 2.5, an upper bound for the distance k-domination number of a tree is characterized. Namely, if T is an n-vertex tree with distance k-domination Let T n,k,γ k be the set of all n-vertex trees with distance k-domination number γ k and S n-kγ k +1 be the star of order nkγ k + 1 with pendent vertices v 1 , v 2 , . . . , v n-kγ k . Denote by T n,k,γ k the tree formed from S n-kγ k by attaching a path P k-1 to v 1 and attaching a path  Figure 3. Then T n,k,γ k ∈ T n,k,γ k . Even more . It implies that the upper bound on the distance k-domination number mentioned in the above paragraph is sharp.
The Zagreb indices of T n,k,γ k are computed as and For k = 1, the distance k-domination number γ 1 (G) is the domination number γ (G). Furthermore, the upper bounds on the Zagreb indices of an n-vertex tree with domination number were studied in [1], so we only consider k ≥ 2 in the following.

Lemma 2.6 ([52]) T be a tree on (k + 1)n vertices. Then γ k (T) = n if and only if at least one of the following conditions holds:
(1) T is any tree on k + 1 vertices; (2) T = R • k for some tree R on n ≥ 1 vertices, where R • k is the graph obtained by taking one copy of R and |V (R)| copies of the path P k-1 of length k -1 and then joining the ith vertex of R to exactly one end vertex in the ith copy of P k-1 .

Lemma 2.7 Let T be an n-vertex tree with distance k-domination number
and with equality if and only if T ∼ = T n,k,γ k .
. By the definition of the first Zagreb index, we have The equality holds if and only if R ∼ = S γ k , that is, T ∼ = T n,k,γ k . We have The equality holds if and only if R ∼ = S γ k . As a consequence, T ∼ = T n,k,γ k .

Lemma 2.8 Let G be a graph which has a maximum value of the Zagreb indices among all n-vertex connected graphs with distance k-domination number and S
. , x r are the pendent vertices adjacent to u and y 1 , y 2 , . . . , y t are the pendent vertices adjacent to v, where r ≥ 1 and t ≥ 1. Let D be a minimum distance k-dominating set of G. If In addition, we have either

Main results
In this section, we give upper bounds on the Zagreb indices of a tree with given order n and distance k-domination number γ k . If P = v 0 v 1 · · · v d is a diameter path of an n-vertex tree Let T i n,k,2 be the tree obtained from the path P 2k+2 = v 0 · · · v 2k+1 by joining n -2(k + 1) pendent vertices to v i , where i ∈ {1, . . . , 2k}. Proof Assume that T ∈ T n,k,2 is the tree that maximizes the Zagreb indices and P  Proof Assume that T ∈ T n,k,3 . We complete the proof by induction on n. By Lemma 2.4, we have n ≥ (k + 1)γ k . This lemma is true for n = (k + 1)γ k by Lemma 2.7. Suppose that n > 3(k + 1) and the statement holds for n -1 in the following.
Let D be a minimum distance k-dominating set of T and P = v 0 v 1 · · · v d be a diameter path of T. Then d ≥ 2k + 2. Otherwise, {v k , v k+1 } is a distance k-dominating set, a con- . . , u m are the pendent vertices of T and S T = {u i | 1 ≤ i ≤ m, γ k (Tu i ) = γ k (T)}. We have the following claim. 1 and v d-k are different from each other, a contradiction to γ k (T) = 3. As a consequence, d = 2k + 2 and thus Recalling that m ≤ γ k = 3, we have m = 3, which implies that T k+1 is a path with end vertices v k+1 and u 3 . If d(v k+1 , u 3 ) > k, then u 3 cannot be dominated by the vertices in D. If d(v k+1 , u 3 ) < k, then D \ {v k+1 } is a distance k-dominating set, a contradiction. Therefore, d(v k+1 , u 3 ) = k. We conclude that |V (T)| = 3(k + 1), which contradicts n > 3(k + 1), so Claim 1 is true.
Considering S T = ∅ for T ∈ T n,k,3 , the tree among T n,k,3 that maximizes the Zagreb indices must be in the set {T * ∈ T n,k,3 | |N T * (S T * )| = 1} by Lemma 2.8. To determine the extremal trees among T n,k,3 , we assume that T ∈ {T * ∈ T n,k,3 | |N T * (S T * )| = 1} in what follows.
The equality holds if and only if Tu i ∼ = T n-1,k,γ k and d T (s) = = nkγ k , i.e., T ∼ = T n,k,γ k .
Note that |A| + |B| = n -1d T (s) and |A| ≤ γ k . Therefore, By the above inequality and the definition of M 2 , we have Proof Let T ∈ T n,k,γ k and P = v 0 v 1 · · · v d be a diameter path of T.
then we suppose that T ∈ {T * ∈ T n,k,γ k | |N T * (S T * )| = 1} by Lemma 2.8 for establishing the maximum Zagreb indices of trees among T n,k,γ k .
Let T be the tree obtained from T by applying Transformation I on T i repeatedly for i = 1, . . . , k such that Figure 4). Then T ∈ T n,k,γ k . By Lemma 2.  and Then D is a minimum distance k-dominating set of T and d T (v i ) = 2 for i = 1, . . . , k. Assume that PN k,D (x) is the set of all private k-neighbors of x with respect to D in T . It is clear that the vertices in k By the definition of the first Zagreb index, we get If i 1 = 1, then with equality if and only if m = 0, that is, T ∼ = T . If i 1 = 1 and i 1 = k, then Also, M 2 (T ) -M 2 ( T ) = 0 if and only if at least one of the following conditions holds: If i 1 = 1 and i 1 = k, then As a result, M 2 (T ) -M 2 ( T ) = 0 if and only if at least one of the following conditions holds: (1) m = 0, which implies that T ∼ = T ; In what follows, we prove M 1 (T ) ≤ (nkγ k )(nkγ k + 1) + 4(kγ k -1) and M 2 (T ) ≤ (nkγ k )[n -(k -1)γ k ] + (4k -2)γ k -4 with equality if and only if T ∼ = T n,k,γ k by induction on γ k . The statement is true for γ k = 3 and n ≥ (k + 1)γ k by Lemma 3.3. Assume that γ k ≥ 4, the statement holds for γ k -1 and all the n ≥ (k + 1)(γ k -1).