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Abstract

Graph theory has provided chemist with a variety of useful tools, such as topological indices. A topological index Top(G) of a graph G is a number with the property that for every graph H isomorphic to G, Top(H) = Top(G). In this article, we present exact expressions for some topological indices of carbon compound graphene. We compute first Zagreb index, second Zagreb index, first multiple Zagreb index, second multiple Zagreb index, augmented Zagreb index, harmonic index and hyper-Zagreb index of graphene. These are some topological indices based on degrees.

Keywords

Zagreb indices multiple Zagreb indices augmented Zagreb index harmonic index hyper-Zagreb index graphene

Introduction

Graphene is an atomic-scale honeycomb lattice made of carbon atoms. It is one of the most promising nanomaterials because of its unique combination of excellent properties, which opens a way for its exploitation in a wide spectrum of applications ranging from electronics to optics, sensors and biodevices. Also, it is the most effective material for electromagnetic interference shielding.

A graph invariant is any function on a graph that does not depend on a labelling of its vertices. A topological index is a graph invariant applicable in chemistry. By IUPAC terminology, a topological index is a numerical value associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In an exact phrase, if graph denotes the class of all finite graphs, then a topological index is a function Top from graph into real numbers with this property that Top(G) = Top(H), if G and H are isomorphic. Obviously, the number of vertices and the number of edges are topological indices.

Topological indices are the molecular descriptors that describe the structures of chemical compounds, and they help us to predict certain physico–chemical properties such as boiling point, enthalpy of vaporization, stability, and so on. Molecules and molecular compounds are often modelled by molecular graph. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Note that hydrogen atoms are often omitted. All molecular graphs considered in this article are finite, connected, loopless and without multiple edges. Let G = $(V, E)$ be a graph with vertex set V and edge set E. The degree of a vertex $u \in E (G)$ is denoted by d_u and is the number of vertices that are adjacent to u. The edge connecting the vertices u and v is denoted by uv. Recently, Sridhara G et al.¹ determined ABC index, ABC₄ index, Randic connectivity index, sum connectivity index, GA index and GA₅ index of graphene. In this article, we determine the topological indices such as Zagreb indices, multiple Zagreb indices, augmented Zagreb index, harmonic index and hyper-Zagreb index of graphene.

Zagreb indices

Much studied and in chemistry much applied graph invariants are the pair of molecular descriptors (or topological index), known as the first Zagreb index $M_{1} (G)$ and second Zagreb index $M_{2} (G)$ . They first appeared in the topological formula for total ϕ-energy of conjugated molecules that has been derived in 1972 by Gutman and Trinajstić.² Soon after, these indices have been used as branching indices. Later, the Zagreb indices found applications in QSPR and QSAR studies. The Zagreb indices are included in a number of programs used for the routine computation of topological indices, such as POLLY, OASIS, DRAGON, CERIUS, TAM, DISSIM, and so on. $M_{1} (G) and M_{2} (G)$ are, in fact, measures of branching of the molecular carbon–atom skeleton³ and can thus be viewed as molecular structure descriptors. The Zagreb indices and their variants have been used to study molecular complexity, chirality, ZE-isomerism and heterosystems. Overall, Zagreb indices exhibited a potential applicability for deriving multilinear regression models. Details on the chemical applications of the two Zagreb indices can be found in the books by Todeschini and Consonni.^4,5 Further studies on Zagreb indices can be found in the works of Braun et al.,⁶ Gutman and Das,⁷ Zhou and Gutman⁸ and Zhou and Trinajstić.^9,10

Definition 1.1

For a simple connected graph G, the first and second Zagreb indices were defined as follows

M_{1} (G) = \sum_{e = u v \in E (G)} (d_{u} + d_{v}), M_{2} (G) = \sum_{e = u v \in E (G)} d_{u} d_{v}

where d_v denotes the degree (number of first neighbours) of vertex v in G.

Multiple Zagreb indices

In 2012, Ghorbani and Azimi¹¹ defined the multiple Zagreb topological indices of a graph G based on degree of vertices of G.

Definition 1.2

For a simple connected graph G, the first and second multiple Zagreb indices were defined as follows

{PM}_{1} (G) = \prod_{e = u v \in E (G)} (d_{u} + d_{v}), {PM}_{2} (G) = \prod_{e = u v \in E (G)} d_{u} d_{v} .

Properties of the first and second multiple Zagreb indices may be found in the work by Eliasi et al.¹² and Gutman.¹³

Augmented Zagreb index

The augmented Zagreb index was introduced by Furtula et al.¹⁴ This graph invariant has proven to be a valuable predictive index in the study of the heat of formation in octanes and heptanes and is a novel topological index in chemical graph theory, whose prediction power is better than atom–bond connectivity index. Some basic investigations implied that AZI index has better correlation properties and structural sensitivity among the very well-established degree-based topological indices.

Definition 1.3

Let G = $(V, E)$ be a graph and d_u be the degree of a vertex u, then augmented Zagreb index is denoted by $AZI (G)$ and is defined as

AZI (G) = \sum_{u v \in E} {[\frac{d_{u} d_{v}}{d_{u} + d_{v} - 2}]}^{3}

Further studies can be found in the work of Huang et al.¹⁵ and the references cited there in.

Harmonic index

The Harmonic index was introduced by Zhong.¹⁶ It has been found that the harmonic index correlates well with the Randic index and the π-electron energy of benzenoid hydrocarbons.

Definition 1.4

Let G = $(V, E)$ be a graph and d_u be the degree of a vertex u, then Harmonic index is defined as follows

H (G) = \sum_{e = u v \in E (G)} \frac{2}{d_{u} + d_{v}}

Further studies on $H (G)$ can be found in the work by Zhou and Trinajstić⁹ and Wu et al.¹⁷

Hyper-Zagreb index

Shirdel et al.¹⁸ introduced a new distance based on Zagreb indices of a graph G named hyper-Zagreb index.

Definition 1.5

The hyper-Zagreb index is defined as follows

HM (G) = \sum_{e = u v \in E (G)} {(d_{u} + d_{v})}^{2}

Main results

Theorem 2.1

The first Zagreb index of graphene with ‘t’ rows of benzene rings and ‘s’ benzene rings in each row is given by

$M_{1} (G) = {\begin{array}{l} 26 s - 2 & if & t = 1 \\ 8 t + 8 s + 18 t s - 10 & if & t \neq 1 \end{array}$

Proof

Consider a graphene with t rows and s benzene rings in each row. Let $m_{i, j}$ denotes edges connecting the vertices of degrees d_i and d_j. Two-dimensional structure of graphene (as shown in Figure 1) contains only $m_{2, 2}, m_{2, 3} and m_{3, 3}$ edges. In the above figure, $m_{2, 2} and m_{3, 3}$ edges are coloured in green and red, respectively. The number of $m_{2, 2}, m_{2, 3} and m_{3, 3}$ edges in each row is mentioned in the following table:

Row $| m_{2, 2} |$ $| m_{2, 3} |$ $| m_{3, 3} |$

1 3 2s 3s − 2

2 1 2 3s − 1

3 1 2 3s − 1

4 1 2 3s − 1

. . . . . . . . . . . .

t 3 2s s − 1

Total t + 4 4s + 2t − 4 3ts − 2s − t − 1

Figure 1.
Two dimensional structure of graphene with “t” rows and “s” benzene rings in each row.

∴ Clearly from the above table, $| m_{2, 2} | = (t + 4)$ edges, $| m_{2, 3} | = (4 s + 2 t - 4)$ edges and $| m_{3, 3} | = (3 t s - 2 s - t - 1)$ edges.

Case 1

The first Zagreb index of graphene for $t \neq 1$ is

$\begin{array}{l} M_{1} (G) = \sum_{e = u v \in E (G)} (d_{u} + d_{v}) \\ = | m_{2,2} | (2 + 2) + | m_{2,3} | (2 + 3) + | m_{3,3} | (3 + 3) \\ = (t + 4) (4) + (4 s + 2 t - 4) (5) + (3 t s - 2 s - t - 1) (6) \\ = 4 t + 16 + 20 s + 10 t - 20 + 18 t s - 12 s - 6 t - 6 \\ = 8 t + 8 s + 18 t s - 10 \\ ∴ M_{1} (G) = 8 t + 8 s + 18 t s - 10, for t \neq 1 . \end{array}$

Case 2

For $t = 1, | m_{2, 2} | = 6, | m_{2, 3} | = (4 s - 4) and | m_{3, 3} | = (s - 1)$ edges as shown in the following Figure 2.

Figure 2.
Two dimensional structure of graphene with one row of “s” benzene rings.

The first Zagreb index of graphene for t = 1 is

$\begin{array}{l} M_{1} (G) = \sum_{e = u v \in E (G)} (d_{u} + d_{v}) \\ = | m_{2, 2} | (2 + 2) + | m_{2, 3} | (2 + 3) + | m_{3, 3} | (3 + 3) \\ = 6 (4) + (4 s - 4) (5) + (s - 1) (6) \\ ∴ M_{1} (G) = 26 s - 2, for t = 1. \end{array}$

Theorem 2.2

The second Zagreb index of graphene with t rows of benzene rings and s benzene rings in each row is given by

$M_{2} (G) = {\begin{array}{l} 33 s - 9 & if & t = 1 \\ 7 t + 6 s + 27 t s - 17 & if & t \neq 1 \end{array}$

Proof. Case 1

The second Zagreb index of graphene for t ≠ 1 is

$\begin{array}{l} M_{2} (G) = \sum_{e = u v \in E (G)} d_{u} d_{v} \\ = | m_{2, 2} | (2.2) + | m_{2, 3} | (2.3) + | m_{3, 3} | (3.3) \\ = (t + 4) (4) + (4 s + 2 t - 4) (6) + (3 t s - 2 s - t - 1) (9) \\ = 4 t + 16 + 24 s + 12 t - 24 + 27 t s - 18 s - 9 t - 9 \\ ∴ M_{2} (G) = 7 t + 6 s + 27 t s - 17, for t \neq 1. \end{array}$

Case 2

The second Zagreb index of graphene for t = 1 is

$\begin{array}{l} M_{2} (G) = \sum_{e = u v \in E (G)} d_{u} d_{v} \\ = | m_{2, 2} | (2.2) + | m_{2, 3} | (2.3) + | m_{3, 3} | (3.3) \\ = 6 (4) + (4 s - 4) (6) + (s - 1) (9) \\ ∴ M_{2} (G) = 33 s - 9, for t = 1. \end{array}$

Theorem 2.3

The first multiple Zagreb index of graphene with t rows of benzene rings and s benzene rings in each row is given by

$P M_{1} (G) = {\begin{array}{l} (1.092267) \times 5^{4 s} \times 6^{s} & if & t = 1 \\ 2^{(2 t + 8)} \times 5^{(4 s + 2 t - 4)} \times 6^{3 t s - 2 s - t - 1} & if & t \neq 1 \end{array}$

Proof. Case 1

The first multiple Zagreb index of graphene for t ≠ 1 is

$\begin{array}{l} {PM}_{1} (G) = \prod_{e = u v \in E (G)} (d_{u} + d_{v}) \\ = \prod_{e = u v \in m_{2, 2}} (d_{u} + d_{v}) \prod_{e = u v \in m_{2, 3}} (d_{u} + d_{v}) \prod_{e = u v \in m_{3, 3}} (d_{u} + d_{v}) \\ = 4^{(t + 4)} \times 5^{(4 s + 2 t - 4)} \times 6^{3 t s - 2 s - t - 1} \\ = 2^{(2 t + 8)} \times 5^{(4 s + 2 t - 4)} \times 6^{3 t s - 2 s - t - 1}, for t \neq 1 \end{array}$

Case 2

The first multiple Zagreb index of graphene for t = 1 is

$\begin{matrix} {PM}_{1} (G) = \prod_{e = u v \in E (G)} (d_{u} + d_{v}) \\ = \prod_{e = u v \in m_{2, 2}} (d_{u} + d_{v}) \prod_{e = u v \in m_{2, 3}} (d_{u} + d_{v}) \prod_{e = u v \in m_{3, 3}} (d_{u} + d_{v}) \\ = 4^{6} \times 5^{(4 s - 4)} \times 6^{s - 1} \\ = (1.092267) \times 5^{4 s} \times 6^{s}, for t = 1. \end{matrix}$

Theorem 2.4

The second multiple Zagreb index of graphene with t rows of benzene rings and s benzene rings in each row is given by

${PM}_{2} (G) = {\begin{array}{l} (0.351166) \times 6^{4 s} \times 9^{s} & if & t = 1 \\ 2^{(2 t + 8)} \times 6^{(4 s + 2 t - 4)} \times 3^{6 t s - 2 t - 4 s - 2} & if & t \neq 1 \end{array}$

Proof. Case 1

The second multiple Zagreb index of graphene for t ≠ 1 is

$\begin{matrix} {PM}_{2} (G) = \prod_{e = u v \in E (G)} d_{u} d_{v} \\ = \prod_{e = u v \in m_{2, 2}} (d_{u} d_{v}) \prod_{e = u v \in m_{2, 3}} (d_{u} d_{v}) \prod_{e = u v \in m_{3, 3}} (d_{u} d_{v}) \\ = 4^{(t + 4)} \times 6^{(4 s + 2 t - 4)} \times 9^{3 t s - 2 s - t - 1} \\ = 2^{(2 t + 8)} \times 6^{(4 s + 2 t - 4)} \times 3^{6 t s - 2 t - 4 s - 2}, for t \neq 1 \end{matrix}$

Case 2

The second multiple Zagreb index of graphene for t = 1 is

$\begin{matrix} {PM}_{2} (G) = \prod_{e = u v \in E (G)} d_{u} d_{v} \\ = \prod_{e = u v \in m_{2, 2}} (d_{u} d_{v}) \prod_{e = u v \in m_{2, 3}} (d_{u} d_{v}) \prod_{e = u v \in m_{3, 3}} (d_{u} d_{v}) \\ = 4^{6} \times 6^{(4 s - 4)} \times 9^{s - 1} \\ = (0.351166) \times 6^{4 s} \times 9^{s}, for t = 1. \end{matrix}$

Theorem 2.5

The augmented Zagreb index of graphene with t rows of benzene rings and s benzene rings in each row is given by

$AZI(G) = {\begin{array}{l} (43.390625) s + (4.609375) & if & t = 1 \\ (12.609375) t + (9.21875) s + (34.171875) t s - (11.390625) & if & t \neq 1 \end{array}$

Proof. Case 1

The augmented Zagreb index of graphene for t ≠ 1 is

$\begin{matrix} AZI (G) = {\prod_{e = u v \in E (G)} [\frac{d_{u} d_{v}}{d_{u} + d_{v} - 2}]}^{3} \\ = | m_{2, 2} | {(\frac{2.2}{2 + 2 - 2})}^{3} + | m_{2, 3} | {(\frac{2.3}{2 + 3 - 2})}^{3} + | m_{3, 3} | {(\frac{3.3}{3 + 3 - 2})}^{3} \\ = (t + 4) (8) + (4 s + 2 t - 4) (8) + (3 t s - 2 s - t - 1) {(\frac{9}{4})}^{3} \\ = (8 t + 32) + (32 s + 16 t - 32) + (3 t s - 2 s - t - 1) (\frac{729}{64}) \\ = \frac{807 t + 590 s + 2187 t s - 729}{64} \end{matrix}$

$∴ AZI (G) = (12.609375) t + (9.21875) s + (34.171875) t s - (11.390625), for t \neq 1 .$

Case 2

The augmented Zagreb index of graphene for t = 1 is

$\begin{matrix} AZI (G) = {\sum_{e = u v \in E (G)} [\frac{d_{u} d_{v}}{d_{u} + d_{v} - 2}]}^{3} \\ = | m_{2, 2} | {(\frac{2.2}{2 + 2 - 2})}^{3} + | m_{2, 3} | {(\frac{2.3}{2 + 3 - 2})}^{3} + | m_{3, 3} | {(\frac{3.3}{3 + 3 - 2})}^{3} \\ = 6 (8) + (4 s - 4) (8) + (s - 1) {(\frac{9}{4})}^{3} \\ = 48 + 32 s - 32 + (s - 1) (\frac{729}{64}) \\ = \frac{2777 + 295}{64} \end{matrix}$

$∴ AZI (G) = (43.390625) + (4.609375), for t = 1 .$

Theorem 2.6

The harmonic index of graphene with t rows of benzene rings and s benzene rings in each row is given by

$H (G) = {\begin{array}{l} (1.93333) s + (1.0667) & if & t = 1 \\ (0.9667) t + (0.9334) s + t s + (0.0667) & if & t \neq 1 \end{array}$

Proof. Case 1

The harmonic index of graphene for t ≠ 1 is

$\begin{matrix} H (G) = \prod_{e = u v \in E (G)} \frac{2}{d_{u} + d_{v}} \\ = | m_{2, 2} | (\frac{2}{2 + 2}) + | m_{2, 3} | (\frac{2}{2 + 3}) + | m_{3, 3} | (\frac{2}{3 + 3}) \\ = (t + 4) (\frac{1}{2}) + (4 s + 2 t - 4) (\frac{2}{5}) + (3 t s - 2 s - t - 1) (\frac{2}{6}) \\ = \frac{29 t + 28 s + 30 t s + 2}{30} \end{matrix}$

$∴ H (G) = (0.9667) t + (0.9334) s + t s + (0.0667), for t \neq 1 .$

Case 2

The harmonic index of graphene for t = 1 is

$\begin{matrix} H (G) = \prod_{e = u v \in E (G)} \frac{2}{d_{u} + d_{v}} \\ = | m_{2, 2} | (0.5) + | m_{2, 3} | (0.4) + | m_{3, 3} | (0.3333) \\ = 6 (\frac{1}{2}) + (4 s - 4) (\frac{2}{5}) + (s - 1) (\frac{2}{6}) \\ = \frac{29 s + 16}{15} \end{matrix}$

$∴ H (G) = (1.9333) s + (1.0667), for t = 1 .$

Theorem 2.7

The hyper-Zagreb index of graphene with t rows of benzene rings and s benzene rings in each row is given by

$HM (G) = {\begin{array}{l} 136 s - 40 & if & t = 1 \\ 30 t + 28 s + 108 t s - 72 & if & t \neq 1 \end{array}$

Proof. Case 1

The hyper-Zagreb index of graphene for t ≠ 1 is

$\begin{matrix} HM (G) = \sum_{e = u v \in E (G)} {(d_{u} + d_{v})}^{2} \\ = | m_{2, 2} | {(2 + 2)}^{2} + | m_{2, 3} | {(2 + 3)}^{2} + | m_{3, 3} | {(3 + 3)}^{2} \\ = (t + 4) (16) + (4 s + 2 t - 4) (25) + (3 t s - 2 s - t - 1) (36) \end{matrix}$

$∴ H M (G) = 30 t + 28 s + 108 t s - 72, for t \neq 1 .$

Case 2

The hyper-Zagreb index of graphene for t = 1 is

$\begin{matrix} HM (G) = \sum_{e = u v \in E (G)} {(d_{u} + d_{v})}^{2} \\ = | m_{2, 2} | {(2 + 2)}^{2} + | m_{2, 3} | {(2 + 3)}^{2} + | m_{3, 3} | {(3 + 3)}^{2} \\ = 6 (16) + (4 s - 4) (25) + (s - 1) (36) \end{matrix}$

$∴ H M (G) = 136 s - 40, for t = 1 .$

Conclusion

The problem of finding the general formula for first Zagreb index, second Zagreb index, first multiple Zagreb index, second multiple Zagreb index, augmented Zagreb index, harmonic index and hyper-Zagreb index of graphene is solved here analytically without using computers.

Row	$\| m_{2, 2} \|$	$\| m_{2, 3} \|$	$\| m_{3, 3} \|$
1	3	2s	3s − 2
2	1	2	3s − 1
3	1	2	3s − 1
4	1	2	3s − 1
. . .	. . .	. . .	. . .
t	3	2s	s − 1
Total	t + 4	4s + 2t − 4	3ts − 2s − t − 1

Footnotes

Authors' Contributions

All the authors worked together for the preparation of the manuscript and all of us take the full responsibility for the content of the article. However, the first and third authors typed the article and all of us read and approved the final manuscript.

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

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Some results on topological indices of graphene

Abstract

Keywords

Introduction

Zagreb indices

Definition 1.1

Multiple Zagreb indices

Definition 1.2

Augmented Zagreb index

Definition 1.3

Harmonic index

Definition 1.4

Hyper-Zagreb index

Definition 1.5

Main results

Theorem 2.1

Proof

Case 1

Case 2

Theorem 2.2

Proof. Case 1

Case 2

Theorem 2.3

Proof. Case 1

Case 2

Theorem 2.4

Proof. Case 1

Case 2

Theorem 2.5

Proof. Case 1

Case 2

Theorem 2.6

Proof. Case 1

Case 2

Theorem 2.7

Proof. Case 1

Case 2

Conclusion

Footnotes

Authors' Contributions

Declaration of conflicting interests

Funding

References