#### COMPUTING SANSKRUTI INDEX OF TURC4C8(S) NANOTUBE

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**COMPUTING SANSKRUTI INDEX OF TURC _{4}C_{8}(S) NANOTUBE**

Y. Y. Gao ^{1}, M. S. Sardar ^{2}, S. M. Hosamani ^{3} and M. R. Farahani ^{* 4}

College of Pharmacy and Biological Engineering ^{1}, Chengdu University, Chengdu - 610106, China. University of Management and Technology (UMT) ^{2}, Lahore, Pakistan.

Department of Mathematics ^{3}, Rani Channamma University, Belgavi - 591156, Karnataka, India.

Department of Applied Mathematics ^{4}, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran.

**ABSTRACT:** Among topological descriptors connectivity indices are very important and they have a prominent role in chemistry. One of them is Sanskruti index defined as , where Su is the summation of degrees of all neighbours of vertex u in G. In this chapter we compute this new topological index for TURC_{4}C_{8}(S) nanotube.

Keywords: |

Topological Index, Connectivity index, Sanskruti Index, TURC_{4}C_{8}(S), Nanotube

**INTRODUCTION: **Let G be a simple connected graph in chemical graph theory. The vertices and edges of a graph also correspond to the atoms and bonds of the molecular graph, respectively. If e is an edge / bond of G, connecting the vertices /atoms u and v, then we write e = uv and say "u and v are adjacent". A simple graph is an un-weighted, undirected graph without loops or multiple edges. And also a connected graph is a graph such that there is a path between all pairs of vertices. Clearly, a molecular graph is a simple connected graph. A topological index is a numeric quantity from the structural graph of a molecule and is invariant on the automorphism of the graph.

And computing topological indices of molecular graphs from chemical graph theory is an important branch of mathematical chemistry ^{1 - 3}. One of the best known and widely used is the Randić connectivity index and introduced in 1975 by Milan Randić ^{1}, who has shown this index to reflect molecular branching.

The Sanskruti index S(G) of a graph G is defined in ^{25-28} as follows:

Where S_{u} is the summation of degrees of all neighbours of vertex u in G. The goal of this chapter is to study this new index and computing Sanskruti index of famous nano - structure TURC_{4}C_{8}(S) nanotubes. Our notation is standard and mainly taken from standard books of chemical graph theory ^{3}. One can see the references ^{4-11}, for more details about topological and connectivity indices

**Preliminaries: **Consider the molecular graph G = TURC_{4}C_{8}(S) nanotube and suppose that there are rs cycle C_{8} and C_{4} in its structure. Let us denote this graph simply by TUC_{4}C_{8}[r; s]. Obviously TUC_{4}C_{8}[r; s] nanotube has 8rs + 2r vertices and 12rs + r edges. For further study and more detail of this nanotube, see the paper series ^{4-8, 10} and the general representation of this nano structure is shown in **Fig. 1** and **Fig. 2**.

The goal of this section is computing the Sanskruti index of a lattice of TUC_{4}C_{8}[r; s], with r rows and s columns in following theorem.

**Theorem 2.1:** Let G be the 2 - Dimensional Lattice of TURC_{4}C_{8}[r; s] nanotube (r; s > 1). Then the Sanskruti index of G is equal to:

**FIG. 1: THE 3 DIMENSIONAL LATTICE (OR CYLINDER) OF TURC _{4}C_{8}(S) NANOTUBE ^{19}**

^{ }** **

**FIG. 2: DIMENSIONAL LATTICE OF TUC _{4}C_{8}[R; S] ^{19}**

**Proof:** Consider now 2 dimensional graph of lattice G = TUC4C8[r; s] (r; s > 1) depicted in **Fig. 1**. Summation of degrees of edge endpoints of this graph have ve types e(5;5); e(5;8); e(8;8); e(8;9) and e(9;9) that are shown in **Fig. 2** by red, blue, yellow, green and black colors. In other word for all edge e = uv of the types e(5;5); S(v)=S(u)=5 and for an edge f = vw of the types e(5;8); S(v)=5 and S(w)=8 and other types are analogous. Also the number of edges of the types e(5;5) and e(5;8) are equal to 2r and 22r, respectively and for other types see following table.

**TABLE 1: ****SUMMATION OF DEGREES OF EDGE ENDPOINTS**

Summation of degrees of edge endpoints |
^{e}(5;5) |
^{e}(5;8) |
^{e}(8;8) |
^{e}(8;9) |
^{e}(9;9) |

Number of edges of this type |
2r |
4r |
2r |
4r |
12r-11r |

**CONCLUSION:** In chemical graph theory, mathematical chemistry and mathematical physics, molecular descriptors, topological and connectivity indices are very important and useful and have more applications which characterize a molecular graph topology. In this work, a new connectivity topological index called "Sanskruti index" of TURC_{4}C_{8}(S) nanotube was determined. Further works in this line are soon to be communicated ^{9-24}.

**ACKNOWLEDGEMENT: **Nil.

**CONFLICTS OF INTEREST: **Nil.

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**How to cite this article:**Gao YY, Sardar MS, Hosamani SM and Farahani MR: Computing sanskruti index of TURC

_{4}C_{8}(s) nanotube. Int J Pharm Sci Res 2017; 8(10): 4423-25.doi: 10.13040/IJPSR.0975-8232.8(10).4423-25.

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# Article Information

48

4423-4425

371

358

English

IJPSR

Y. Y. Gao, M. S. Sardar, S. M. Hosamani and M. R. Farahani *

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

MrFarahani88@gmail.com

26 February, 2017

25 April, 2017

12 May, 2017

10.13040/IJPSR.0975-8232.8(10).4423-25

01 October, 2017