The First Eccentric Zagreb Index of Linear Polycene Parallelogram of Benzenoid

Let G = (V,E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges, e = uv∈E(G), d(u) is degree of vertex u. Then the first Zagreb polynomial and the first Zagreb index Zg1(G,x) and Zg1(G) of the graph G are defined as ( ) ( ) ∑ u v u d v E G d x + ∈ and ( ) ( ) ∑ e u G v uv E d d = ∈ + respectively. Recently Ghorbani and Hosseinzadeh introduced the first Eccentric Zagreb index as ( ) ( ) ( ) ( ) ( ) ∑ uv E G Zg G ecc v ecc u 1 ∈ + = , that ecc(u) is the largest distance between u and any other vertex v of G. In this paper, we compute this new index (the first Eccentric Zagreb index or third Zagreb index) of an infinite family of linear Polycene parallelogram of benzenoid.


Introduction
By a graph, we mean a finite, undirected, simple graph.We denote the vertex set and the edge set of a graph G by V(G) and E(G), respectively.And the number of first neighbors of vertex u in G (the degree of u) is denoted by d(u).For notation and graph theory terminology not presented here, we follow [1]- [3].All of the graphs in this paper are simple and a topological index of a graph is a number related to a graph which is invariant under graph automorphisms and is a numeric quantity from the structural graph of a molecule.One of the best known and widely used is the Zagreb topological index Zg 1 introduced by I. Gutman and N. Trinajstić in 1972 as [1] ∑ Also, we know another definition of the first Zagreb index as the sum of the squares of the degrees of all vertices of G.
where d u denotes the degree of u.Mathematical properties of the first Zagreb index for general graphs can be found in [4]- [8].Let x,y∈V(G), then the distance d(x,y) between x and y is defined as the length of any shortest path in G connecting x and y [9]- [11].
In other words, The radius and diameter of a graph G are defined as the minimum and maximum eccentricity among vertices of G, respectively.In other words, ( ) Recently in 2012, M. Ghorbani and M. A. Hosseinzadeh introduced a new version of first Zagreb index (the Eccentric version and ecc(v) denotes the eccentricity of vertex v) as follows [12]: In this study, we call this eccentric version of the first Zagreb index by the third Zagreb index and denote by ( ) ( ) ( ) And in continue, a formula of the third Zagreb index for an infinite family of linear Polycene parallelogram of benzenoid by using the Cut Method is obtained.

Results and Discussion
In this sections, we compute the third Zagreb index M 3 (G) for linear Polycene parallelogram of benzenoid P(n,n) (∀n ≥ 1).This family of benzenoid graph has 2n(n+2) vertices/atoms and ( ) ( ) ( )  edges (bonds) [13]- [23].The general representation of linear Po- lycene parallelogram of benzenoid P(n,n) is shown in Proof.∀n∈ℕ, let P(n,n) be the linear Polycene parallelogram of benzenoid, as shown in Figure 1.To achieve our aims, we use of the Cut Method.Definition of the Cut Method and some of its properties are presented in [24].Thus, we encourage readers to look at Therefore, by using above results and [14]- [23], we have the following computations for the third Zagreb index of the linear Polycene parallelogram of benzenoid P(n,n)

Zg 1 (
G,x) and Zg 1 (G) of the graph G are defined as Ghorbani and Hosseinzadeh introduced the first Eccentric Zagreb index as

Figure 1 . 1 .
Now, we can exhibit the closed formula of the third Zagreb index M 3 (H k ) in the following theorem.Theorem Considering the linear Polycene parallelogram of benzenoid P(n,n) (∀n∈ℕ), then its third Zagreb index is equal to

Figure 1
and see all cuts of the linear Polycene parallelogram of benzenoid P(n,n).So according to Figure 1, one can see that the eccentric vertices with degree two are between 2n+1, 2n+2,  , 4n−6, 4n−4, 4n−2, 4n−1 or the number set And also, the eccentric vertices with degree two are between 2n, 2n+1 to 4n−4, 4n−3 or in the number set