Computing Some Degree-Based Topological Indices of Graphene

Graphene is one of the most promising nanomaterial 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 bio devices. Inspired by recent work on Graphene of computing topological indices, here we compute new topological indices viz. Arithmetic-Geometric index (AG2 index), SK3 index and Sanskruti index of a molecular graph G and obtain the explicit formulae of these indices for Graphene.


Introduction
A molecular graph is a representation of a chemical compound having atoms as vertices and the bonds between atoms correspond to the edge of the graph.The collection of vertices of a graph, says G, is denoted by V (G) and the set of edges is denoted by E (G).The degree of a vertex v of a graph is the number of vertices of G adjacent to v, denoted by dG (v) or simply as dv [3].
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-dimentional basis it is called structural graphs [7,8].
Topological invariant of a graph is a single number descriptor which is correlated to certain chemical, thermodynamical and biological behavior of the chemical compounds.Several topological indices have been defined over past decades which depend on degree of vertices.

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 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.Now we focus on computation of topological indices of Graphene [1,4].
Motivated by previous research on Graphene [4,6,[11][12][13][14], here we compute three new topological indices viz.Arithmetic-Geometric index (AG2 index), SK3 index and Sanskruti index of a molecular graph G and obtain the explicit formulae of these indices for Graphene.

Arithmetic-Geometric (AG 2 ) Index
Let G = (V, E) be a molecular graph, and SG(u) is the degree of the vertex u, then AG2 index of G is defined as where, AG2 index is considered for distinct vertices.The above equation is the sum of the ratio of the Arithmetic mean and Geometric mean of u and v, where SG (u) (or SG (v)) is the summation of degrees of all neighbours of vertex u (or v) in G.

SK 3 Index
The SK3 index of a graph G = (V, E) is defined as where SG (u) (or SG (v)) is the summation of degrees of all neighbours of vertex u (or v) in G.

Sanskruti Index
Recently, Hosamani [9], studied a novel topological index, namely the Sanskruti index S (G) of a molecular graph G, which is denoted as where SG (u) (or SG (v)) is the summation of degrees of all neighbours of vertex u (or v) in G.

The AG 2 index of Graphene is
Let e i, j denote the number of edges with i=Su and j=Sv.It is easy to see that the summation of degrees of edge endpoints of Graphene has nine edge types e 4,5, e 5,5, e 5,7, e 5,8, e 6,7, e 7,9, e 8,8, e 8,9 and e 9,9 that are enumerated Table 1.For convenience these edge type are colored by grey, yellow, red, purple, blue, green, light blue, brown, and black, respectively, as shown in Figure 1.Case 1.The AG 2 index of Graphene for t ≠1 is .

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Proof: Let e i,j denote the number of edges with i=Su and j=Sv.It is easy to see that the summation of degrees of edge endpoints of Graphene has nine edge types e 4,5, e 5,5, e 5,7, e 5,8, e 6,7, e 7,9, e 8,8, e 8,9 and e 9,9 that are enumerated Table1.For convenience these edge type are colored by grey, yellow, red, purple, blue, green, light blue, brown, and black, respectively, as shown in Figure1.
Case 1.The SK 3 index of Graphene for t ≠1 is Case 2. For t=1 and s>1, Graphene has five types of edges, namely e 4,4, e 4,5, e 5,7, e 6,7, and e 7,7 .These edges are colored in orange, pink, red, blue, and lavender, respectively, as shown in Figure 2. The number of edges of these types is shown in Table 2.
Proof: Let e i, j denote the number of edges with i=Su and j=Sv.It is easy to see that the summation of degrees of edge endpoints of Graphene has nine edge types e 4,5, e 5,5, e 5,7, e 5,8, e 6,7, e 7,9, e 8,8, e 8,9 and e 9,9 that are enumerated Table 1.For convenience these edge type are colored by grey, yellow, red, purple, blue, green, light blue, brown, and black, respectively, as shown in Figure 1.
Case 1.The Sanskruti index of Graphene for t ≠1 is . 2

Conclusion
In this paper we have computed the Arithmetic-Geometric (AG 2 ) index, SK 3 index and Sanskruti index S (G) of Graphene is obtained without using computer.

3 S
290.72s -210.68Case 3.For t=1 and s=1, we have only 6 edges of the type e 4, 4 as shown in Figure

Table 1 .
Nine edge types of Graphene

Table 2 Table 2 .
0588t + 2.0277s + 3ts -1.0032 Case 2. For t=1 and s>1, Graphene has five types of edges, namely e 4,4, e 4,5, e 5,7, e 6,7, and e 7,7 .These edges are colored in orange, pink, red, blue, and lavender, respectively, as shown in Figure2.The number of edges of these types is shown in The Number of Edges For t=1 and s=1, we have only 6 edges of the type e 4, 4 as shown in Figure3.
Graphene has five types of edges, namely e 4,4, e 4,5, e 5,7, e 6,7, and e 7,7 .These edges are colored in orange, pink, red, blue, and lavender, respectively, as shown in Figure2.The number of edges of these types is shown in Table2.