Fourth atom-bond connectivity index of an infinite class of Nanostar Dendrimer

The atom-bond connectivity (ABC) index of a graph G is a connectivity topological index was defined as  where d v denotes the degree of vertex v of G . In 2010, M. Ghorbani et. al . introduced a new version of atom-bond connectivity index as  where  In this paper, we compute a cloused formula of ABC4 index of an infinite class of Nanostar Dendrimer D 3 [n] . A Dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers.


INTRODUCTION
Let G=(V;E) be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge sets of which 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 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". A connected graph is a graph such that there is a path between all pairs of vertices.
Chemical graph theory is an important branch of mathematics chemistry. A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity [1][2][3][4][5][6][7][8].
The chemical graph theory is an important branch of mathematical chemistry. In this branch, there are many molecular descriptors (or Topological Index), that have very useful properties to study of chemical molecules. A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.
First connectivity index introduced in 1975 by Milan Randić [9], who has shown this index to reflect molecular branching and called the branching index, that later became the well-known Randić connectivity index and defined as: In 2009, Furtula et al. [10,11] introduced atom-bond connectivity (ABC) index, which it has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. This index is defined as follows: For further research on these connectivity indices see paper series [13][14][15][16][17][18].
Dendrimers are one of the main objects of Nano biotechnology. Here a dendrimer is a synthetic 3-dimensional macromolecule that is prepared in a step-wise fashion from simple branched monomer units, the nature and functionality of which can be easily controlled and varied.
Dendrimers are now considered to be one of the prime nanometer-scale building blocks for the construction of nanoscale objects, molecular devices and molecular 'machines', advanced drug-delivery systems, etc.
Dendrimers are one of the main objects of Nano biotechnology and is a part of a new group of macromolecules that seem photon funnels just like artificial antennas and also, it is a great resistant of photo bleaching. The Nanostar dendrimer promises to have great applications but first the structure and the energy transfer mechanism must be understood. For further study on this topic, we encourage the reader to consult papers [18][19][20][21][22][23][24][25][26][27]. In this paper, we continue this work to compute the fourth atom-bond connectivity index of this infinite class of Dendrimer D3[n].

Main Results and Discussions
In this section we compute the truncated ABC4 index of an infinite class of nanostar dendrimers and we have following theorems, immediately.
be the nanostar dendrimer . n   (Figures 1 and 2). This nanostar dendrimer, we define an element as Figure 3 by "Leaf", every Leaf consist of a cycle C6 or chemically Benzene and add 3(2 n ) leafs to D3[n-1] in the n th growth.

Fig. 3 "Leaf", the added graph in each branch of D3[n].
Therefore, there exist the number of leafs(C6) is equal to
From Figure 4, one can see that for every vertex vV1 (hydrogen (H)