Note on the Reformulated Zagreb Indices of Two Classes of Graphs

/e 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 sumof degrees of its two end verticesminus 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.


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 α � 2. 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][18][19][20]). For a graph G, the first Zagreb index M 1 and the second Zagreb index M 2 [21] are defined as For an edge e � uv, the edge degree of e is referred as the sum of degrees of its two end vertices minus 2 and is denoted by d(e) � d(u) + d(v) − 2. e ∼ f indicates the edges e and f are adjacent.
For a given G, let L(G) be its line graph. Observe that two edges are adjacent in G if and only if the corresponding two vertices are adjacent in L(G). e edge version of the Zagreb indices [22], motivated by the above property, was proposed by Miličeviv́et al. in 2004 through the edge degree instead of vertex degree, that is, e reformulated Zagreb indices, particularly its bounds, have attracted recently the attention of many mathematicians (see, [12,[22][23][24][25][26][27][28][29][30]).
In order to describe this more clearly in the sequel, we now introduce some notations. Let G p n be the set of connected graphs with p( ≥ 2) pendant vertices. Evidently, if G ∈ G p n , then there will be a connected subgraph H 0 with order n − p for which G can be reconstructed by linking p vertices to some vertices H 0 . For convenience, we call H 1 as the core of G. Since H 0 is connected, it has two extremal cases, i.e., H 0 � K n− p and H 0 � T n− p . Let A n ∈ G p n be the graph with core K n− p , and let all pendants of A n have a common neighbor in K n− p . Let B n be the set of all graphs for which each of its element will be changed to a tree by deleting some of its vertex. at is to say, if G belongs to B n , then there is a vertex v 0 ∈ V(G) such that G − v 0 is isomorphic to a tree. We call the vertex v 0 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 B n be the graph with two vertices having degree n − 1 and other vertices owning degree 2. Obviously, B n ∈ B n and the two vertices possessing degree n − 1 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.

Main Results
In the section, we will research the maximal properties regarding the reformulated Zagreb index on G p n and B n , respectively. Meanwhile, the graphs attaining the bounds are obtained.
Based on the definition of EM 1 , the following result holds obviously. Figure 1, and G ′ is regarded as the graph from G by shifting all

Lemma 1. G and G ′ denote the two graphs as shown in
Proof. Let G and G ′ be the two graphs as shown in Figure 1, and v s and v t be two vertices owning ℓ pendants and k pendants, respectively. e common neighbors of v s and v t are labeled as u 1 , u 2 , . . . , u n− p− 2 , and these n − p vertices induce a complete subgraph K n− p of G. We write In order to show EM 1 (G ′ ) > EM 1 (G), we can confirm that EM 1 (G ′ ) − EM 1 (G) > 0. In fact, we arrive at e proof hence is complete.
Proof. Let G be a graph with n vertices and p pendants and having the maximum with respect to EM 1 . Let H 0 denote the core of G. Clearly, H 0 is connected. In fact, H 0 � K n− p . On the contrary, suppose that H 0 is not a complete subgraph of G. at is to say, there are some nonadjacent vertex pairs in H 0 . After connecting these pairs of H 0 , we obtain a new graph G 1 . Evidently, G 1 ∈ G p n . From Proposition 1, EM 1 (G 1 ) > EM 1 (G), which is contradicted with the maximum of G.
We now show that all pendants of G are adjacent to the same vertex in H 0 . If not, assume that there are two vertices v s , v t ∈ V(H 0 ) possessing pendants. In other words, We obtain a new graph G ′ ∈ G p n by removing all pendants of v s and joining them to v t . en, by Lemma 1, EM 1 (G ′ ) > EM 1 (G). Hence, all pendants in G own a common neighbor. In other words, G � A n . In addition, by calculation, erefore, we complete the proof. Proof. Let G and G ′ are two graphs with n vertices as shown in Figure 2.
We now consider the difference EM 1 (G ′ ) − EM 1 (G) and deduce that erefore, the result holds. Proof. Let G be a graph in B n . Hence, there is a vertex v 0 ∈ V(G) such that T � G − v 0 is a tree with n − 1 vertices. Assume that G is the maximal graph with respect to EM 1 . We firstly claim that Let the new graph G 1 is obtained from G by connecting v 0 to all of its nonadjacent vertices in G. Hence, by Proposition 1, EM 1 (G 1 ) > EM 1 (G), which is contradicted with the choice of G. We next claim that the tree T is isomorphic to a star with n − 1 vertices. If not, we find an edge e � v s v t in E(T) . From Lemma 2, we will obtain a new graph G 2 ∈ B n and EM 1 (G 2 ) is more than EM 1 (G), a contradiction. Based on the above discussion, we deduce that G � B n . Furthermore, by direct calculation, EM 1 (B n ) � 2n 3 − 4n 2 − 10n + 12.
Consequently, the proof is complete.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Journal of Chemistry