Computation of Topological Indices of Graphene

Graphene is an atomic scale honeycomb lattice made of carbon atoms. It is the world’s first 2D material which was isolated from graphite in the year 2004 by Professor Andre Geim and Professor Kostya Novoselov. 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 industries 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. Topological indices are the molecular descriptors that describe the structures of chemical compounds and they help us to predict certain physicochemical properties like boiling point, enthalpy of vaporization, stability, and so forth. In this paper, we determine the topological indices like atom-bond connectivity index, fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, geometric-arithmetic connectivity index, and fifth geometric-arithmetic connectivity index of Graphene. All molecular graphs considered in this paper are finite, connected, loopless, and without multiple edges. Let G = (V, E) be a graph with n vertices andm edges.The degree of a vertexu ∈ V(G) is denoted byd u and is the number of vertices that are adjacent to u. The edge connecting the vertices u and V is denoted by uV. Using these terminologies, certain topological indices are defined in the following manner. The atom-bond connectivity index, ABC index, is one of the degree based molecular descriptors, which was introduced by Estrada et al. [1] in late 1990s, and it can be used formodelling thermodynamic properties of organic chemical compounds; it is also used as a tool for explaining the stability of branched alkanes [2]. Some upper bounds for the atombond connectivity index of graphs can be found in [3]. The atom-bond connectivity index of chemical bicyclic graphs and connected graphs can be seen in [4, 5]. For further results onABC index of trees, see the papers [6–9] and the references cited therein.


Introduction
Graphene is an atomic scale honeycomb lattice made of carbon atoms. It is the world's first 2D material which was isolated from graphite in the year 2004 by Professor Andre Geim and Professor Kostya Novoselov. 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 industries 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.
Topological indices are the molecular descriptors that describe the structures of chemical compounds and they help us to predict certain physicochemical properties like boiling point, enthalpy of vaporization, stability, and so forth. In this paper, we determine the topological indices like atom-bond connectivity index, fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, geometric-arithmetic connectivity index, and fifth geometric-arithmetic connectivity index of Graphene.
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 V is denoted by V. Using these terminologies, certain topological indices are defined in the following manner.
The atom-bond connectivity index, ABC index, is one of the degree based molecular descriptors, which was introduced by Estrada et al. [1] in late 1990s, and it can be used for modelling thermodynamic properties of organic chemical compounds; it is also used as a tool for explaining the stability of branched alkanes [2]. Some upper bounds for the atombond connectivity index of graphs can be found in [3]. The atom-bond connectivity index of chemical bicyclic graphs and connected graphs can be seen in [4,5]. For further results on ABC index of trees, see the papers [6][7][8][9] and the references cited therein. Definition 1. Let = ( , ) be a molecular graph, and is the degree of the vertex ; then ABC index of is defined as The fourth atom-bond connectivity index, ABC 4 ( ) index, was introduced by Ghorbani and Hosseinzadeh [10] in 2010. Further studies on ABC 4 ( ) index can be found in [11,12].

Definition 3. For the graph
Randic index is defined as Sum connectivity index belongs to a family of Randic like indices and it was introduced by Zhou and Trinajstić [14]. Further studies on Sum connectivity index can be found in [15]. The geometric-arithmetic index, GA( ) index, of a graph was introduced by Vukičević and Furtula [16]. Further studies on GA index can be found in [17][18][19].
Definition 6. For a graph , the fifth geometric-arithmetic index is defined as is the sum of the degrees of all neighbors of the vertex in , similarly V .

Main Results
Theorem 7. The atom-bond connectivity index of Graphene with " " rows of benzene rings and " " benzene rings in each row is given by (1) Proof. Consider a Graphene with " " rows and " " benzene rings in each row. Let , denote the number of edges connecting the vertices of degrees and . Two-dimensional structure of Graphene (as shown in Figure 1) contains only 2,2 , 2,3 , and 3,3 edges. In Figure 1 2,2 and 3,3 edges are colored in green and red, respectively. The number of 2,2 , 2,3 , and 3,3 edges in each row is mentioned in Table 1.

Theorem 10. The Sum connectivity index of Graphene is
Proof.
6 Journal of Nanomaterials Theorem 11. The geometric-arithmetic index of Graphene with " " rows and " " benzene rings in each row is given by Proof.
Theorem 12. The fifth geometric-arithmetic index of Graphene is

Conclusion
The problem of finding the general formula for ABC index, ABC 4 index, Randic connectivity index, Sum connectivity index, GA index, and GA 5 index of Graphene is solved here analytically without using computers.