COMPUTING SANSKRUTI INDEX OF TURC4C8(S) NANOTUBE

COMPUTING SANSKRUTI INDEX OF TURC₄C₈(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 TURC₄C₈(S) nanotube.

Topological Index, Connectivity index, Sanskruti Index, TURC₄C₈(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 TURC₄C₈(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 = TURC₄C₈(S) nanotube and suppose that there are rs cycle C₈ and C₄ in its structure. Let us denote this graph simply by TUC₄C₈[r; s]. Obviously TUC₄C₈[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 TUC₄C₈[r; s], with r rows and s columns in following theorem.

Theorem 2.1: Let G be the 2 - Dimensional Lattice of TURC₄C₈[r; s] nanotube (r; s > 1). Then the Sanskruti index of G is equal to:

FIG. 1: THE 3 DIMENSIONAL LATTICE (OR CYLINDER) OF TURC₄C₈(S) NANOTUBE ¹⁹

FIG. 2: DIMENSIONAL LATTICE OF TUC₄C₈[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

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 TURC₄C₈(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 TURC₄C₈(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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