Research Article | Open Access
Computation of Topological Indices of Graphene
We compute ABC index, ABC4 index, Randic connectivity index, Sum connectivity index, GA index, and GA5 index 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 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 . Using these terminologies, certain topological indices are defined in the following manner.
The atom-bond connectivity index, index, is one of the degree based molecular descriptors, which was introduced by Estrada et al.  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 . Some upper bounds for the atom-bond connectivity index of graphs can be found in . The atom-bond connectivity index of chemical bicyclic graphs and connected graphs can be seen in [4, 5]. For further results on index of trees, see the papers [6–9] and the references cited therein.
Definition 1. Let be a molecular graph, and is the degree of the vertex ; then index of is defined as .
Definition 2. Let be a graph; then its fourth index is defined as , where is sum of the degrees of all neighbours of vertex in . In other words, , similarly for .
The first and oldest degree based topological index is Randic index  denoted by and it was introduced by Milan Randic in 1975. It provides a quantitative assessment of branching of molecules.
Definition 3. For the graph Randic index is defined as .
Definition 4. For a simple connected graph , its Sum connectivity index is defined as .
Definition 5. Let be a graph and let be an edge of ; then, .
The fifth geometric-arithmetic index, , was introduced by Graovac et al.  in 2011.
Definition 6. For a graph , the fifth geometric-arithmetic index is defined as , where is the sum of the degrees of all neighbors of the vertex in , similarly .
2. 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
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 , , and edges. In Figure 1 and edges are colored in green and red, respectively. The number of , , and edges in each row is mentioned in Table 1.
Graphene contains edges, edges, and edges.
Case 1. The atom-bond connectivity index of Graphene for isCase 2. For , , , and edges as shown in Figure 2:
Theorem 8. The fourth atom-bond connectivity index of Graphene is
Proof. Let denote the number of edges of Graphene with and . It is easy to see that the summation of degrees of edge endpoints of Graphene has nine edge types , , , , , , , , and that are enumerated in Table 2. For convenience these edge types are colored by grey, yellow, red, purple, blue, green, lightblue, brown, and black, respectively, as shown in Figure 3.
Case 1. The fourth atom-bond connectivity index of Graphene for isCase 2. For and , Graphene has five types of edges, namely, , , , , and . These edges are colored in orange, pink, red, blue, and lavender, respectively, as shown in Figure 4. The number of edges of these types is shown in Table 3: Case 3. For and , we have only 6 edges of the type as shown in Figure 5:
Theorem 9. The Randic connectivity index of Graphene is
Case 1. For ,Case 2. For ,
Theorem 10. The Sum connectivity index of Graphene is
Case 1. For ,Case 2. For ,
Theorem 11. The geometric-arithmetic index of Graphene with “” rows and “” benzene rings in each row is given by
Case 1. For , Case 2. For ,
Theorem 12. The fifth geometric-arithmetic index of Graphene is
The problem of finding the general formula for index, index, Randic connectivity index, Sum connectivity index, index, and index of Graphene is solved here analytically without using computers.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are thankful to both anonymous referees for their valuable comments and useful suggestions. The second author is also thankful to the University Grants Commission, Government of India, for the financial support under the Grant MRP(S)-0535/13-14/KAMY004/UGC-SWRO.
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