Abstract
The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.
1. Introduction
Some constants are used to characterize some properties of the graph of a molecule, which are usually called topological indices. One of the most famous topological indices is the Randić index (Randić connectivity index), proposed by Randic [1] in 1975 (for details, see [2, 3]). Soon later, a lot of mathematicians focused on the structure and application of Randić connectivity index. In 1977, Kier and Kall [4] extended the concept of molecular connectivity index and defined the zeroth-order general Randić index. Note that the first Zagreb index is the zeroth-order general Randić index for . For more results of the zeroth-order general Randić index and first Zagreb index, we refer to [5, 6, 7]. In addition, Zagreb indices have been explored as molecular descriptors in QSPR and QSAR (see [8, 9, 10, 11, 12, 13, 14, 15, 16, 17–20]). For a graph G, the first Zagreb index and the second Zagreb index [21] are defined as
For an edge , the edge degree of e is referred as the sum of degrees of its two end vertices minus 2 and is denoted by . indicates the edges e and f are adjacent.
For a given G, let be its line graph. Observe that two edges are adjacent in G if and only if the corresponding two vertices are adjacent in . The edge version of the Zagreb indices [22], motivated by the above property, was proposed by Miličeviv́ et al. in 2004 through the edge degree instead of vertex degree, that is,
The reformulated Zagreb indices, particularly its bounds, have attracted recently the attention of many mathematicians (see, [12, 22–30]).
In order to describe this more clearly in the sequel, we now introduce some notations. Let be the set of connected graphs with pendant vertices. Evidently, if , then there will be a connected subgraph with order for which G can be reconstructed by linking p vertices to some vertices . For convenience, we call as the core of G. Since is connected, it has two extremal cases, i.e., and . Let be the graph with core , and let all pendants of have a common neighbor in . Let be the set of all graphs for which each of its element will be changed to a tree by deleting some of its vertex. That is to say, if G belongs to , then there is a vertex such that is isomorphic to a tree. We call the vertex as the key of G. Note that, for a given graph, its key may not be unique, e.g., G is a cycle, and every vertex is a key of G. Let be the graph with two vertices having degree and other vertices owning degree 2. Obviously, and the two vertices possessing degree are keys.
In this paper, we determined the two upper bounds of reformulated Zagreb indices of two kinds of graphs and characterized completely extremal graphs.
2. Main Results
In the section, we will research the maximal properties regarding the reformulated Zagreb index on and , respectively. Meanwhile, the graphs attaining the bounds are obtained.
Based on the definition of , the following result holds obviously.
Proposition 1. Let G be a connected graph.(i)If , then (ii)If , then
Lemma 1. G and denote the two graphs as shown in Figure 1, and is regarded as the graph from G by shifting all pendants of to . If and , then .
(a)
(b)
Proof. Let G and be the two graphs as shown in Figure 1, and and be two vertices owning pendants and k pendants, respectively. The common neighbors of and are labeled as , and these vertices induce a complete subgraph of G. We write for short. Obviously, , , , and . In order to show , we can confirm that . In fact, we arrive atThe proof hence is complete.
Theorem 1. If , then Furthermore, the above equality is attained only if
Proof. Let G be a graph with n vertices and p pendants and having the maximum with respect to . Let denote the core of G. Clearly, is connected. In fact, . On the contrary, suppose that is not a complete subgraph of G. That is to say, there are some nonadjacent vertex pairs in After connecting these pairs of , we obtain a new graph . Evidently, . From Proposition 1, , which is contradicted with the maximum of G.
We now show that all pendants of G are adjacent to the same vertex in If not, assume that there are two vertices possessing pendants. In other words, We obtain a new graph by removing all pendants of and joining them to . Then, by Lemma 1, . Hence, all pendants in G own a common neighbor. In other words, . In addition, by calculation,Therefore, we complete the proof.
Lemma 2. Let G be a graph with key and let be a tree having an edge with . If is obtained by deleting the branches of and adding them to , then
Proof. Let G and are two graphs with n vertices as shown in Figure 2. Suppose that . It is clear that , and . We now consider the difference and deduce thatTherefore, the result holds.
(a)
(b)
Theorem 2. Let . Then, with equality if and only if
Proof. Let G be a graph in . Hence, there is a vertex such that is a tree with vertices. Assume that G is the maximal graph with respect to . We firstly claim that . Otherwise, if is a tree for , we have . Let the new graph is obtained from G by connecting to all of its nonadjacent vertices in G. Hence, by Proposition 1, , which is contradicted with the choice of G.
We next claim that the tree T is isomorphic to a star with vertices. If not, we find an edge in such that . Suppose now that . From Lemma 2, we will obtain a new graph and is more than , a contradiction. Based on the above discussion, we deduce that . Furthermore, by direct calculation, .
Consequently, the proof is complete.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant nos. 11401348 and 11561032), Shandong Provincial Natural Science Foundation (no. ZR201807061145), and Postdoctoral Science Foundation of China.