Journal of Mathematics
Volume 2016 (2016), Article ID 4341919, 6 pages
http://dx.doi.org/10.1155/2016/4341919

Research Article

Computation of New Degree-Based Topological Indices of Graphene

V. S. Shigehalli and Rachanna Kanabur

Department of Mathematics, Rani Channamma University, Belagavi, Karnataka 591156, India

Received 24 June 2016; Revised 26 August 2016; Accepted 7 September 2016

Academic Editor: Wai Chee Shiu

Copyright © 2016 V. S. Shigehalli and Rachanna Kanabur. 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.

Abstract.

Graphene is one of the most promising nanomaterials because of its unique combination of superb properties, which opens a way for its exploitation in a wide spectrum of applications ranging from electronics to optics, sensors, and biodevices. Inspired by recent work on Graphene of computing topological indices, here we propose new topological indices, namely, Arithmetic-Geometric index ( index), SK index, SK₁ index, and SK₂ index of a molecular graph and obtain the explicit formulae of these indices for Graphene.

1. Introduction

A topological index of a chemical compound is an integer, derived following a certain rule, which can be used to characterize the chemical compound and predict certain physiochemical properties like boiling point, molecular weight, density, refractive index, and so forth [1, 2].

A molecular graph is a simple graph having vertices and edges. The vertices represent nonhydrogen atoms and the edges represent covalent bonds between the corresponding atoms. In particular, hydrocarbons are formed only by carbon and hydrogen atom and their molecular graphs represent the carbon skeleton of the molecule [1, 2].

Molecular graphs are a special type of chemical graphs, which represent the constitution of molecules. They are also called constitutional graphs. When the constitutional graph of a molecule is represented in a two-dimensional basis, it is called structural graph [1, 2].

All molecular graphs considered in this paper are finite, connected, loopless, and without multiple edges. Let be a graph with vertices and edges. The degree of a vertex is denoted by and is the number of vertices that are adjacent to . The edge connecting the vertices and is denoted by [3].

2. Computing the Topological Indices of Graphene

Graphene is an atomic scale honeycomb lattice made of carbon atoms. Graphene is 200 times stronger than steel, one million times thinner than a human hair, and world’s most conductive material. So it has captured the attention of scientists, researchers, and industrialists worldwide. It is one of the most promising nanomaterials because of its unique combination of superb 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 (EMI) shielding. Now we focus on computation of topological indices of Graphene [4–6].

Motivated by previous research on Graphene, here we introduce four new topological indices and computed their corresponding topological index value of Graphene [7–13].

In Figure 1, the molecular graph of Graphene is shown.

Figure 1

2.1. Motivation

By looking at the earlier results for computing the topological indices for Graphene, here we introduce new degree-based topological indices to compute their values for Graphene.

In the upcoming sections, topological indices and their computation of topological indices for Graphene are discussed.

Definition 1 (Arithmetic-Geometric (AG₁) index). Let be a molecular graph and be the degree of the vertex ; then AG₁ index of is defined as

where index is considered for distinct vertices.
The above equation is the sum of the ratio of the Arithmetic mean and Geometric mean of and , where (or denote the degree of the vertex (or ).

Definition 2 (SK index). The SK index of a graph is defined as , where and are the degrees of the vertices and in , respectively.

Definition 3 (SK₁ index). The index of a graph is defined as , where and are the product of the degrees of the vertices and in , respectively.

Definition 4 (SK₂ index). The SK₂ index of a graph is defined as , where and are the degrees of the vertices and in , respectively.

3. Main Results

Theorem 5. The index of Graphene having “” rows of Benzene rings with “” Benzene rings in each row is given by

Proof. Consider a Graphene having “” rows with “” Benzene rings in each row. Let denote the number of edges connecting the vertices of degrees and . Two-dimensional structure of Graphene (Figure 1) contains only , , and edges. The number of , , and edge in each row is mentioned in Table 1.
Therefore Graphene contains edges, edges, and edges.

Now consider the following cases.
Case 1. The Arithmetic-Geometric index of Graphene for is

Case 2. , , , and , edges as shown in Figure 2:

Table 1

Row

1
3

2
1
2

3
1
2

4
1
2

3

Total

Figure 2

Theorem 6. The SK index of Graphene having “” rows of Benzene rings with “” Benzene rings in each row is given by

Proof. Consider Graphene having “” rows with “” Benzene rings in each row. Let denote the number of edges connecting the vertices of degrees and . Two-dimensional structure of Graphene (Figure 1) contains only , , and edges. The number of , , and edge in each row is mentioned in Table 1.
Therefore, Graphene contains edges, edges, and edges.

Now consider the following cases.
Case 1. The SK index of Graphene for is

Case 2. , , , and , edges as shown in Figure 2:

For ,

Theorem 7. The index of Graphene having “” rows of Benzene rings with “” Benzene rings in each row is given by

Proof. Consider Graphene having “” rows with “” Benzene rings in each row. Let denote the number of edges connecting the vertices of degrees and . Two-dimensional structure of Graphene (Figure 1) contains only , , and edges. The number of , , and edge in each row is mentioned in Table 1.
Therefore, Graphene contains edges, edges, and edges.

Now consider the following cases.
Case 1. The SK₁ index of Graphene for is

Case 2. , , , and , edges as shown in Figure 2:

For ,

Theorem 8. The index of Graphene having “” rows of Benzene rings with “” Benzene rings in each row is given by

Proof. Consider Graphene having “” rows with “” Benzene rings in each row. Let denote the number of edges connecting the vertices of degrees and . Two-dimensional structure of Graphene (Figure 1) contains only , , and edges. The number of , , and edge in each row is mentioned in Table 1.
Therefore, Graphene contains edges, edges, and edges.

Now consider the following cases.
Case 1. The SK₂ index of Graphene for is

Case 2. , , , and , edges as shown in Figure 2:

For ,

3.1. Conclusion

A generalized formula for Arithmetic-Geometric index (AG₁ index), SK index, SK₁ index, and SK₂ index of Graphene has been obtained without using computer.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

References

M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Babeș-Bolyai University, Cluj-Napoca, Romania, 2001.

N. Trinajstić, Chemical Graph Theory, Mathematical Chemistry Series, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1992.

F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, USA, 1969.

A. Madanshekaf and M. Moradi, “The first geometric-arithmetic index of some nanostar dendrimers,” Iranian Journal of Mathematical Chemistry, vol. 5, no. 1, supplement 1, pp. 1–6, 2014.

G. Sridhara, M. R. R. Kanna, and R. S. Indumathi, “Computation of topological indices of graphene,” Journal of Nanomaterials, vol. 2015, Article ID 969348, 8 pages, 2015.

S. M. Hosamani, “Computing Sanskruti index of certain nanostructures,” Journal of Applied Mathematics and Computing, 2016.

I. Gutman, “Degree-based topological indices,” Croatica Chemica Acta, vol. 86, no. 4, pp. 351–361, 2013.

K. Lavanya Lakshmi, “A highly correlated topological index for polyacenes,” Journal of Experimental Sciences, vol. 3, no. 4, pp. 18–21, 2012.

S. M. Hosamani and B. Basavanagoud, “New upper bounds for the first Zagreb index,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 74, no. 1, pp. 97–101, 2015.

S. M. Hosamani and I. Gutman, “Zagreb indices of transformation graphs and total transformation graphs,” Applied Mathematics and Computation, vol. 247, pp. 1156–1160, 2014.

S. M. Hosamani, S. H. Malaghan, and I. N. Cangul, “The first geometric-arithmetic index of graph operations,” Advances and Applications in Mathematical Sciences, vol. 14, no. 6, pp. 155–163, 2015.

V. S. Shegehalli and R. Kanabur, “Arithmetic-Geometric indices of some class of Graph,” Journal of Computer and Mathematical Sciences, vol. 6, no. 4, pp. 194–199, 2015.

V. S. Shegehalli and R. Kanabur, “Arithmetic-Geometric indices of Path Graph,” Journal of Computer and Mathematical Sciences, vol. 6, no. 1, pp. 19–24, 2015.