Topological Indices of the Pent-Heptagonal Nanosheets VC5C7 and HC5C7

Deng, Fei; Zhang, Xiujun; Alaeiyan, Mehdi; Mehboob, Abid; Farahani, Mohammad Reza

doi:https://doi.org/10.1155/2019/9594549

Advances in Materials Science and Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 9594549 | https://doi.org/10.1155/2019/9594549

Topological Indices of the Pent-Heptagonal Nanosheets VC₅C₇ and HC₅C₇

Fei Deng,¹Xiujun Zhang,²Mehdi Alaeiyan,³Abid Mehboob,⁴and Mohammad Reza Farahani³

Academic Editor: Stefano Bellucci

Received18 Jan 2019

Accepted23 Apr 2019

Published23 Jun 2019

Abstract

In this paper, we computed the topological indices of pent-heptagonal nanosheet. Formulas for atom-bond connectivity index, fourth atom-bond connectivity index, Randić connectivity index, sum-connectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometric-arithmetic index, fifth geometric-arithmetic index, Sanskruti index, forgotten index, and harmonic index of pent-heptagonal nanosheet have been derived.

1. Introduction

Nanostructures, which have scale less than 100 nm, include nanotubes, nanosheets, and nanoparticles. Nanosheets (two-dimensional nanostructures) have large surface and sharp edge, which make them play an important role in many applications such as energy storage [1–3], optoelectronics [4], catalysis [5], and bioelectronics [6]. Graphene [7], silicone [8], and borophene [9] are typical nanosheets. Graphene is the basic nanosheet that exhibits the honeycomb lattice structure. Because of the unique structural, electrical, optical, and mechanical properties, graphene nanosheets drew dramatic attention of academic and industrial research [10].

People found some defects in graphene nanosheets made through experiments. In Figure 1, the 5–7 defect in graphene is illustrated. As nanoscale-level defects introduced into graphene could be extremely useful and exploited to generate novel, innovative, and useful materials and devices [10], the 5–8–5 [11, 12] and 5–7–7–5 defects are obtained by several approaches [13–16]. Naturally, the property of C₅C₇ nanostructures has become an active area of research [17–22].

In chemical graph theory, topological indices are very useful to describe molecules or nanostructures. The vertices of the graph correspond to the atoms of molecules, and the edges correspond to chemical bonds.

In this paper, we only consider undirected graphs without loops and multiple edges. Let G be a graph. We denote V(G) and E(G) as the vertex set and edge set of G, respectively. We denote by the degree of the vertex of graph G and the set of neighbors of . We denote by the set of edges with degrees of end vertices a and b, i.e.,

Let the sum of degrees of neighboring vertices of u be denoted by , which is defined as

We define

In 1998, Estrada et al. [23] defined the atom-bond connectivity index as

The ABC₄ index is defined as

The Randić connectivity index (see [24]) is defined on the ground of vertex degrees as

The sum-connectivity index is defined (see [25]) as

The first Zagreb index is defined as the sum of squares of the vertex degrees and of vertices u and in graph G (see [26]). The mathematical formula of first Zagreb index is

The second Zagreb index is defined as the sum of squares of the vertex degrees and of vertices u and in graph G (see [26]). The mathematical formula of second Zagreb index is

Augmented Zagreb index AZI(G) (see [27]) is defined as

Modified Zagreb index (see [28]) is defined as

In 2013, Shirdel et al. [28] introduced hyper Zagreb index as

The geometric-arithmetic index (see [29]) is defined as

The is Defined as

For a detailed survey about topological indices and applications of topological indices, we refer [30–38] and references therein. Recently, motivated by the previous research on topological descriptors and their applications, Hosamani [39] proposed the Sanskruti index which can be utilized to guess the bioactivity of chemical compounds and shows good correlation with entropy of octane isomers.

The Sanskruti index of graph G is defined as

The forgotten index is defined as the sum of cubic of the vertex degrees in G. Mathematical formula of forgotten index is [40]

The harmonic index (see [41]) is defined as

2. C₅C₇ Nanosheets and Their Structures

A C₅C₇ net is a trivalent decoration made by alternating C₅ and C₇. It can cover either a cylinder or a torus and certainly able to cover a sheet.

The topological indices of nanostructures have been studied extensively. Graovac et al. [42] discussed the GA index of TUC₄C₈(S) nanotube in 2009. Graovac et al. [42] computed fifth geometric-arithmetic index for nanostar dendrimers in 2011. Asadpour et al. [43] computed Randić, Zagreb, and ABC indices of TUC₄C₈(S) and TUC₄C₈(R) nanotubes and V-phenylenic nanotorus in 2011. Al-Fozan et al. [44] computed the Szeged index of C₄C₈(S) and H-naphtalenic nanosheets (2n, 2m) in 2014. Azari and Iranmanesh [45] computed in Harary index of Pm × Pn (nanosheets), C₄ nanotubes, and C₄ nanotori in 2014. Ashrafi and Loghman [46] discussed the Padmakar–Ivan (PI) index of TUC₄C₈(S) nanotubes in 2006.

The topological indices of C₅C₇ nanostructures have also been studied. Iranmanesh et al. [47] introduced a GAP program to compute the Szeged and PI indices of VC₅C₇[p,q] and HC₅C₇[p,q] nanotubes in 2008. Farahani and Zhang [48–53] discussed Randić connectivity index and sum connectivity index of the pent-heptagonal nanotube VAC₅C₇(S) in 2013. For the nanosheet VC₅C₇(m, n), we denote the number of pentagons in the first row by n; in this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by m (see, e.g.,Figure 2(a) with (m, n) = (2, 4)).

For the nanosheet HC5C7(m, n), we denote the number of heptagons in the first row by n. In this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by m (see, e.g., Figure 2(b) with (m, n) = (2, 4)).

(a)

(b)

The nanosheet VC₅C₇(2, 4) has 16mn + 2m + 5n vertices and 24mn + 4n edges. Moreover, it has 6m + 7n vertices with degree 2 and 16mn − 4m − 2n vertices with degree 3. The edge partitions according to different and are presented in Tables 1 and 2, respectively. We computed some topological indices according to different types of and . As an example, Figures 3(a) and 3(b) show the edge partition of the nanosheet VC₅C₇(2, 4) by different and , respectively.

(a)

(b)

The nanosheet HC₅C₇(2, 4) has 16mn + 2m + 4n vertices and 24mn + 3n edges. Moreover, it has 6m + 6n vertices with degree 2 and 16mn − 4m − 2n vertices with degree 3. We computed some topological indices according to different types of and . The edge partitions according to different and are presented in Tables 3 and 4, respectively. As an example, Figures 4(a) and 4(b) show the edge partition of the nanosheet HC₅C₇(2, 4) by different and , respectively.

(a)

(b)

3. Main Results

Theorem 1. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of ABC index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of ABC index, we have

Theorem 2. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 2. By the definition of ABC₄ index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of ABC₄ index, we have

Theorem 3. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the Randić connectivity index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the Randić connectivity index, we have

Theorem 4. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the sum connectivity index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the sum connectivity index, we have

Theorem 5. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the first Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the first Zagreb index, we have

Theorem 6. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the second Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the second Zagreb index, we have

Theorem 7. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the augmented Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the augmented Zagreb index, we have

Theorem 8. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the modified Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the modified Zagreb index, we have

Theorem 9. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the hyper Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the hyper Zagreb index, we have

Theorem 10. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the first geometric-arithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the first geometric-arithmetic index, we have

Theorem 11. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 2. By the definition of the fifth geometric-arithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of the fifth geometric-arithmetic index, we have

Theorem 12. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 2. By the definition of the Sanskruti index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of the Sanskruti index, we have

Theorem 13. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the forgotten topological index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the forgotten topological index, we have

Theorem 14. (i)Let G be a VC₅C₇(m, n) nanosheet, then(ii)Let G be a HC₅C₇(m, n) nanosheet, then

Proof. (i)The proof is calculated based on the edge partition given in Table 1. By the definition of the harmonic index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the harmonic index, we have

4. Conclusions

QSARs represent predictive models derived from the application of statistical tools correlating to the biological activity (including desirable therapeutic effects and undesirable side effects) of chemicals (drugs/toxicants/environmental pollutants) with descriptors representative of molecular structure and/or properties. QSARs are being applied in many disciplines like risk assessment, toxicity prediction, and regulatory decisions in addition to drug discovery and lead optimization. In this paper, some important topological indices of the pent-heptagonal nanosheet that is covered by pentagons and heptagons are calculated, and formulas for their atom-bond connectivity index, fourth atom-bond connectivity index, Randić connectivity index, sum-connectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometric-arithmetic index, fifth geometric-arithmetic index, Sanskruti index, forgotten index, and harmonic index of two types of pent-heptagonal nanosheets are derived. Our calculated results help in understanding topology of understudy nanotubes. For instance, the first Zagreb index is used to be known as pi-electronic energy. The augmented Zagreb index can be used to investigate the heat of formation. Randić index is the mostly applied and investigated degree-based topological index.

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 paper.

Acknowledgments

This work was supported by the National Key Research and Development Program under grant 2017YFB0802300, Applied Basic Research (Key Project) of Sichuan Province under grant 2017JY0095, Nature Science Key Project of Sichuan Provincial Department of Education under grant 17ZA0079, and Automotive Creative Design Pilot Area of Chengdu University and Longquanyi District under grant 2015-CX00-00010-ZF.

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Advances in Materials Science and Engineering

Topological Indices of the Pent-Heptagonal Nanosheets VC5C7 and HC5C7

Abstract

1. Introduction

2. C5C7 Nanosheets and Their Structures

3. Main Results

4. Conclusions

Data Availability

Conflicts of Interest

Acknowledgments

References

Copyright

Topological Indices of the Pent-Heptagonal Nanosheets VC₅C₇ and HC₅C₇

2. C₅C₇ Nanosheets and Their Structures