COMPUTING SANSKRUTI INDEX OF DENDRIMER NANOSTARS

Let G = (V ;E) be a simple connected graph. The Sanskruti index was introduced by Hosamani [7] and 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 give explicit formulas for the Sanskruti index of an infinite class of dendrimer nanostars.


Introduction and Preliminaries
Let G = (V ; E) be a simple connected graph.In chemical graph theory, the sets of vertices and edges of G are denoted by V = V (G) and E = E(G), respectively.A molecular graph is a simple finite graph such that its vertices correspond to the atoms and the edges to the bonds.A general reference for the notation in graph theory is [1]- [5].
In chemical graph theory, we have many different topological index of arbitrary molecular graph G.A topological index of a graph is a number related to a graph which is invariant under graph automorphisms.Obviously, every topological index defines a counting polynomial and vice versa.
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 R(G) = uv∈E(G) dudv introduced in 1975 by Milan Randić (see [6]), who has shown this index to reflect molecular branching.
The Sanskruti index S(G) of a graph G is defined as follows (see [7]):    Proof.Consider Nanostar Dendrimer D 3 [n] for every n ≥ 0, (see Figures 1 and 2).From Figure 2 and Ref. [17], one can see that the number of vertices/atoms in this nanostar is equal to |V (D 3 Since all vertices/atoms of nanostar Dendrimer have degree 3, 2 and 1 (hydrogen (H) atom), we divide the vertex/atom set of D 3 [n] in three partitions as By according to the 2−Dimensional of dendrimer D 3 [n] in Figure 2, one can see that New we can divide the edge/bond set E(D 3 [n]) in three partitions as follow: And also, summation of degrees of edge endpoints of this nanostar have six types e (3,5) , e (5,5) , e ′ (5,5) ,e (5,7) , e (7,9) and e (9,9) that are shown in Figure 2 by red, yellow, green, blue, hoary and black colors.Since for all edge e = uv of the types e (3,5) , S v = 3 (for all hydrogen H atom) and S u = 5 and for an edge xy of the types e ′ (5,5) , S x = S y = 5, such that vertices x and y are one of adjacent vertices of degree 2 and other types are analogous.From Figure 2, the number of edges of these edge types are shown in following table.Thus, by using above Table and

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where S u is the summation of degrees of all neighbors of vertex u in G.In this paper, we shall give explicit computing formulas for the Sanskruti index of an infinite class of Nanostars Dendrimer D 3[n].For further study, we encourage the reader to consult papers[9]-[26].In this paper, for every infinite integer n D 3 [n] denotes the n th growth of nanostar dendrimer.In following figures, a kind of 3 th growth of dendrimer and D 3 [0] are shown.Here our notations are standard and mainly taken from standard books of chemical graph theory[1]-[5].

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Main Results and Discussion Let D 3 [n] denote a kind of dendrimer nanostars with n growth stages, see for example Figures 1 and 2, the goal of this paper is to compute a closed formula of this new Connectivity index "Sanskruti index" of D 3 [n] for every n ≥ 0 as follows: Theorem 2.1.The Sanskruti index S(G) of Nanostar Dendrimer D 3 [n] for every n ≥ 0 is equal to S(D 3 [n] [n])| = 24(2 n ) − 20 and also the number of edges/bonds is |E(D 3 [n])| = 24(2 n+1 − 1).

Figure 3 ,
we can deduce the following formula for Sanskruti index S(G) of Nanostar Dendrimer D 3 [n] ∀n ≥ 0, as follow:S(D 3 [n]) = uv∈E(G) S u S v S u + S v − 2