THE FIRST ECCENTRIC ZAGREB INDEX OF THE N TH GROWTH OF NANOSTAR DENDRIMER D 3 [ N ]

THE FIRST ECCENTRIC ZAGREB INDEX OF THE N GROWTH OF NANOSTAR DENDRIMER D3[N ] Zeinab Foruzanfar1, Mohammad Reza Farahani2 , Abdul Qudair Baig3, Wasim Sajjad4 1Department of Engineering Sciences and Physics Buein Zahra Technical University Buein Zahra, Qazvin, IRAN 2Department of Applied Mathematics Iran University of Science and Technology (IUST) Narmak, Tehran, 16844, IRAN 3Department of Mathematics COMSATS Institute of Information Technology Attock Campus, PAKISTAN 4Department of Mathematics University of Sargodha Mandi Bahauddin Campus Mandi Bahauddin, PAKISTAN


Introduction
Chemical graph theory is a branch of mathematical chemistry which has an important effect on the development of the chemical sciences.A topological index is a numerical value associated with the chemical constitution of a certain chemical compound aiming to correlate various physical and chemical properties, or some biological activity in it.Carbon nanostructures have found many potential industrial applications such as energy storage, gas sensors, biosensors, nanoelectronic devices and chemical probes [29], just to name a few.Carbon allotropes such as carbon nanocones and carbon nanotubes have been proposed as possible molecular gas storage devices [1,35].
Let G = (V, E) be a simple connected molecular graph, the vertex and edge sets of a graph G are denoted by V (G) and E(G), respectively.Throughout this paper, graph means simple connected graph [23,24,34].If x, y ∈ V (G) then the distance d(u, v) between u and v is defined as the length of a minimum path connecting u and v.The eccentricity ǫ u of a vertex u in G is the largest distance between u and any other vertex of G.
The Eccentric connectivity index of the molecular graph G, was proposed by Sharma, Goswami and Madan [33] as, where d u is the degree of the vertex u and ǫ u is the eccentricity of the vertex u.
The Zagreb topological index Zg 1 was introduced by I. Gutman and N. Trinajstic in 1972 [23,24] as the sum of the squares of the degrees of all vertices of G Zg where d u denotes the degree of u.Mathematical properties of the First Zagreb index for general graphs can be found in [23,24,31,34].
where ǫ u is the eccentricity of the vertex u and ǫ v is the eccentricity of the vertex v.
In this study, we consider an infinite family of Nanostar Dendrimers and compute its First Eccentric Zagreb index and we call this First Eccentric Zagreb index Zg * 1 (G) by the Third Zagreb index and denote by Zg 3 (G).

Results And Discussion
In this sections, we compute the third Zagreb index Zg 3 (G) of an infinite family of Nanostar Dendrimers, we denote the n th growth of Nanostar Dendrimer (∀n ≥ 1) by From Figure 1, one can see that the general representation of this family of Nanostar has 21(2 n+1 ) − 20 vertices/atoms and 24(2 n+1 − 1) bonds/edges [10]- [19].Also, the Nanostar Dendrimer D 3 [n] has a core depicted in Figure 2 and a repeat element cycle C 6 that we named by Leaf, and obviously the n th growth of Nanostar Dendrimer has of leafs, see Figure 2.

Conclusion
In this paper, we have discussed the Eccentric connectivity index, First Zagreb index and Third Zagreb index.We have considered an infinite family of

Figure 2 :
Figure 2: The added graph in each branch and D 3 [0] is the primal structure of Nanostar Dendrimer D 3 [n].