COMPUTING SANSKRUTI INDEX OF TURC4C8(S) NANOTUBEHTML Full Text
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 Su 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
|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||
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.
CONFLICTS OF INTEREST: Nil.
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- Hosamani SM: Computing Sanskruti index of certain nanostructures. Journal of Applied Math-ematics and Computing. In press.
- Sardar MS, Zafar S and Farahani MR: Computing Sanskruti index of the Polycyclic Aromatic Hydrocarbons. Geology, Ecology, and Landscapes (TGEL) 2017; 1(1): 37-40.
- Gao YY, Sardar MS, Zafar S and Farahani MR. Sanskruti index of Benzenoid molecular graphs. Applied Mathematics (scirp), 7403538. In press, 2017.
- Y. Gao, M.S. Sardar, S. Zafar, M.R. Farahani. Computing Sanskruti index of Dendrimer Nanostars. International Journal of Pure and Applied Mathematics. In press, 2017.
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.
All © 2013 are reserved by International Journal of Pharmaceutical Sciences and Research. This Journal licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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.
26 February, 2017
25 April, 2017
12 May, 2017
01 October, 2017