Multiplicative Zagreb Indices of Molecular Graphs

Zhang, Xiujun; Awais, H. M.; Javaid, M.; Siddiqui, Muhammad Kamran

doi:https://doi.org/10.1155/2019/5294198

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Mathematical Tools for Solving Problems of Chemical Structure Generation

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Research Article | Open Access

Volume 2019 | Article ID 5294198 | https://doi.org/10.1155/2019/5294198

Multiplicative Zagreb Indices of Molecular Graphs

Xiujun Zhang,¹H. M. Awais,²M. Javaid,²and Muhammad Kamran Siddiqui³

Academic Editor: Juan L. G. Guirao

Received01 Nov 2019

Accepted19 Nov 2019

Published06 Dec 2019

Abstract

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as and , respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

1. Introduction

The numerical demonstration of a molecular graph can be assumed as a single number, commonly known as topological index (TI). There are many interesting and significant results about TIs to study the different properties of chemical compounds such as chromatographic retention times, heat of formation and evaporation, flash point, viscosity, freezing, boiling and melting point, octanol-water partition coefficient, surface tension, stability, temperature, density, weight, polarizability, connectivity, and solubility. Many medicines, crystalline and nanomaterials, that are used in numerous pharmaceutical industries are examined with the assistance of different TIs, see [1–7]. TIs also study QSPR and QSAR models that join molecular graphs to their molecular characteristics by means of statistical tools. For additional information, see [8–17].

In 1947, to investigate the paraffin’s boiling point, Wiener utilized the distance-based TI [18]. Gutman and Trinajstić [19] determined a pair of degree-based first and second Zagreb indices. After this, different impressive TIs are introduced in molecular graph theory [20,21], but the degree-based TIs are famous than others, see [22].

In graph theory, the various operations on different graphs show an important role in the creation of advanced families of graphs, see [23,24]. Yan et al. [13] gave the idea of four operations and of graphs and computed the Wiener index of the resultant graphs obtained by using these operations. Eliasi and Taeri [25] introduced the -sum graphs by using the cartesian product of graphs and , where and are the two simple graphs and is obtained by using . They also determined the Wiener index of these F₁-sum graphs. Additionally, Deng et al. [26], Imran and Akhtar [27], Shirdel et al. [28], and Liu et al. [29] determined the 1st and 2nd Zagreb indices, F-index, Hyper-Zagreb index, and the 1st general Zagreb index of F₁-sum graphs.

Recently, Liu et al. [30] defined the generalized form of the aforesaid four operations . They also constructed the generalized F-sum graphs ( for ) and obtained the 1st and 2nd Zagreb indices. In this paper, we calculate the upper bounds of the 1st and 2nd multiplicative Zagreb indices of the -sum graphs. The remaining article is organized as follows: Section 2 contains few definitions and terminologies, Section 3 covers the main results, and Section 4 includes applications and closing comments.

2. Preliminaries

A molecular graph has the node (vertex) set and the edge set . The vertices in the molecular graphs are denoted as atoms, and bonds are denoted as edges. The order and size of a chemical structure is denoted as and . The degree is the total edges that are incident on a node p. Throughout the paper, we study the finite, indirected, simple (without loops, isolated vertices, and multiple edges), and connected graphs. The well-known TIs are discussed as follows.

Definition 1. Let be a graph, then the first and second Zagreb indices are defined as follows:Gutman and Trinajstić [19] defined these indices which are utilized to determine the structural base different characteristics of graphs like molecular complexity, energy, ZE-isomerism, chirality, connectivity, and heterosystems, as well as branching , see [15,31–36].

Definition 2. Let be a graph, then the first and second multiplicative Zagreb indices areDuring the past two decades, various noteworthy applications regarding multiplicative Zagreb indices have been explained in detail, see [37–46] and the references cited therein. The resultant graphs under the four new generalized subdivision operations defined in [30] are given as follows:(i) is the generalized subdivided graph .(ii) is the generalized semitotal (point) graph.(iii) is the generalized semitotal (line) graph.(iv) is the generalized total graph. For more details, see Figure 1.

(a)

(b)

(c)

(d)

(e)

Definition 3. Let and be two graphs, then and be a new graph gained after using on having an edge set and vertex set . So, the generalized F-sum graph () is a graph having the vertex set in such a way that two vertices and of are adjacent if and only if [ and ] or [ and ].
Thus, the generalized F-sum graph contains copies of new graphs that are labeled with the vertex set of . For further details, see Figures 2 and 3.

(a)

(b)

(c)

(d)

(a)

(b)

3. Main Results

Now, we calculate the key results of the multiplicative Zagreb indices for the different classes on graphs.

Theorem 1. Let , be two graphs with , , , and For and and , we have

Proof. (a)Consider as a degree of a node in .Now,Hence,Now,Consequently,

Theorem 2. Let , be two graphs with , , , and For and , we have

Proof. (a)Consider as a degree of a node in :Now,Now, for , then if and only if ; for , we get and for , we have . Now,Hence,Now,Now, we considerTherefore, we have

Theorem 3. Let , be two graphs with , , , and For and and , we have

Proof. Now,Consider Then,Now we takeLet thenTherefore, we haveNext,ConsiderThen,Let thenwhere r is the total neighbors that we consider mutual nodes of u and in .
Therefore,

Theorem 4. Let , be two graphs with , , , and For and and , we have

4. Conclusion

For and , we consider that and are specific examples of alkane known as paths having orders a and b, respectively. Then, the following results are the direct outcomes of the above four theorems.

In this paper, for the integer , we computed upper bonds of the 1st and 2nd multiplicative Zagreb indices of the generalized F-sum graphs [ for ] that was defined by Liu et al. [30]. However, the problem is still open to compute the other topological indices of the generalized F-sum graphs.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

This research was supported by the Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095 and the Soft Science Project of Sichuan Province under grant 2017ZR0041.

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Copyright © 2019 Xiujun Zhang 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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