Some New Upper Bounds for the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 11.3618 11.4652" width="11.3618pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-90" d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"/></g></svg>-</span>Index of Graphs

Maji, Durbar; Ghorai, Ganesh; Shami, Faria Ahmed

doi:https://doi.org/10.1155/2022/4346234

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Volume 2022 | Article ID 4346234 | https://doi.org/10.1155/2022/4346234

Some New Upper Bounds for the -Index of Graphs

Durbar Maji,¹Ganesh Ghorai,¹and Faria Ahmed Shami²

Academic Editor: Naeem Jan

Received25 Nov 2021

Revised01 Jan 2022

Accepted03 Jan 2022

Published30 Jan 2022

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 [1], 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 [4] 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 [5] to search the boiling points of alkanes. After long years, Gutman and Trinajstic [6] investigated two oldest s. They are defined as and , respectively. The concept of first general was considered by Li and Zheng [7]. It is defined aswhere . For , it becomes forgotten topological index proposed by Furtula et al. [8]. It is presented as

Liu et al. ([9]) introduced the reformulated -index of as follows:

In [10], Milicevic et al. introduced the first reformulated Zagreb index of a graph . It is defined as

where . Recently, Alameri et al. [11] 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 [12–17].

2. Preliminaries

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

Lemma 1. [18]). 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. [19]). If is an vertex graph with , then .

Lemma 3. [20]). 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. [2]). If contains the maximum value of the s among all s of -vertices with and . If , then .

Lemma 5. [2]). 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. [2]). Let be a tree of order with and . Then .

Lemma 7. [21]). Suppose is any edge of with vertices. Then, for any integer (i)(ii)(iii)(iv).

Lemma 8. [22]). (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 ([23]). 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 [24], 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 [25], 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 Setting and and , then by Jensen’s theorem .

Theorem 8. Letbe a graph oforder andsize. Then,. The equality occurred whenis regular graph.

Proof. We prove the theorem using the following inequalities.
If , and for , thenwhere is a constant with some positive constants . If for some , then the equality holds if and only if and for every . Setting , and and also , , we haveso , i.e., . This completes the proof.

Theorem 9. For anvertex graph, we have. The equality is satisfied whenis regular.

Proof. Let be positive real numbers, and let be positive rational numbers. Then, by Jensen’s inequality ([26]) if . The equality holds iff . Considering for , then we have , that is, .

Theorem 10. Letbe a (n, m) graph. Then,. The equality is attained whenis regular.

Proof. Suppose and are the positive real numbers such that for . Then, by Diaz-Metacalf inequality [27], ant the equality is attained if and only if and . Now taking and and , , we getThus, .

Theorem 11. Letbe a graph whose number of vertices isand edges. Then,The equality is attained iffandand also, whereis the largest integer greater than or equal to.

Proof. Let and be positive real numbers for which there exist real constants and such that and for , respectively. Then, we have (discrete) Gruss inequality ([27]) .
The equality controls iff and .
By setting and for every , we have and .
Then, the inequality becomesSo, . This completes our claim.

Corollary 2. Since, therefore.

Theorem 12. Letbe agraph. Then,with equality holds if and only if is biregular.

Proof. We have from [28] that , where be the -coindex of . From [29], . Define by . Since , we have .
After simplification, we get the required result. □

Theorem 13. Letbe agraph, we haveThe equality holds when be a biregular graph and also , where be a positive real number.

Proof. Define by .Setting , and , then . Thus,

Theorem 14. Letbe agraph. Then,The equality occurs when is a tetra-regular graph.

Proof. Suppose that where are the positive real numbers.Setting and , then . Therefore, we get the required result. The equality is satisfied when is a tetra-regular graph.

Corollary 3. Letbe a graph withvertices andedges. Then, and also with equality holding when is a biregular graph.

Proof. Consider an auxiliary function , where and are the real numbers. Thus,Taking , then and . Also for and , we have . Thus, .

Corollary 4. Letbe a graph of orderand size. Then,where the equality is satisfied iff is a biregular graph.

Proof. Define by Since

Corollary 5. Ifis a graph withvertices andedges, the upper bounds of theare given by. The equality holds ifis a regular graph.

Proof. Similarly, it is to be proved by defining . Obviously, .

In 2005, Klavzar et al. [30] introduced the generalized Sierpinski graph . It is obtained from by adding a new vertex , called the special vertex of , and edges joining with all extreme vertices of .

Theorem 15. Letbe a graph of orderand sizeand letbe its generalized Sierpinski graph with dimension. Then, theofis given byThe upper bound is achieved iff is a -regular graph.

Proof. The of can be defined asBy applying Lemma 7, we have

4. Some for under Some Graph Operations

In this section, we derive some for under several graph operations. Let be a graph with the vertex set and the edge set for . For each and , we get and .

4.1. Cartesian Product

The Cartesian product ([31]) of and , denoted by , is the graph with vertex set and its degree distribution is .

Theorem 16. Theofsatisfies the following inequality:with equality occurring whenandare regular graphs.

Proof. By the definition of -index, we haveThe inequality must be equality if for any and .

4.2. Join

The degree of a vertex for the join [32] of and , denoted by , is given by

Theorem 17. Theon the-index of two graphsandfor join is given byThe equality holds when and are regular graphs.

Proof. By the definition of the , we get

4.3. Composition

For the composition of two graphs and [11], the degree of a vertex is given by .

Theorem 18. Theofforis given by. The equality carries for and regular graphs.

Proof. From the definition of -index, we have

4.4. Corona Product

For Corona product [33] of and , denoted by , the degree of a vertex is given bywhere is the -th copy of the graph .

Theorem 19. Letbe the corona product ofand.satisfies the following inequalities. The equality holds when and are regular.

Proof. From definition of , we have

4.5. Strong Product

Consider as a degree distribution of a vertex in the strong product [11] .

Theorem 20. The sharp of for is given byThe equality occurs when and are regular.

5. Application

As an application, we compute the of Fullerene, by using Theorem 11. Fullerenes are the molecules such as cage-like polyhedra, containing solely carbon atoms. Fullerenes contain the networks of pentagons and hexagons. Here, we consider the fullerene such that molecules made up entirely of (natural number) carbon atoms contain twelve pentagonal sides and hexagonal faces, where . For the graph representing fullerene which is given in [34], we have . The number of edges(m) in fullerene is . Thus, by Theorem 11, = .

6. Conclusion

The is one of the new chemical descriptors, which passes the test of having a highly correlation with the physiochemical properties it claims to describe in [11]. It comes as no surprise. Then, we determine some new for the using various parameters such as order, size, maximum degree, minimum degree, distance -domination number, and some other topological indices. Furthermore, some sharp for the based on graph binary operations are obtained. At last, we consider an application for index of Fullerene. The appeal of computing the is of course their generality and simple proofs. Along in this line, determining new lower bounds for is considered to be studied in the future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The second author acknowledges the support of DST-FIST, New Delhi (India) (Sanction No. SR/FST/MS- I/2018/21) for carrying out this work.

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Copyright © 2022 Durbar Maji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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