On the Sanskruti Index of Circumcoronene Series of Benzenoid

Author(s)
Yingying Gao^{1},
Mohammad Reza Farahani^{2}^{*},
Muhammad Shoaib Sardar^{3},
Sohail Zafar^{3}

Affiliation(s)

^{1}
Colleage of Pharmacy and Biological Engineering, Chengdu University, Chengdu, China.

^{2}
Department of Applied Mathematics of Iran University of Science and Technology (IUST), Narmak, Tehran, Iran.

^{3}
Department of Mathematics, University of Management and Technology (UMT), Lahore, Pakistan.

ABSTRACT

Let*G* = (*V*; *E*) be a simple connected graph. The sets of vertices and edges of G are denoted by *V* = *V*(*G*) and *E* = *E*(*G*), respectively. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The Sanskruti index *S*(*G*) is a topological index was defined as where *S*_{u} is the summation of degrees of all neighbors of vertex *u* in *G*. The goal of this paper is to compute the Sanskruti index for circumcoronene series of benzenoid.

Let

1. Introduction and Preliminaries

Let $G=\left(V;E\right)$ be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge sets of it are represented by $V=V\left(G\right)$ and $E=E\left(G\right)$ , respectively. In chemical graphs, the vertices correspond to the atoms of the molecule, and the edges represent to the chemical bonds. Note that hydrogen atoms are often omitted. If e is an edge of G, connecting the vertices u and v, then we write $e=uv$ and say “u and v are adjacent”. A connected graph is a graph such that there is a path between all pairs of vertices.

Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. Chemical graph theory is a branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena [1] [2] [3] . This theory had an important effect on the development of the chemical sciences.

In mathematical chemistry, numbers encoding certain structural features of organic molecules and derived from the corresponding molecular graph, are called graph invariants or more commonly topological indices.

Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. One of the best known and widely used is the connectivity index, introduced in 1975 by Milan Randić [4] , who has shown this index to reflect molecular branching.

$R\left(G\right)={\displaystyle \underset{uv\in E\left(G\right)}{\sum}}\frac{1}{\sqrt{{d}_{u}{d}_{v}}}$

where ${d}_{u}$ denotes G degree of vertex u. One of the important classes of connectivity indices is Sanskruti index $S\left(G\right)$ defined as [5]

$S\left(G\right)={\displaystyle \underset{uv\in E\left(G\right)}{\sum}}{\left(\frac{{S}_{u}{S}_{v}}{{S}_{u}+{S}_{v}-2}\right)}^{3}.$

Here our notation is standard and mainly taken from standard books of chemical graph theory [1] [2] [3] .

2. Main Results and Discussions

In this section, we compute the Sanskruti index $S\left(G\right)$ for circumcoronene series of benzenoid. The circumcoronene series of benzenoid is family of molecular graph, which consist several copy of benzene ${C}_{6}$ on circumference. The first terms of this series are ${H}_{1}=\text{benzene},$ ${H}_{2}=\text{coronene},$ ${H}_{3}=\text{circumcoronene},$ ${H}_{4}=\text{circumcircumcoronene}$ , see Figure 1 and Figure 2 where they are shown, also for more study and historical details of this benze- noid molecular graphs see the paper series [6] - [15] .

At first, consider the circumcoronene series of benzenoid ${H}_{k}$ for all integer number $k\ge 1$ . From the structure of ${H}_{k}$ (Figure 2) and references [17] - [23] , one can see that the number of vertices/atoms in this benzenoid molecular

Figure 1. The three graphs ${H}_{1}\mathrm{,}{H}_{2}\mathrm{,}{H}_{3}$ of the circumcoronene series of benzenoid [16] .

Figure 2. The general representation ${H}_{k}$ of the circumcoronene series of benzenoid [16] .

graph is equal to $\left|V\left({H}_{k}\right)\right|=6{k}^{2}$ and the number of edges/bonds is equal to $\left|E\left({H}_{k}\right)\right|=\frac{3\times 6k\left(k-1\right)+2\times 6k}{2}=9{k}^{2}-3k$ . Because, the number of vertices/

atoms as degrees 2 and 3 are equal to $6k$ and $6k\left(k-1\right)$ and in circumco- ronene series of benzenoid molecule, there are two partitions ${V}_{2}=v\in V\left(G\right)|{d}_{v}=2$ and ${V}_{3}=v\in V\left(G\right)|{d}_{v}=3$ of vertices. These partitions imply that there are three partitions ${E}_{4}\mathrm{,}{E}_{5}$ and ${E}_{6}$ of edges set of molecule ${H}_{k}$ with size 6, $12\left(k-1\right)$ and $9{k}^{2}-15k+6$ , respectively. Clearly, we mark the members of ${E}_{4}\mathrm{,}{E}_{5}$ and ${E}_{6}$ by red, green and black color in Figure 2.

From Figure 2, one can see that the summation of degrees of vertices of molecule benzenoid ${H}_{k}$ are in four types, as follow:

• ${S}_{v}={S}_{u}=2+3=5\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}u,v\in {V}_{2}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}uv\in {E}_{4}$

• ${S}_{u}={d}_{v}+{d}_{v}=6\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}u\in {V}_{2},v\in {V}_{3}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}uv\in {E}_{5}$

• ${S}_{u}={d}_{v}+{d}_{v}+3=7\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}u\in {V}_{3},v\in {V}_{2}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}uv\in {E}_{5}$

• ${S}_{u}={S}_{v}={d}_{v}+{d}_{u}+3=9\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}u,v\in {V}_{3}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}uv\in {E}_{6}$

So, the Sanskruti index for circumcoronene series of benzenoid ${H}_{k}\left(k\ge 1\right)$ will be

$\begin{array}{c}S\left({H}_{k}\right)={\displaystyle \underset{uv\in E\left(G\right)}{\sum}}{\left(\frac{{S}_{u}{S}_{v}}{{S}_{u}+{S}_{v}-2}\right)}^{3}\\ =\left(6\right){\left(\frac{5\times 5}{5+5-2}\right)}^{3}+\left(6\right){\left(\frac{5\times 7}{5+7-2}\right)}^{3}+\left(2\times 6\left(k-2\right)\right){\left(\frac{6\times 7}{6+7-2}\right)}^{3}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(6\left(k-1\right)\right){\left(\frac{7\times 9}{7+9-2}\right)}^{3}+\left(9{k}^{2}-21k+12\right){\left(\frac{9\times 9}{9+9-2}\right)}^{3}\\ =\frac{6561}{8}{k}^{2}-\frac{5141425023}{4499456}k+\frac{6499681847}{40495104}.\end{array}$

3. Conclusion

In this paper, we discuss the Sanskruti index. We consider the molecular graph “circumcoronene series of benzenoid” and we compute its Sanskruti index.

Cite this paper

Gao, Y. , Farahani, M. , Sardar, M. and Zafar, S. (2017) On the Sanskruti Index of Circumcoronene Series of Benzenoid.*Applied Mathematics*, **8**, 520-524. doi: 10.4236/am.2017.84041.

Gao, Y. , Farahani, M. , Sardar, M. and Zafar, S. (2017) On the Sanskruti Index of Circumcoronene Series of Benzenoid.

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https://doi.org/10.1002/9783527613106

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https://doi.org/10.1021/ja00856a001

[5] Hosamani, S.M. (2016) Computing Sanskruti Index of Certain Nanostructures. Journal of Applied Mathematics and Computing, 1-9.

[6] Brunvoll, J., Cyvin, B.N. and Cyvin, S.J. (1987) Enumeration and Classification of Benzenoid Hydrocarbons. Symmetry and Regular Hexagonal Benzenoids. Journal of Chemical Information and Computer Sciences, 27, 171-177.

https://doi.org/10.1021/ci00056a006

[7] Chepoi, V. and Klavzar, S. (1998) Distances in Benzenoid Systems: Further Developments. Discrete Mathematics, 192, 27-39.

https://doi.org/10.1016/S0012-365X(98)00064-8

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[9] Diudea, M.V. (2003) Capra A Leapfrog Related Map Operation. Studia Universitatis Babes-Bolyai, 4, 3-21.

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http://match.pmf.kg.ac.rs/electronic_versions/Match63/n2/match63n2_411-424.pdf

[12] Klavzar, S. and Gutman, I. (1997) Bounds for the Schultz Molecular Topological Index of Benzenoid Systems in Terms of Wiener Index. Journal of Chemical Information and Computer Sciences, 37, 741-744.

https://doi.org/10.1021/ci9700034

[13] Klavzar, S. (2008) A Bird’s Eye View of the Cut Method and a Survey of Its Applications in Chemical Graph Theory. MATCH Communications in Mathematical and in Computer Chemistry, 60, 255-274.

[14] Klavzar, S., Gutman, I. and Mohar, B. (1995) Labeling of Benzenoid Systems Which Reects the Vertex-Distance Relations. Journal of Chemical Information and Computer Sciences, 35, 590-593.

https://doi.org/10.1021/ci00025a030

[15] Klavzar, S. and Gutman, I. (1996) A Comparison of the Schultz Molecular Topological Index with the Wiener Index. Journal of Chemical Information and Computer Sciences, 36, 1001-1003.

https://doi.org/10.1021/ci9603689

[16] Farahani, M.R. (2014) A New Version of Zagreb Index of Circumcoronene Series of Benzenoid. New Frontiers in Chemistry (AWUT), 23, 141-147.

[17] Farahani, M.R. (2013) On the Schultz Polynomial, Modified Schultz Polynomial, Hosoya Polynomial and Wiener Index of Circumcoronene Series of Benzenoid. Journal of Applied Mathematics & Informatics, 31, 595-608.

https://doi.org/10.14317/jami.2013.595

[18] Farahani, M.R., Kato, K. and Vlad, M.P. (2012) Omega Polynomials and Cluj-Il-menau Index of Circumcoronene Series of Benzenoid. Studia Universitatis Babes-Bolyai, 57, 177-182.

[19] Farahani, M.R. (2013) Third-Connectivity and Third-Sum-Connectivity Indices of Circumcoronene Series of Benzenoid Hk. Acta Chimica Slovenica, 60, 198-202.

[20] Farahani, M.R. (2012) Computing Ω(G, x) and Θ(G, x) Polynomials of an Infinite Family of Benzenoid. Acta Chimica Slovenica, 59, 965-968.

[21] Farahani, M.R. and Rajesh Kanna, M.R. (2015) Fourth Zagreb Index of Circumcoronene Series of Benzenoid. Leonardo Electronic Journal of Practices and Technologies, 27, 155-161.

http://lejpt.academicdirect.org/A27/155_161.pdf

[22] Farahani, M.R., Rajesh Kanna, M.R., Jamil, M.K. and Imran, M. (2016) Computing the M-Polynomial of Benzenoid Molecular Graphs. Science International (Lahore), 28, 3251-3255.

[23] Gao, W. (2015) The Fourth Geometric-Arithmetic Index of Benzenoid Series. Journal of Chemical and Pharmaceutical Research, 7, 634-639.

[1] West, D.B. (1996) An Introduction to Graph Theory. Prentice-Hall, Upper Saddle River, NJ.

[2] Trinajstic, N. (1992) Chemical Graph Theory. CRC Press, Boca Raton, FL.

[3] Todeschini, R. and Consonni, V. (2000) Handbook of Molecular Descriptors. Wiley, Weinheim.

https://doi.org/10.1002/9783527613106

[4] Randic, M. (1975) Characterization of Molecular Branching. Journal of the American Chemical Society, 97, 6609-6615.

https://doi.org/10.1021/ja00856a001

[5] Hosamani, S.M. (2016) Computing Sanskruti Index of Certain Nanostructures. Journal of Applied Mathematics and Computing, 1-9.

[6] Brunvoll, J., Cyvin, B.N. and Cyvin, S.J. (1987) Enumeration and Classification of Benzenoid Hydrocarbons. Symmetry and Regular Hexagonal Benzenoids. Journal of Chemical Information and Computer Sciences, 27, 171-177.

https://doi.org/10.1021/ci00056a006

[7] Chepoi, V. and Klavzar, S. (1998) Distances in Benzenoid Systems: Further Developments. Discrete Mathematics, 192, 27-39.

https://doi.org/10.1016/S0012-365X(98)00064-8

[8] Dias, J.R. (1996) From Benzenoid Hydrocarbons to Fullerene Carbons. MATCH Communications in Mathematical and in Computer Chemistry, 4, 57-85.

[9] Diudea, M.V. (2003) Capra A Leapfrog Related Map Operation. Studia Universitatis Babes-Bolyai, 4, 3-21.

[10] Dress, A. and Brinkmann, G. (1996) Phantasmagorical Fulleroids. MATCH Communications in Mathematical and in Computer Chemistry, 33, 87-100.

[11] Ilic, A., Klavzar, S. and Stevanovic, D. (2010) Calculating the Degree Distance of Partial Hamming Graphs. MATCH Communications in Mathematical and in Computer Chemistry, 63, 411-424.

http://match.pmf.kg.ac.rs/electronic_versions/Match63/n2/match63n2_411-424.pdf

[12] Klavzar, S. and Gutman, I. (1997) Bounds for the Schultz Molecular Topological Index of Benzenoid Systems in Terms of Wiener Index. Journal of Chemical Information and Computer Sciences, 37, 741-744.

https://doi.org/10.1021/ci9700034

[13] Klavzar, S. (2008) A Bird’s Eye View of the Cut Method and a Survey of Its Applications in Chemical Graph Theory. MATCH Communications in Mathematical and in Computer Chemistry, 60, 255-274.

[14] Klavzar, S., Gutman, I. and Mohar, B. (1995) Labeling of Benzenoid Systems Which Reects the Vertex-Distance Relations. Journal of Chemical Information and Computer Sciences, 35, 590-593.

https://doi.org/10.1021/ci00025a030

[15] Klavzar, S. and Gutman, I. (1996) A Comparison of the Schultz Molecular Topological Index with the Wiener Index. Journal of Chemical Information and Computer Sciences, 36, 1001-1003.

https://doi.org/10.1021/ci9603689

[16] Farahani, M.R. (2014) A New Version of Zagreb Index of Circumcoronene Series of Benzenoid. New Frontiers in Chemistry (AWUT), 23, 141-147.

[17] Farahani, M.R. (2013) On the Schultz Polynomial, Modified Schultz Polynomial, Hosoya Polynomial and Wiener Index of Circumcoronene Series of Benzenoid. Journal of Applied Mathematics & Informatics, 31, 595-608.

https://doi.org/10.14317/jami.2013.595

[18] Farahani, M.R., Kato, K. and Vlad, M.P. (2012) Omega Polynomials and Cluj-Il-menau Index of Circumcoronene Series of Benzenoid. Studia Universitatis Babes-Bolyai, 57, 177-182.

[19] Farahani, M.R. (2013) Third-Connectivity and Third-Sum-Connectivity Indices of Circumcoronene Series of Benzenoid Hk. Acta Chimica Slovenica, 60, 198-202.

[20] Farahani, M.R. (2012) Computing Ω(G, x) and Θ(G, x) Polynomials of an Infinite Family of Benzenoid. Acta Chimica Slovenica, 59, 965-968.

[21] Farahani, M.R. and Rajesh Kanna, M.R. (2015) Fourth Zagreb Index of Circumcoronene Series of Benzenoid. Leonardo Electronic Journal of Practices and Technologies, 27, 155-161.

http://lejpt.academicdirect.org/A27/155_161.pdf

[22] Farahani, M.R., Rajesh Kanna, M.R., Jamil, M.K. and Imran, M. (2016) Computing the M-Polynomial of Benzenoid Molecular Graphs. Science International (Lahore), 28, 3251-3255.

[23] Gao, W. (2015) The Fourth Geometric-Arithmetic Index of Benzenoid Series. Journal of Chemical and Pharmaceutical Research, 7, 634-639.