Computation of New Degree-Based Topological Indices of Graphene

Hindawi Publishing Corporation Journal of Mathematics Volume 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 Correspondence should be addressed to Rachanna Kanabur; [email protected] Received 24 June 2016; Revised 26 August 2016; Accepted 7 September 2016 Academic Editor: Wai Chee Shiu Copyright © 2016 V. S. Shigehalli and R. 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. 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 (AG1 index), SK index, SK1 index, and SK2 index of a molecular graph 𝐺 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 𝐺 = (𝑉, 𝐸) is a simple graph having 𝑛 = |𝑉| vertices and 𝑚 = |𝐸| edges. The vertices V𝑖 ∈ 𝑉 represent nonhydrogen atoms and the edges (V𝑖 , V𝑗 ) ∈ 𝐸 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 𝐺 = (𝑉, 𝐸) be a graph with 𝑛 vertices and 𝑚 edges. The degree of a vertex 𝑢 ∈ 𝑉(𝐺) is denoted by 𝑑𝑢 and is the number of vertices that are adjacent to 𝑢. The edge connecting the vertices 𝑢 and V is denoted by 𝑢V [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. 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. 2 Journal of Mathematics 1 2 3 4 5 s−2 s−1 s 2 3 ··· 4 .. . .. . t−3 t−2 ··· t−1 t Figure 1 In the upcoming sections, topological indices and their computation of topological indices for Graphene are discussed. Definition 1 (Arithmetic-Geometric (AG1 ) index). Let 𝐺 = (𝑉, 𝐸) be a molecular graph and 𝑑𝑢 be the degree of the vertex 𝑢; then AG1 index of 𝐺 is defined as AG1 (𝐺) = 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) , 𝑢,V∈𝐸(𝐺) 2√𝑑𝐺 (𝑢) ⋅ 𝑑𝐺 (V) ∑ (1) where AG1 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 𝑑𝐺(𝑢) (or 𝑑𝐺(V)) denote the degree of the vertex 𝑢 (or V). Definition 2 (SK index). The SK index of a graph 𝐺 = (𝑉, 𝐸) is defined as SK(𝐺) = ∑𝑢,V∈𝐸(𝐺) ((𝑑𝐺(𝑢) + 𝑑𝐺(V))/2), where 𝑑𝐺(𝑢) and 𝑑𝐺(V) are the degrees of the vertices 𝑢 and V in 𝐺, respectively. Definition 3 (SK1 index). The SK1 index of a graph 𝐺 = (𝑉, 𝐸) is defined as SK1 (𝐺) = ∑𝑢,V∈𝐸(𝐺) ((𝑑𝐺(𝑢) ⋅ 𝑑𝐺(V))/2), where 𝑑𝐺(𝑢) and 𝑑𝐺(V) are the product of the degrees of the vertices 𝑢 and V in 𝐺, respectively. Definition 4 (SK2 index). The SK2 index of a graph 𝐺 = (𝑉, 𝐸) is defined as SK2 (𝐺) = ∑𝑢,V∈𝐸(𝐺) ((𝑑𝐺(𝑢) + 𝑑𝐺(V))/2)2 , where 𝑑𝐺(𝑢) and 𝑑𝐺(V) are the degrees of the vertices 𝑢 and V in 𝐺, respectively. Table 1 Row 1 2 3 4 .. . 𝑡 Total 𝑚2,2 3 1 1 1 .. . 3 𝑡+4 𝑚2,3 2𝑠 2 2 2 .. . 3𝑠 4𝑠 + 2𝑡 − 4 𝑚3,3 3𝑠 − 2 3𝑠 − 1 3𝑠 − 1 3𝑠 − 1 .. . 𝑠−1 3𝑡𝑠 − 2𝑠 − 𝑡 − 1 3. Main Results Theorem 5. The 𝐴𝐺1 index of Graphene having “𝑡” rows of Benzene rings with “𝑠” Benzene rings in each row is given by 𝐴𝐺1 (𝐺) 6√6𝑠𝑡 + (20 − 4√6) 𝑠 + 10𝑡 − (20 − 6√6) { { , 𝑖𝑓 𝑡 ≠ 1 { { 2√6 ={ { (2√6 + 20) 𝑠 + 6√6 − 10 { { , 𝑖𝑓 𝑡 = 1. { 2√6 (2) Proof. Consider a Graphene having “𝑡” rows with “𝑠” Benzene rings in each row. Let 𝑚𝑖,𝑗 denote the number of edges connecting the vertices of degrees 𝑑𝑖 and 𝑑𝑗 . Two-dimensional structure of Graphene (Figure 1) contains only 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edges. The number of 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edge in each row is mentioned in Table 1. Journal of Mathematics 3 Therefore Graphene contains 𝑚2,2 = (𝑡 + 4) edges, 𝑚2,3 = (4𝑠 + 2𝑡 − 4) edges, and 𝑚3,3 = (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) edges. AG1 (𝐺) = = 5 (1) + (4𝑠 − 2) ( 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) , 𝑢,V∈𝐸(𝐺) 2√𝑑𝐺 (𝑢) ⋅ 𝑑𝐺 (V) = 𝑠 + 3 + (4𝑠 − 2) ( ∑ = 2+3 2+2 AG1 (𝐺) = 𝑚2,2 ( ) + 𝑚2,3 ( ) 2√2.2 2√2.3 + 𝑚3,3 ( 3+3 ) 2√3.3 = = (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) ( −𝑡−1 (2√6 + 20) 𝑠 + 6√6 − 10 (3) 5 ) 2√6 2√6 (3𝑡𝑠 − 2𝑠 + 3) + 5 (4𝑠 + 2𝑡 − 4) = 2√6 = = 6√6𝑡𝑠 − 4√6𝑠 + 6√6 + 20𝑠 + 10𝑡 − 20 2√6 6√6 + (20 − 4√6) 𝑠 + 10𝑡 − (20 − 6√6) 2√6 . Case 1. The Arithmetic-Geometric index of Graphene for 𝑡 ≠ 1 is SK (𝐺) 2√6 . (4) Case 2. 𝑡 = 1, 𝑚2,2 = 𝑡 + 4, 𝑚2,3 = 4𝑠 − 2, and 𝑚3,3 = 𝑠 − 2, edges as shown in Figure 2: 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) , AG1 (𝐺) = ∑ 𝑢,V∈𝐸(𝐺) 2√𝑑𝐺 (𝑢) ⋅ 𝑑𝐺 (V) 2+2 2+3 AG1 (𝐺) = 𝑚2,2 ( ) + 𝑚2,3 ( ) 2√2.2 2√2.3 + 𝑚3,3 ( 3+3 ) 2√3.3 5 4 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 4 2√6 6 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) 6 (5) (6) Proof. Consider Graphene having “𝑡” rows with “𝑠” Benzene rings in each row. Let 𝑚𝑖,𝑗 denote the number of edges connecting the vertices of degrees 𝑑𝑖 and 𝑑𝑗 . Two-dimensional structure of Graphene (Figure 1) contains only 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edges. The number of 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edge in each row is mentioned in Table 1. Therefore, Graphene contains 𝑚2,2 = (𝑡 + 4) edges, 𝑚2,3 = (4𝑠 + 2𝑡 − 4) edges, and 𝑚3,3 = (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) edges. SK (𝐺) = AG1 (𝐺) = . 18𝑡𝑠 + 8𝑠 + 8𝑡 − 10 { , 𝑖𝑓 𝑡 ≠ 1 { SK (𝐺) = { 26𝑠 − 2 2 { , 𝑖𝑓 𝑡 = 1. { 2 Now consider the following cases. 6√6 + (20 − 4√6) 𝑠 + 10𝑡 − (20 − 6√6) 2√6 Theorem 6. The SK index of Graphene having “𝑡” rows of Benzene rings with “𝑠” Benzene rings in each row is given by 5 ) + 3𝑡𝑠 − 2𝑠 2√6 = (3𝑡𝑠 − 2𝑠 + 3) + (4𝑠 + 2𝑡 − 4) ( 5 ) √ 2 6 2√6 (𝑠) + 6√6 + 20𝑠 − 10 2√6 4 5 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 4 2√6 6 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) 6 5 ) + (𝑠 − 2) (1) √ 2 6 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) , 2 𝑢,V∈𝐸(𝐺) = 𝑚2,2 ( ∑ 2+2 2+3 3+3 ) + 𝑚2,3 ( ) + 𝑚3,3 ( ) 2 2 2 4 5 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 6 + (3𝑡𝑠 − 2𝑠 − 1) ( ) 2 5 = 2 (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) ( ) 2 + 3 (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) 5 = 2𝑡 + 8 + (4𝑠 + 2𝑡 − 4) ( ) + 9𝑡𝑠 − 6𝑠 − 3𝑡 − 3 2 = = 4𝑡 + 16 + 20𝑠 + 10𝑡 − 20 + 18𝑡𝑠 − 12𝑠 − 6𝑡 − 6 2 18𝑡𝑠 + 8𝑠 + 8𝑡 − 10 . 2 Now consider the following cases. (7) 4 Journal of Mathematics 1 2 3 4 ··· s−2 s−1 9 + (3𝑡𝑠 − 2𝑠 − 1) ( ) 2 s = 2 (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) (3) 9 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) 2 Figure 2 Case 1. The SK index of Graphene for 𝑡 ≠ 1 is SK (𝐺) = 18𝑡𝑠 + 8𝑠 + 8𝑡 − 10 . 2 (8) Case 2. 𝑡 = 1, 𝑚2,2 = 𝑡 + 4, 𝑚2,3 = 4𝑠 − 2, and 𝑚3,3 = 𝑠 − 2, edges as shown in Figure 2: SK (𝐺) = 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) , 2 𝑢,V∈𝐸(𝐺) ∑ 2+2 2+3 SK (𝐺) = 𝑚2,2 ( ) + 𝑚2,3 ( ) 2 2 3+3 ) + 𝑚3,3 ( 2 4 5 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 6 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) . 2 For 𝑡 = 1, 5 = 5 (2) + (4𝑠 − 2) ( ) + (𝑠 − 2) (3) 2 5 = 10 + (4𝑠 − 2) ( ) + 3𝑠 − 6 2 20 + 20𝑠 − 10 + 6𝑠 − 12 26𝑠 − 2 = = . 2 2 9 = 2𝑡 + 8 + 12𝑠 + 6𝑡 − 12 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) 2 = = 4𝑡 + 16 + 24𝑠 + 12𝑡 − 24 + 27𝑡𝑠 − 18𝑠 − 9𝑡 − 9 2 27𝑡𝑠 + 7𝑡 + 6𝑠 − 17 . 2 Now consider the following cases. Case 1. The SK1 index of Graphene for 𝑡 ≠ 1 is SK1 (𝐺) = (9) SK1 (𝐺) = SK1 (𝐺) 𝑑𝐺 (𝑢) ⋅ 𝑑𝐺 (V) , 2 𝑢,V∈𝐸(𝐺) 2×3 3×3 2×2 ) + 𝑚2,3 ( ) + 𝑚3,3 ( ) 2 2 2 4 6 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 = 𝑚2,2 ( (13) 𝑑𝐺 (𝑢) ⋅ 𝑑𝐺 (V) , 2 𝑢,V∈𝐸(𝐺) ∑ SK1 (𝐺) = 𝑚2,2 ( (10) 27𝑡𝑠 + 7𝑡 + 6𝑠 − 17 { , 𝑖𝑓 𝑡 ≠ 1 { 𝑆𝐾1 (𝐺) = { 33𝑠 − 10 2 (11) { , 𝑖𝑓 𝑡 = 1. 2 { Proof. Consider Graphene having “𝑡” rows with “𝑠” Benzene rings in each row. Let 𝑚𝑖,𝑗 denote the number of edges connecting the vertices of degrees 𝑑𝑖 and 𝑑𝑗 . Two-dimensional structure of Graphene (Figure 1) contains only 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edges. The number of 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edge in each row is mentioned in Table 1. Therefore, Graphene contains 𝑚2,2 = (𝑡 + 4) edges, 𝑚2,3 = (4𝑠 + 2𝑡 − 4) edges, and 𝑚3,3 = (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) edges. ∑ 27𝑡𝑠 + 7𝑡 + 6𝑠 − 17 . 2 Case 2. 𝑡 = 1, 𝑚2,2 = 𝑡 + 4, 𝑚2,3 = 4𝑠 − 2, and 𝑚3,3 = 𝑠 − 2, edges as shown in Figure 2: 2×3 2×2 ) + 𝑚2,3 ( ) 2 2 + 𝑚3,3 ( 3×3 ) 2 4 6 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 (14) 9 + (3𝑡𝑠 − 2𝑠 − 1) ( ) 2 Theorem 7. The 𝑆𝐾1 index of Graphene having “𝑡” rows of Benzene rings with “𝑠” Benzene rings in each row is given by SK1 (𝐺) = (12) = 2 (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) (3) For 𝑡 = 1, 9 + (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) ( ) . 2 9 = 2 (1 + 4) + (4𝑠 − 2) 3 + (𝑠 − 2) ( ) 2 9 = 10 + 12𝑠 − 6 + (𝑠 − 2) ( ) 2 = (15) 20 + 24𝑠 − 12 + 9𝑠 − 18 33𝑠 − 10 = . 2 2 Theorem 8. The 𝑆𝐾2 index of Graphene having “𝑡” rows of Benzene rings with “𝑠” Benzene rings in each row is given by 108𝑡𝑠 + 30𝑡 + 28𝑠 − 72 { , 𝑖𝑓 𝑡 ≠ 1 { 𝑆𝐾2 (𝐺) = { 136𝑠 − 42 4 { , 𝑖𝑓 𝑡 = 1. { 4 (16) Journal of Mathematics 5 Proof. Consider Graphene having “𝑡” rows with “𝑠” Benzene rings in each row. Let 𝑚𝑖,𝑗 denote the number of edges connecting the vertices of degrees 𝑑𝑖 and 𝑑𝑗 . Two-dimensional structure of Graphene (Figure 1) contains only 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edges. The number of 𝑚2,2 , 𝑚2,3 , and 𝑚3,3 edge in each row is mentioned in Table 1. Therefore, Graphene contains 𝑚2,2 = (𝑡 + 4) edges, 𝑚2,3 = (4𝑠 + 2𝑡 − 4) edges, and 𝑚3,3 = (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) edges. SK2 (𝐺) = SK2 (𝐺) = 𝑚2,2 ( ∑ 𝑢,V∈𝐸(𝐺) ( 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) 2 ), 2 (17) = 4𝑡 + 16 + (4𝑠 + 2𝑡 − 4) ( = = 25 ) + 9 (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) 4 25 ) + 27𝑡𝑠 − 18𝑠 − 9𝑡 − 9 4 Case 1. The SK2 index of Graphene for 𝑡 ≠ 1 is (18) Case 2. 𝑡 = 1, 𝑚2,2 = 𝑡 + 4, 𝑚2,3 = 4𝑠 − 2, and 𝑚3,3 = 𝑠 − 2, edges as shown in Figure 2: ∑ 𝑢,V∈𝐸(𝐺) SK2 (𝐺) = 𝑚2,2 ( ( 𝑑𝐺 (𝑢) + 𝑑𝐺 (V) 2 ), 2 2+3 2 2+2 2 ) + 𝑚2,3 ( ) 2 2 + 𝑚3,3 ( 3+3 2 ) 2 4 2 5 2 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 6 2 + (3𝑡𝑠 − 2𝑠 − 1) ( ) 2 25 ) + 9 (𝑠 − 2) 4 25 ) + 9𝑠 − 18 4 (20) 80 + 100𝑠 − 50 + 36𝑠 − 72 136𝑠 − 42 = . 4 4 3.1. Conclusion. A generalized formula for ArithmeticGeometric index (AG1 index), SK index, SK1 index, and SK2 index of Graphene has been obtained without using computer. References Now consider the following cases. SK2 (𝐺) = (19) The authors declare that there are no competing interests regarding the publication of this paper. 108𝑡𝑠 + 30𝑡 + 28𝑠 − 72 . 4 108𝑡𝑠 + 30𝑡 + 28𝑠 − 72 . 4 = 4 (1 + 4) + (4𝑠 − 2) ( 25 ) 4 Competing Interests 16𝑡 + 64 + 100𝑠 + 50𝑡 − 100 + 108𝑡𝑠 − 72𝑠 − 36𝑡 − 36 4 SK2 (𝐺) = For 𝑡 = 1, = 4 2 5 2 = (𝑡 + 4) ( ) + (4𝑠 + 2𝑡 − 4) ( ) 2 2 = 4 (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) ( + 9 (3𝑡𝑠 − 2𝑠 − 𝑡 − 1) . = 20 + (4𝑠 − 2) ( 2+3 2 3+3 2 2+2 2 ) + 𝑚2,3 ( ) + 𝑚3,3 ( ) 2 2 2 6 2 + (3𝑡𝑠 − 2𝑠 − 1) ( ) 2 = 4 (𝑡 + 4) + (4𝑠 + 2𝑡 − 4) ( [1] M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Babes, -Bolyai University, Cluj-Napoca, Romania, 2001. [2] N. Trinajstić, Chemical Graph Theory, Mathematical Chemistry Series, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1992. [3] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, USA, 1969. [4] A. Madanshekaf and M. Moradi, “The first geometricarithmetic index of some nanostar dendrimers,” Iranian Journal of Mathematical Chemistry, vol. 5, no. 1, supplement 1, pp. 1–6, 2014. [5] 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. [6] S. M. Hosamani, “Computing Sanskruti index of certain nanostructures,” Journal of Applied Mathematics and Computing, 2016. [7] I. Gutman, “Degree-based topological indices,” Croatica Chemica Acta, vol. 86, no. 4, pp. 351–361, 2013. [8] K. Lavanya Lakshmi, “A highly correlated topological index for polyacenes,” Journal of Experimental Sciences, vol. 3, no. 4, pp. 18–21, 2012. [9] 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. [10] 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. [11] 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. 6 [12] 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. [13] 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. 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