#### Abstract

In mathematical chemistry, the topological indices with highly correlation factor play a leading role specifically for developing crucial information in QSPR/QSAR analysis. Recently, there exists a new graph invariant, namely, -index of graph proposed by Alameri as the sum of the fourth power of each and every vertex degree of that graph. The approximate range of the descriptors is determined by obtaining the bounds for the topological indices of graphs. In this paper, firstly, some upper bounds for the -index on trees with several types of domination number are studied. Secondly, some new bounds are also presented for this index of graphs in terms of relevant parameters with other topological indices. Additionally, a new idea on bounds for the -index by applying binary graph operations is computed.

#### 1. Introduction

In this paper, we only consider the molecular graphs , which are simple and connected. In chemical graph theory, molecules or molecular compounds are often modelled by chemical structure as . The atoms of a molecular compound are to be represented as the vertices of the , whereas the edges represent the chemical bonds. Let be a with as the vertex set and as the edge set, such that and . The degree of , denoted by , is the total number of edges, which are associated with . Obviously, , where and . A set satisfying the condition , is called a dominating set of . When satisfies the condition , for some , where denotes the distance between and , is said to be a distance -domination set of . The number ([2, 3]) of , denoted by , is the minimum cardinality among all sets. The notations and denote the diameter of and the eccentricity of , respectively. A path is called a diameter path of when the length of is equal to . We follow the book  for the graph theoretical definitions and notations.

The graph invariant is a number that is uniquely determined by a graph. The subset of s is topological indices, which are used to predict several properties such as physical, chemical, pharmaceutical, and biological activities of chemical species. In 1947, the great chemist holder Wiener initiated a first-time idea about the topological index . He presented the first , namely, Wiener index  to search the boiling points of alkanes. After long years, Gutman and Trinajstic  investigated two oldest s. They are defined as and , respectively. The concept of first general was considered by Li and Zheng . It is defined aswhere . For , it becomes forgotten topological index proposed by Furtula et al. . It is presented as

Liu et al. () introduced the reformulated -index of as follows:

In , Milicevic et al. introduced the first reformulated Zagreb index of a graph . It is defined as

where . Recently, Alameri et al.  introduced a named -index and defined as

The is the special case of the first general Zagreb index for .

In this study, we obtain some new upper bounds for the in terms of different graph parameters, on for tree of vertex and s. We arrange the remaining work as follows: Section 2 contains the for the on trees with . Section 3 contains some for in behavior of some relevant parameters. Section 4 collects some for under several graph operations. Finally, Section 5 presents the conclusions of the obtained results. To know more related to this field, readers are referred to .

#### 2. Preliminaries

To establish the main results, the following lemmas are required.

Lemma 1. ). Let be a vertex tree with , a nonpendant edge. Suppose the union of and is equal to , where for . Let be a new tree obtained by taking an edge joining transformation (EJT) of on . It is attained by identifying with and also joining a pendent vertex to the . In short, we denote . Then, we get .

Lemma 2. ). If is an vertex graph with , then .

Lemma 3. ). Let and be two trees with and vertices, respectively. Then, holds iff at least one of following conditions is satisfied:(1) is any -vertex tree.(2) is equal to obtained by taking and copies of and then link with the vertex of to exactly one end vertex in the copy of .

Lemma 4. ). If contains the maximum value of the s among all s of -vertices with and . If , then .

Lemma 5. ). Suppose and be two vertices in such that and pendent vertices adjacent to and respectively. Define and . Then either is greater than or is greater than .

Lemma 6. ). Let be a tree of order with and . Then .

Lemma 7. ). Suppose is any edge of with vertices. Then, for any integer (i)(ii)(iii)(iv).

Lemma 8. ). (Radon’s inequality) Let and be two sequences of positive real numbers. For any ,where the equality occurs for for some constant , for all .

#### 3. Main Results and Discussions

##### 3.1. Some for the on Trees with Number

In this section, we establish some sharp for the of graphs on the trees as to the number, . The set of all vertex trees with and the star of order with pendent vertices are denoted as and , respectively.

Theorem 1. Letbe a tree of orderand it contains; then theofcan be expressed as. The equality holds for, where.

Proof. Let be a tree with a such that . For , there exists a set of , a contradiction. In case , also denoted by the tree obtained from by (Lemma 1) on the edge for some , then ; therefore, . But , a contradiction. Thus, it is only s for .
In this case, we consider a tree obtained from the path by attaching pendent vertices to , where . Moreover, for and also for . Consequently, , for some . Therefore, the for the tree can be directly computed as .

Theorem 2. Consider anvertex treethat belongs to. Then,. The equality occurs as.

Proof. Given that . Obviously , by Lemma 2. Also, the equality of Lemma 3 holds the results. Now, we proceed to prove the theorem by induction hypothesis on . Assume that and the statement is true for . Our main goal is to reach .
Let and be a and minimum set of , respectively. Actually, we have to prove . Otherwise, is a set, a contradiction. Let us assume that such that and .
Choose as the pendent vertices of and also for . If , then for .
When , then . Taking , so for , a contradiction. Therefore for . Clearly , since .
Remark that and . So, . Likewise, . For , the vertices are distinguished, a contradiction. So, and .
On the other side, if , then and is a set, which is an inconsistency. Therefore, and also . When , then for some , an impropriety. So, . Thus, is a path of which ended vertices are and . That is, . Hence, , which contradicts .
Assume that is a unique vertex which is a pendent vertex with . Note that , by Lemma 6. So, by the and the definition of , we getTherefore, the equality arrives if and only if and , that is, .

Theorem 3. Letbe a tree havingvertices and. If, we have.The equality is attained when .

Proof. Given that for the tree of vertices with number, . By Lemma 3, we get for some tree on vertices. Let us consider . Then, . Therefore, since for every tree (assume ) containing -vertices with vertex set occurs . By the definition of the , we can expressfor equalities that imply .

Theorem 4. Consideras an-vertex tree whosenumber is. Then,.
The equality occurs for .

Proof. Let be a tree containing a such that that maximized the of graphs. The main goal is to establish the maximization of with respect to . Let us consider to be a minimum set of and also define . If , then . Also, for , by Lemma 4, . In case, , as , we get that implies , a contradiction. Therefore, we consider that , and thus , from Theorem 2.
By Lemma 1, applying of on any nonpendent edge of repeatedly for , it is to be constructed a tree from such that , where is the component of having , for . Then, we have and also , where the equality holds .
Now let for some .
Then, by Lemma 5, we get with equality if and only if .
Again, define by . In fact, let .
Then, for and also will be the minimum set of . It implies that all the vertices in can be determined by . Therefore, will be a set of . Suppose that is the set of all private -neighbors of upon in . Then, . Thus, will be a minimum set of the tree . Therefore, .
So, from the definition of , we haveIt means that , where the equality holds either i.e. or .
So far, we have proved with equality iff by induction on . We have from Theorem 2 that the assertion is mathematics for as well as .
Now let us consider the affirmation contains for and all the vertices .
Because of and , we get by the where the equality holds iff and also and otherwise with for . Therefore, we can conclude that with either equality iff or with . Besides, .

Here, we determine some on the of trees containing -vertices with domination number (). The number of a graph is said to be the domination number of that graph if . If is an -vertex tree containing a such that , then denote by the component of containing . To compute our main outcome, at first, we will focus on the following definition.

Consider to be a tree constructed from a star with involvement of a pendant edge to its pendant vertices. Note that , a class of vertex trees and domination number . Also, occurs iff .

Corollary 1. If, then with equality holding for .

Proof. For , it occurs that . The equality holds for , and . But for , the above inequality is strict. Now, we consider a diameter path and a minimum dominating set of with . To prove the theorem, we will take the way of on . Let us consider that Theorem 1 is true for and also the statement is to be proved as well as truth by replacing from .When , then by the we have (since , by Lemma 6) . The equality holds iff the pendant vertex is adjacent to the vertex of degree , that is, .
Otherwise, assume that . So, it will be which also implies that belongs to every minimum dominating set, i.e., . Therefore, we can obtain by the where the equality holds and pendant vertex is adjacent to the vertex of degree 2, that is, .

##### 3.2. Some for the of Graphs with respect to Some Standard Parameters and Others

In this section, we establish the some sharp for the of to some graph parameters such as and others such as . Let . If , then is said to be -regular. If , then is biregular and so on. Motivating the proof technique as in , we obtain an for in the following theorem.

Theorem 5. Considerto be agraph, i.e.,containsvertices andedges. Then, whereis the integer defined by relationand the equality holds iff at most one vertex ofhas different degree fromand.

Proof. Consider as the number of vertices of degree in . From the definition of , we can writeObviously,After calculation, we getUsing (15) and (16), we haveActually, the term will be negative for . So, the will be maximum if for . Therefore, (15) and (16) become to and . These two equations require thatIf the requirement is not true, we choose such that and for all , except for . Then, (15) and (16) become and which satisfy conditions 3 and 4. In , there survives a vertex graph of which one vertex with degree 0 when and are both odd. If we add the edges to this graph, the vertex degrees increase one at a time up to . There occurs that implies that the degree of one more vertex may be increased up to . Therefore, there exists a graph of order and size along with a unique vertex of degree that different from and .
Suppose now that the graph contains two vertices of degrees and for . If the sum of vertex degrees remains the same by reducing the first vertex degree by 1 and increasing the second vertex degree by 1, the value of the is replaced byIt means that condition 8 is not true, and it will be the optimal choice of the quantities for such that (except for ). Therefore, we can conclude from (17) that

Theorem 6. Ifis agraph, thenwhere the equality holds iff either is regular or semiregular bipartite graph.

Proof. Using Lemma 8, for , setting and for the graph in (1), we haveAlso,since .
From (23) and (24), .

Theorem 7. Suppose a graphthat containsvertices andedges. Then,, with equalities .

Proof. If , , …, are positive numbers sets with elements in each set and are positive numbers such that , then by Jensen’s theorem . We know