COMPUTING SANSKRUTI INDEX OF V-PHENYLENIC NANOTUBES AND NANOTORI

Among topological descriptors connectivity topological indices are very important and they have a prominent role in chemistry. One of them is Sanskruti index defined as S(G) = ∑ uv∈E(G)( SuSv Su+Sv−2 ) where Su is the summation of degrees of all neighbors of vertex u in G. In this paper we compute this topological index for V-phenylenic nanotube and nanotori. AMS Subject Classification: 05C90, 05C35, 05C12


Introduction and Preliminaries
Let G = (V ; E) 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 (G) and E = E(G), respectively.In chemical graphs, the vertices correspond to the atoms of the molecule, and the edges represent to the chemical bonds.Also, 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.
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]- [5]).This theory had an important effect on the development of the chemical sciences.
Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.In other words, if G be the connected graph, then we can introduce many connectivity topological indices for it, by distinct and different definition.A connected graph is a graph such that there is a path between all pairs of vertices.One of the best known and widely used is the connectivity index, introduced in 1975 by Milan Randić [6], who has shown this index to reflect molecular branching and defined as follows: The Sanskruti index S(G) of a graph G is defined as follows (see [7]- [11]): where S u is the summation of degrees of all neighbors of vertex u in G.In Refs [12]- [27] some topological indices of V −phenylenic nanotube and V −phenylenic nanotori are computed.In this paper, we continue this work to compute the Sanskruti index of molecular graphs related to V −phenylenic nanotube and nanotori.Our notation is standard and mainly taken from Refs.

Main Results and Discussion
The goal of this section is to computing the Sanskruti index of V − phenylenic nanotube and nanotori.The novel phenylenic and naphthylenic lattices pro-posed can be constructed from a square net embedded on the toroidal surface.Phenylenes are polycyclic conjugated molecules, composed of four membered ring (=square) and six-membered rings (=hexagons) such that every four membered ring (4-membered cycle) is adjacent to two 6-membered cycles, and no two six-membered rings are mutually adjacent.Each four-membered ring lies between two six-membered rings, and each hexagon is adjacent only two fourmembered rings.Because of such structural features phenylenes are very interesting conjugated species [28]- [33].The rapid development of the experimental study of phenylenes motivated a number of recent theoretical studies of thee conjugated π-electron systems [33].Following M. V. Diudea [5] we denote a V -Phenylenic nanotube and V -Phenylenic nanotorus by G = V P HX[m, n] and H = V P HY [m, n], respectively.The general representation of these nano structures are shown in Figure 1 and Figure 2.For more information and background materials, refer to paper series [12]- [33] again.Now we have following theorems, immediately.Proof.Consider the V −phenylenic nanotube G = V P HX[m, n] with 6mn vertices and 9mn − m edges (Figure 1).In V − phenylenic molecule, there are two partitions ), since the degree of an arbitrary vertex/ atom of a molecular graph is equal to 2 or 3. Next, the two partitions of E(G) are Also, two adjacent vertices v 1 ,v 2 of a vertex v ∈ V 2 have degree three, then S v = 2×3 = 6 and two edges vv 1 and vv 2 belong to E 5 (and Also, for all vertices u in first and end row of V −phenylenic nanotube with degree three, Finally, for other vertices S w = 9, because all other vertices and their edges belong to V 3 and E 6 , respectively.So, the Sanskruti index S(G) of V P HX[m, n](m, n ≥ 1) will be Proof.The proof is easily, since by considering the V −phenylenic nanotori H = V P HY [m, n] with 6mn vertices and 9mn edges (Figure 2).We see that this nanotori is a Cubic graph and all vertices belong to V 3 and ∀v ∈ V (V P HY [m, n]) S v = 9.This implies that all edges belong =(9mn).( 9.9 9 + 9 − 2 ) 3 = 4782969 4096 mn.

Conclusions
In this report, we study some properties of a new connectivity index of (molecular) graphs that called Sanskruti index.This connectivity index was defined as follows: where S u is the summation of degrees of all neighbors of vertex u in G.In continue, closed analytical formulas for S(G) of a physico chemical structure of phenylenic nanotubes and nanotorus are given.These nano structures are V −Phenylenic Nanotube V P HX[m, n] and V -phenylenic nanotorus V P HY [m, n].

Theorem 2 . 1 .
∀m, n ∈ N , the Sanskruti index S(G) of V − Phenylenic Nanotube V P HX[m, n] is equal to S(V P HX[m, n])

Figure 1 :
Figure 1: The Molecular Graph of V −Phenylenic Nanotube V P HX[m, n].