New Results on the Forgotten Topological Index and Coindex

Abstract

The -coindex (forgotten topological coindex) for a simple connected graph is defined as the sum of the terms over all nonadjacent vertex pairs of , where and are the degrees of the vertices and in , respectively. The -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total -electron energy. Therefore, considering the importance of the -index and -coindex, in this paper, we study these indices, and we present new bounds for the -index and -coindex.

1. Introduction

Suppose be a simple graph with vertex set and edge set . The integers and are the order and the size of the graph , respectively; we say is a -graph. The open neighborhood of vertex is , and the degree of is . We write and for the maximum and minimum degrees of , respectively. A graph is said to be -regular if all of its vertices have degree . An -semiregular graph is a graph whose each vertex is of degree or , and a -triangular graph is a graph whose each vertex is of degree , or . The complement of a graph is a graph that has the same vertices as and in which two vertices are adjacent if and only if they are not adjacent in . The number of vertex pairs in such that is . A pendant vertex is a vertex of degree one. The number of pendant vertices in is denoted by . We denote by the minimal nonpendant vertex degree.

In [1, 2], the first and second Zagreb indices are defined as the following:respectively. The first and second Zagreb coindices are defined in [3, 4] as

Furtula and Gutman [5] defined the forgotten topological index (-index) as the following:

For more information on the -index, see [6–8].

The -coindex introduced in [9] is as follows:

Gutman in [10] introduced the hyper-Zagreb coindex as follows:

Usha et al. [11] introduced the redefined first Zagreb indices as the following:

Here, we introduce the redefined first Zagreb coindex as the following:

The authors introduced the first general Zagreb coindex in [12], and it is defined as follows:where . The second general Zagreb coindex was introduced in [13] and defined as follows:

Topological indices are numerical quantity derived from a molecular graph which correlate the physicochemical properties of the molecular graph. Recently, topological indices have been studied by many researchers due to their applications in various sciences such as chemistry, physics, and electricity; see [14–16]. Among the topological indices, the first Zagreb index is one of the oldest and most applied topological indices, and for this reason, it is of great importance and has been considered by many researchers today. Furtula and Gutman in [5] recently investigated this index and named this index as “forgotten topological index” or “F-index” and showed that the predictive ability of this index is almost similar to that of the first Zagreb index and for the entropy and acetic factor; both of them yield correlation coefficients greater than 0.95. Therefore, due to the importance of the F-index in this paper, we have decided to study this index.

In [17, 18], some of bounds for the general Zagreb coindices were obtained. Ranjini et al. [19] presented some of the bounds for Zagreb indices and the Zagreb coindices. In [20], some bounds were presented for the -index and -coindex. For more other bounds, see [21–23].

Given the importance of the forgotten topological index and the fact that it has recently attracted the attention of researchers and the interest of many readers, in this paper, we intend to discuss new bounds for this index.

2. Preliminaries

Here, we recall several published results that we will need for proof.

The following result obtained the relationship between the first Zagreb index and the maximum and minimum degrees.

Theorem 1 (see [24]). Let be an -connected graph and . Then,with equality if and only if is isomorphic with a regular graph.

The following result comes from [18].

Theorem 2. Suppose be a -graph and . Then,with equality if and only if is isomorphic with the -regular graph, .

In [19], the authors gave the relation of the second Zagreb coindex, the maximum degree, and the first Zagreb index as follows.

Theorem 3. Let -graph and maximal degree be . Then,

Zhou and Trinajstić [25] proved the following result.

Theorem 4. Suppose be a -graph. Then,with equality if and only if is regular.

Furtula and Gutman [5] showed the following.

Theorem 5. Suppose be a graph of size . Then,with equality if and only if is regular.

In [26], Elumalai et al. obtained the following two results.

Theorem 6. Suppose be a simple graph of order . Then,with equality if and only if is regular, where

Theorem 7. Suppose be a simple graph of order . Then,with equality if and only if is regular, where

In [5], Furtula and Gutman mentioned the following result.

Theorem 8. Suppose be a connected -graph and second Zagreb index . Then,with equality if and only if is the star graph.

The following result was first proved in [27].

Theorem 9. Suppose be a simple graph of order . Then,

3. New Bounds for the -Coindex

In this section, we will obtain some bounds for the -coindex in terms of the maximal degree, minimum degree, order, size, pendant vertex, and the first and the second Zagreb indices.

Theorem 10. Suppose be a -graph. Then,

Proof. By applying the definition of the -index for complement graphs, we have

Theorem 11. Suppose be a -graph. Then,

Proof. For any vertex , we have , and by applying the definition of the -coindex, we haveNow, we give a lower bound for the -coindex in terms of pendant vertices.

Theorem 12. Suppose be a connected graph of order and pendant vertices. Then,

Proof. It is easy to see that our result holds for . Now, we assume that . Here, we let that has exactly one pendant vertex, called , and is the unique neighbor of . Hence,Here, we can let that . Each pair of pendant vertices contributes 2 to . The total contribution of pendant vertex pairs to is . Assume that is a pendant vertex in and is its unique neighbor. Then, for any nonpendant vertex such that , the contribution of vertex pairs to is . The total contribution of such vertex pairs to is . Note that for any nonpendant vertex in ; therefore, we get , as desired. □

Theorem 13. If is a t-regular graph of order , then

Proof. We know that any -regular graph of order has edges.(1)By applying Theorem 10, we have(2)By applying Theorem 9, we can write(3)Similarly, by applying Theorem 11, we obtainNow, we give lower and upper bounds for the -coindex.

Theorem 14. Suppose be a connected -graph, maximum degree be , and nonpendant minimum degree be and with leaves. Then,

Proof. It can be easily seen that the number of vertices pairs is as follows:For any vertex , we have and for nonpendant vertices. Therefore, .
We continue the proof with the following four cases.
Let .

Case 1. If , we have

Case 2. If , hence, we getIt is clear that, for each , . Hence, by applying the definition of the -coindex and above facts, we can writeas desired.

Case 3. If , we get

Case 4. If , we get thatThus,and the proof is completed.
By setting in Theorem 14, we can obtain the following results.

Corollary 1. Suppose be a connected -graph, maximum degree be , and nonpendant minimum degree be . Then,Here, we give a lower bound for the -coindex.

Proposition 1. Suppose be a connected graph of order and size . Then,

Proof. By applying the definition of the -coindex, we haveIn the following result, we obtain the lower bound for the -coindex.

Theorem 15. Suppose be a connected -graph and . Then,

Proof. Clearly, for each , we have . It follows thatand the proof is completed.
Now, we give an upper bound for the -coindex in terms of the second Zagreb coindex and the redefined first Zagreb coindex.

Theorem 16. Suppose be a graph of order . Then,

Proof. By applying the definition of the -coindex, we havewhich leads to the result.
In the following results, we obtain upper bounds for the -coindex in terms of the first and second Zagreb indices and hyper-Zagreb coindex.

Theorem 17. Suppose be a -graph. Then,

The equalities hold if and only if or is -regular or -semiregular.

Proof. By applying the definitions of the -coindex and second Zagreb coindex, we can writeLikewise, we haveandThe equalities hold in (46)–(48) if and only if , for each . This implies that each vertex of has degree or ; that is, or is -regular or -semiregular.
By using Theorems 1, 8, 10, and 9, we get the following result.

Theorem 18. Suppose be a -graph. Then,

The following results were obtained by combining Theorems 1, 4, 5, 6, and 11.

Theorem 19. Suppose be a -graph. Then,

The following results were obtained by combining Theorems 2, 4, 10, and 9.

Theorem 20. Suppose be a -graph. Then,By applying Theorems 2, 8, and 11, we can obtain the following theorem.

Theorem 21. Suppose be a -graph. Then,

The following results were obtained by Theorems 2, 5, 10, and 9.

Theorem 22. Suppose be a -graph. Then,

The following results were obtained by combining Theorems 7, 10, and 9.

Theorem 23. Suppose be a -graph. Then,

4. Conclusion

In this paper, we investigate the relationship between the -coindex and the other topological coindices, such as the first and second Zagreb coindices, the hyper-Zagreb coindex, and the redefined first Zagreb coindex. However, there are still open and challenging problems for researchers, for example, the problem on the relationship among the -coindex and GA-coindex, harmonic coindex, Randić coindex, etc.

Data Availability

The data involved in the examples of our manuscript are included within the article.

Disclosure

The authors would like to declare that the work described was original research that has not been published previously [28].

Conflicts of Interest

The authors declare that they have no conflicts of interest.