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

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.

Introduction

Throughout this paper, all graphs considered are simple, undirected and connected. Let be a simple and connected graph, where is the vertex set and is the edge set of G. The eccentricity of v is defined as . The diameter of G is . A path P is called a diameter path of G if the length of P is . Denote by the set of vertices with distance i from v in G, that is, . In particular, and . A vertex is called a private k-neighbor of u with respect to D if . That is, and for any vertex . 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 [11-13], 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 and the second Zagreb index , 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 [20-25]. Many researchers are interested in establishing the bounds for the Zagreb indices of graphs and characterizing the extremal graphs [1, 26–40].

A set is a dominating set of G if, for any vertex , . The domination number of G is the minimum cardinality of dominating sets of G. For , a set is a distance k-dominating set of G if, for every vertex , for some vertex . The distance k-domination number 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 . 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, 45–47].

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 and Δ, on the distance k-domination number 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.

Lemma 2.1

([24, 48])

If T is an n-vertex tree, different from the star , then for .

In what follows, we present two graph transformations that increase the Zagreb indices.

Transformation I

([49])

Let T be an n-vertex tree () and be a nonpendent edge. Assume that with vertex and . Let be the tree obtained by identifying the vertex u of with vertex v of and attaching a pendent vertex w to the u (=v) (see Figure 1). For the sake of convenience, we denote .

Figure 1

and in Transformation .

Lemma 2.2

Let T be a tree of order n (≥3) and . Then , .

Proof

It is obvious that and Let be a vertex different from u and v. Then This completes the proof. □

Lemma 2.3

([50])

Let u and v be two distinct vertices in G. are the pendent vertices adjacent to u and are the pendent vertices adjacent to v. Define and , as shown in Figure 2. Then either or , .

Figure 2

, and in Lemma  .

Lemma 2.4

([51])

For a connected graph G of order n with , .

Let G be a connected graph of order n. If , then . Otherwise, , a contradiction. Hence, by Lemma 2.4, we have and for any connected graph G of order n if .

Lemma 2.5

Let T be an n-vertex tree with distance k-domination number . Then .

Suppose that . Let be the vertex such that and . Denote by the component of containing the vertex , . Let D be a minimum distance k-dominating set of T, and

Clearly, . If not, is a distance k-dominating set of T, which contradicts . If , then for , so . Therefore, , which implies that . Since , by Lemma 2.4, a contradiction. Thus, . Let and Then , so .

If for some , then is a distance k-dominating set according to the definition of . Thus, we assume that for each . Similarly, suppose that where is a minimum distance k-dominating set of the tree .

We claim that is a distance k-dominating set of T. Let be the vertex such that and . Then and , so, for , we have . Hence, all the vertices in can be dominated by . Therefore, is a distance k-dominating set of T, so the claim is true.

In view of one has 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 number , then .

Let be the set of all n-vertex trees with distance k-domination number and be the star of order with pendent vertices . Denote by the tree formed from by attaching a path to and attaching a path to for each , as shown in Figure 3. Then . Even more noteworthy is the notion that . It implies that the upper bound on the distance k-domination number mentioned in the above paragraph is sharp.

Figure 3

.

.

The Zagreb indices of are computed as and For , the distance k-domination number is the domination number . Furthermore, the upper bounds on the Zagreb indices of an n-vertex tree with domination number were studied in [1], so we only consider in the following.

Lemma 2.6

([52])

T be a tree on vertices. Then if and only if at least one of the following conditions holds:

T is any tree on vertices;

for some tree R on vertices, where is the graph obtained by taking one copy of R and copies of the path of length and then joining the ith vertex of R to exactly one end vertex in the ith copy of .

Lemma 2.7

Let T be an n-vertex tree with distance k-domination number . If , then and with equality if and only if .

When , for some tree R on vertices by Lemma 2.6. Assume that . Then . It is well known that for any n-vertex tree with vertex set . Hence, . By the definition of the first Zagreb index, we have The equality holds if and only if , that is, . We have The equality holds if and only if . As a consequence, . □

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 . If , then .

Suppose that and u and v are two distinct vertices in . are the pendent vertices adjacent to u and are the pendent vertices adjacent to v, where and . Let D be a minimum distance k-dominating set of G. If for some , then is a distance k-dominating set of T. Hence, we assume that , . Similarly, for . Define and . Then . In addition, we have either or , , by a similar proof of Lemma 2.3 and thus omitted here (for reference, see the Appendix). It follows a contradiction, as desired. □

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 . If is a diameter path of an n-vertex tree T, then denote by the component of containing , . By Lemma 2.1, we obtain Theorem 3.1 directly.

Theorem 3.1

Let T be an n-vertex tree and . Then and . The equality holds if and only if .

Let be the tree obtained from the path by joining pendent vertices to , where .

Theorem 3.2

If T is an n-vertex tree with distance k-domination number , then with equality if and only if , where . Also, with equality if and only if , where .

Assume that is the tree that maximizes the Zagreb indices and is a diameter path of T. If , then is a distance k-dominating set of T, a contradiction to . If , define , where . Then . By Lemma 2.2, we have , , a contradiction. Hence, .

If is not a star for some , then there exists an n-vertex tree in such that for by Lemma 2.2, a contradiction. Besides, for some by Lemma 2.3.

Since for and for , we get , . By direct computation, one has , . In addition, and for . Hence, , where . Moreover, . This completes the proof. □

Lemma 3.3

Let tree . Then and with equality if and only if .

Assume that . We complete the proof by induction on n. By Lemma 2.4, we have . This lemma is true for by Lemma 2.7. Suppose that and the statement holds for in the following.

Let D be a minimum distance k-dominating set of T and be a diameter path of T. Then . Otherwise, is a distance k-dominating set, a contradiction. Note that and . Hence, . However, for , so we assume that and . Similarly, and . Suppose that , are the pendent vertices of T and . We have the following claim.

Claim 1

.

Assume that . Namely, for each . If is a minimum distance k-dominating set of the tree , where , then for . Otherwise, or , a contradiction. It follows that .

If for some , then . In view of , we have for , a contradiction. Hence, for .

Since , must be dominated by the vertices in . Bearing in mind that , one has . The same applies to . Hence, . If , then the vertices , , and are different from each other, a contradiction to . As a consequence, and thus .

If , then and is a distance k-dominating set, a contradiction. It follows that . Hence, . Recalling that , we have , which implies that is a path with end vertices and . If , then cannot be dominated by the vertices in D. If , then is a distance k-dominating set, a contradiction. Therefore, . We conclude that , which contradicts , so Claim 1 is true. □

Considering for , the tree among that maximizes the Zagreb indices must be in the set by Lemma 2.8. To determine the extremal trees among , we assume that in what follows.

Let be a pendent vertex such that and s be the unique vertex adjacent to . By Lemma 2.5, . Define and . Then for all . As a consequence, from the proof of Claim 1. By the induction hypothesis, The equality holds if and only if and , i.e., .

Note that and . Therefore, and By the above inequality and the definition of , we have The equality (1) holds if and only if , and for . The equality (2) holds if and only if and . Hence, with equality if and only if . □

Theorem 3.4

Let T be a tree of order n with distance k-domination number (≥3). Then and with equality if and only if .

Let and be a diameter path of T. Define . If , then for . If , then we suppose that by Lemma 2.8 for establishing the maximum Zagreb indices of trees among . If , then , which implies that or . Assume that . Then there is a minimum distance k-dominating set D of T such that from the proof of Lemma 3.3.

Let be the tree obtained from T by applying Transformation I on repeatedly for such that , where is the component of containing , (see Figure 4). Then . By Lemma 2.2, we have , , with equality if and only if .

Figure 4

, , and .

By Lemma 2.3, for some , define and Then one has with equality if and only if and with equality if and only if .

Suppose that , . Let Then D is a minimum distance k-dominating set of and for . Assume that is the set of all private k-neighbors of x with respect to D in . It is clear that the vertices in can be dominated by . Thus, is a distance k-dominating set of tree . In addition, , which means that is a minimum distance k-dominating set of . So . Analogously, .

By the definition of the first Zagreb index, we get so if and only if at least one of the following conditions holds:

, which implies that ;

.

If , then with equality if and only if , that is, . If and , then Also, if and only if at least one of the following conditions holds:

, namely, ;

.

If and , then As a result, if and only if at least one of the following conditions holds:

, which implies that ;

.

In what follows, we prove and with equality if and only if by induction on . The statement is true for and by Lemma 3.3. Assume that , the statement holds for and all the .

In view of and , by the induction hypothesis, we get The equality holds if and only if and . Recalling that for , we have if and only if .

Thus, and if and only if at least one of the following conditions holds:

;

, where . Besides, .

However, the second condition is impossible. If , then and the number of the pendent vertices in is . By the definition of , we have Hence, a contradiction to . Therefore, with equality if and only if .

Note that and . Then The equality holds if and only if and . In consideration of for , the equality holds if and only if .

Hence, if , then , with equality if and only if at least one of the following conditions holds:

;

, where and .

Analogous to the analysis of the first Zagreb index, the second condition above is impossible. Thus, and the equality holds if and only if .

Besides, if , then with equality if and only if immediately. This completes the proof. □

Remark 3.5

Borovićanin and Furtula [1] proved and with equality if and only if , where is the tree obtained from the star by attaching a pendent edge to its pendent vertices. In this paper, we determine the extremal values on the Zagreb indices of trees with distance k-domination number for . Note that the domination number is the special case of the distance k-domination number for and , , , when . Let T be an n-vertex tree with distance k-domination number . Then, by using Theorems 3.1, 3.2 and 3.4 and the results in [1], we have with equality if and only if when , , , when , or when . Moreover, with equality if and only if when and , , , when and , or otherwise.