New Results on the Geometric-Arithmetic Index

Jahanbani, Akbar; Atapour, Maryam; Du, Zhibin

doi:https://doi.org/10.1155/2021/4901484

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Volume 2021 | Article ID 4901484 | https://doi.org/10.1155/2021/4901484

New Results on the Geometric-Arithmetic Index

Akbar Jahanbani,¹Maryam Atapour,²and Zhibin Du³

Academic Editor: Ali Ahmad

Received18 Jun 2021

Accepted20 Jul 2021

Published31 Jul 2021

Abstract

Let be a graph with vertex set and edge set . Let denote the degree of vertex . The geometric-arithmetic index of is defined as . In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometric-arithmetic index and several other topological indices.

1. Introduction

Let be a simple graph (i.e., graph without loops and multiple edges) with vertex set and edge set . The integers and are the order and the size of the graph , respectively. For , we denote by the degree of vertex in . The minimum and maximum degrees of a graph are denoted by and , respectively.

Graph theory has provided chemists with a variety of useful tools, such as topological indices. A topological index of a graph is a number with the property that, for every graph isomorphic to , .

Molecular descriptors play a significant role in mathematical chemistry, especially in QSPR/QSAR investigations. Among them, special place is reserved for so-called topological descriptors. A topological index is a numeric quantity from the structural graph of a molecule.

Usage of topological indices in chemistry began in 1947 when Wiener [1] developed the most widely known topological descriptor, namely, the Wiener index, and used it to determine physical properties of types of alkanes known as paraffin (see, for instance, [2, 3]). The interest of topological indices lies in the fact that they synthesize some of the properties of a molecule into a single number. With this in mind, hundreds of topological indices have been introduced and studied. Topological indices based on the vertex degree play a vital role in mathematical chemistry, and some of them are recognized as tools in chemical research.

Authors are studying various topological descriptors, such as Zagreb indices [4–6], general sum-connectivity index [7, 8], hyper-Zagreb index [9], and harmonic index [10, 11]. Besides the abovementioned ones, there are other topological descriptors based on end vertex degrees of edges of graphs that have found some applications in QSPR/QSAR research [2, 12, 13].

The geometric-arithmetic index of a graph is defined in [13] as

The geometric-arithmetic index has a number of interesting properties, e.g., see [13]. The lower and upper bounds of the geometric-arithmetic index of connected graphs and the characterizations of graphs for which these bounds are best possible can be found in [13–16].

The aim of this paper is to investigate new relationships between the geometric-arithmetic index and other topological indices. In particular, we obtain some lower and upper bounds for the geometric-arithmetic index. Moreover, we improve some known bounds.

2. Preliminaries

Let us recall some remarkable lemmas which will be used in this paper.

The first one is a very straightforward observation.

Lemma 1 (see [17]). Let and be two positive numbers. Then,

The following is the well-known inequality of arithmetic and geometric means.

Lemma 2 (inequality of arithmetic and geometric means, see [18]). Let be positive numbers. Then,

Lemma 3 (see [19]). Let and be two sequences of positive numbers. For any ,

Lemma 4 (see [20]). Let for and be some positive constants. Then,

Lemma 5 (see [21]). If and are positive numbers, where and for each , then

Lemma 6 (the Pólya–Szegö inequality, see p. 62 in [22]). Let and be two sequences of positive numbers, where and , for . Then,

3. Upper Bounds for the Geometric-Arithmetic Index

In this section, we investigate the relationships between geometric-arithmetic index and some topological indices. Moreover, we obtain some upper bounds for the geometric-arithmetic index in terms of order, size, maximum degree, minimum degree, domination number, girth, number of cut edges, and number of pendent vertices.

The first and second Zagreb indices are vertex-degree-based graph invariants defined as

The quantity was first considered in 1972 [6], whereas in 1975 [5]. The general Randić index is defined as follows [23]:where is a real number.

We begin with the establishment of an upper bound for the geometric-arithmetic index in terms of the first Zagreb index and the general Randić index.

Theorem 1. Let be a graph. Then,

Proof. By Lemma 1, we haveas desired.
Using Lemma 1 and an argument similar to the proof of Theorem 1, we can obtain the next result.

Corollary 1. Let be a graph. Then,

From Lemma 1, we get

Again by Lemma 1, we have

Hence, we can see that the bounds in Theorem 1 and Corollary 1 improve the bound:established in [15].

The proof of the following result can be found in [23].

Lemma 7 (see [23]). Let be a graph of size . Then,for .

Using Corollary 1 and Lemma 7, we can drive the next result.

Corollary 2. Let be a graph of size . Then,

Lemma 8. Let and be two positive numbers. Then,

Now, we obtain an upper bound for the geometric-arithmetic index in terms of the first Zagreb index.

Theorem 2. Let be a graph of order , size , and minimum degree . Then,

Proof. Notice thatBy Lemma 8, we haveand this implies the desired bound.
A dominating set of a graph is a vertex subset whose closed neighborhood includes all vertices of the graph. The domination number of a graph is the size of a minimum dominating set.

Theorem 3. (see [24]). Let be a tree of order with domination number . Then,

By Theorems 2 and 3, we have the following result for trees with the given domination number.

Corollary 3. Let be a tree of order with domination number . Then,

Since for every two real numbers , and , we have the next observation.

Lemma 9. Let and be two real numbers, where . Then, .

Next, we establish an upper bound for the geometric-arithmetic index in terms of the second Zagreb index.

Theorem 4. Let be a graph of size with maximum degree . Then,

Proof. By Lemmas 8 and 9, we haveand this implies the desired bound.

In [25], it is proved that, for any tree of order , . Using this and Theorem 4, we obtain the next result.

Corollary 4. Let be a tree of order with maximum degree . Then,

Here, we establish an upper bound for the geometric-arithmetic index in terms of the hyper-Zagreb index.

The hyper-Zagreb index is defined as follows [9]:

Theorem 5. Let be a graph of order , size , and minimum degree . Then,

Proof. By Inequality (21), we haveIt leads to the desired bound.
The next result is proven in [26].

Theorem 6 (see [26]). Let be a graph with vertices and edges. Then,Theorems 5 and 6 lead to the desired result.

Corollary 5. Let be a graph of order , size , and minimum degree . Then,

The redefined third Zagreb index is defined as follows [27]:

Now, we obtain an upper bound for the geometric-arithmetic index in terms of the second Zagreb index, the general Randić index, and the redefined third Zagreb index.

Theorem 7. Let be a graph with maximum degree and minimum degree . Then,

Proof. It is easy to obtainThe desired bound follows.

Theorem 8. Let be a graph of order , size , maximum degree , and minimum degree . Then,

Proof. Now, putting for each edge , , and in Lemma 4, we haveOn the contrary, we haveFinally, we get the bound by using Inequalities (36) and (37).
The sigma index of is defined in [28] asHere, we obtain an upper bound for the geometric-arithmetic index in terms of the first Zagreb index and the sigma index.

Theorem 9. Let be a nontrivial graph with maximum degree . Then,

Proof. For two real numbers and , we have thatBy (40), we obtainand this implies the desired bound.
The general first -index of a graph is defined in [29] aswhere is a real number. In particular, .
Since for every two real numbers and , , and we deduce that, for any graph ,Using these and Theorem 9, we obtain the next result.

Corollary 6. Let be a nontrivial graph with maximum degree . Then,

From , we would like to indicate that the above new bound improves the known bound:which was established in [15].

Now, by using the following result, we want to obtain an upper bound for trees.

Theorem 10 (see [30]). Let be a tree of order with independence number . Then,

Here, by Theorems 9 and 10, we obtain the next result.

Corollary 7. Let be a tree of order with independence number and maximum degree . Then,

4. Lower Bounds for the Geometric-Arithmetic Index

In this section, we first investigate the relationships between the geometric-arithmetic index and some other topological indices, and then, we obtain some lower bounds for the geometric-arithmetic index which improve some well-known bounds.

Theorem 11. Let be a graph of size with minimum degree . Then,

Proof. By Lemmas 1 and 2, we haveThe result follows.

Here, by Theorems 11 and 6, we have the next result.

Corollary 8. Let be a graph of order and size , with minimum degree . Then,

Since for any real numbers and , it holds that ; hence, by this fact and Inequality (49), we can obtain the following result.

Corollary 9. Let be a graph of size with minimum degree . Then,

We start with a lower bound for the geometric-arithmetic index in terms of the general -index.

Theorem 12. Let be a nontrivial graph of size with minimum degree . Then,

Proof. Set , , and for each . By Lemmas 1 and 3, we haveThe proof is completed.

The harmonic index is defined as follows [11]:

Theorem 13. Let be a nontrivial graph of order , size , and minimum degree . Then,

Proof. Notice thatThe result follows.
Applying (56), we obtain the next results.

Corollary 10. Let be a nontrivial graph of order , size , and minimum degree . Then,

Corollary 11. Let be a nontrivial graph of order , size , and minimum degree . Then,

Theorem 14 (see [31]). Let be a connected graph of order . Then,

A cut edge of a graph is an edge whose removal increases the number of connected components of the graph.

Lemma 10 (see [32]). Let be a connected graph of order and cut edges. Then,

Now, by Theorems 13 and 14, and Lemma 10, we can obtain the next result.

Corollary 12. Let be a connected graph of order , cut edges, and minimum degree . Then,

Here, we will use the following particular case of Jensen’s inequality.

Lemma 11. Let be a convex function defined in . For ,

The general sum-connectivity index is defined as follows [8]:

Now, we obtain a lower bound for the geometric-arithmetic index in terms of the general sum connectivity index.

Theorem 15. Let be a graph of size and minimum degree . Then,

Proof. Since is a convex function for , from Lemmas 1 and 11, we haveas desired.

Now, we obtain an upper bound for the geometric-arithmetic index in terms of the sigma index.

Theorem 16. Let be a simple connected graph of size with maximum degree , pendent vertices, and minimum nonpendent vertex degree . Then,

Proof. We partition all the edges into two parts: pendent edges and nonpendent edges, soOn one hand, for the pendent edges, it is not hard to check that decreases in ; thus,Now, we consider the nonpendent edges. It is easy to see that the function gets its maximum value when attains the maximum or minimum value. From for all , we havewhich is equivalent toSet and for each edge , , and in Lemma 6, and we havewhich implies thatFinally, the result follows from (67), (68), and (72).
Next, results are immediate consequences of Theorem 16 with the setting .

Corollary 13. For a graph of size with maximum degree and minimum degree ,

Now, we obtain a lower bound for the geometric-arithmetic index in terms of the second Zagreb index and the general sum connectivity index.

Theorem 17. Let be a graph of size , maximum degree , and minimum degree . Then,

Proof. By Lemma 5 and putting , , and , we haveThis implies thatThe result follows.
Now, we obtain a lower bound for the geometric-arithmetic index in terms of the harmonic index.

Theorem 18. Let be a graph without isolated edges. Then,

Proof. Since for each , , we obtainas desired.
The proof of next results can be found in [33].

Theorem 19 (see [33]). Let be a triangle-free graph of order and the minimum degree . Then,

Theorem 20 (see [33]). Let be a triangle-free graph of order and size . Then,

Applying Theorems 18–20, it leads to the next results.

Corollary 14. Let be a triangle-free graph of order without isolated edges, and the minimum degree . Then,

We can see that Inequality (82) improves the next well-known result for triangle-free graphs [13]. Let be a graph of order and size without isolated vertex. Then,

The eccentricity of is defined aswhere is the length of a shortest path connecting and . The radius and diameter are defined as the minimum and maximum values among over all vertices , respectively.

Xu [34] showed that, for any nontrivial connected graph of order , size , and radius , . Using this and Theorem 18, we obtain the next result.

Corollary 15. Let be a nontrivial connected graph of order , size , and radius . Then,

Theorem 21. Let be a nontrivial connected graph of size and radius . Then,

Proof. Note that, for each vertex , we have . Thus, for each edge ,as desired.

Theorem 22. Let be a nontrivial graph of order , size , and pendent edges without isolated vertex. Then,

Proof. Since , therefore we deduce thatFor each pendent edge , we clearly have . If is a nonpendent edge, then , as any pendent vertex is adjacent to at most one of and . So, ; hence,Thus,The desired result follows.
In [35], Kulli et al. defined the first and second generalized multiplicative Zagreb indices:Here, we obtain a lower bound in terms of the first and second generalized multiplicative Zagreb indices.

Theorem 23. Let be a nontrivial graph of size . Then,

Proof. By Lemma 2, we obtainas desired.

Theorem 24. Let be a graph of size and minimum degree . Then,

Proof. By Lemma 1, we getas desired.
In the sequel, we obtain a lower bound in terms of the first Zagreb index.

Theorem 25. Let be a graph of size , maximum degree , and minimum degree . Then,

Proof. By Lemma 8, we haveand this implies the desired bound.

Zhou [36] proved that, for any triangle-free graph of order and size , . Together with Theorem 25, we get the next result.

Corollary 16. Let be a triangle-free graph of order , size , maximum degree , and minimum degree . Then,

Inequality (98) leads to the following results.

Corollary 17. Let be a graph of size , maximum degree , and minimum degree . Then,

Note that, for every two real numbers and , . Applying this, we obtain a lower bound for the geometric-arithmetic index in terms of the hyper-Zagreb index.

Theorem 26. Let be a graph of size , maximum degree , and minimum degree . Then,

Proof. From the above inequality, we haveand this implies the desired bound.
Here, we obtain a lower bound for the geometric-arithmetic index in terms of the first Zagreb index.

Theorem 27. Let be a graph of size and minimum degree . Then,

Proof. From the fact that for any , we haveand this implies the desired bound.

Theorem 28 (see [37]). Let be a graph of size and diameter . Then,

Now, by Theorems 27 and 28, we have the following result.

Corollary 18. Let be a graph of size , minimum degree , and diameter . Then,

Theorem 29 (see [38]). Let be a graph of size , with triangles and pendent vertex . Then,

Again, by Theorems 27 and 29, we have the following result.

Corollary 19. Let be a graph of size , with triangles, leaf number , and minimum degree . Then,

Theorem 30 (see [39]). Let be a triangle- and quadrangle-free graph with vertices. Then,Also, by Theorems 27 and 30, we have the following result.

Corollary 20. Let be a triangle- and quadrangle-free graph of order , size , and minimum degree . Then,

Data Availability

The data used to support the findings of the study are provided within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interests.

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