^{1}

^{1}

^{*}

^{2}

^{3}

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
*Zg*
_{1}(
*G*,
*x*) and
*Zg*
_{1}(
*G*) of the graph G are defined as Σ
_{uv∈E(G)}x
^{(du+dv)} and Σ
_{e=uv∈E(G)}(
*d*
_{u}+
*d*
_{v}) respectively. Recently Ghorbani and Hosseinzadeh introduced the first Eccentric Zagreb index as
* Zg*
_{1}
^{*}=Σ
_{uv∈E(G)}(
*ecc*(
*v*)+
*ecc*(
*u*)), 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.

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 [

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 [

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 [

Let x,yÎV(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 [

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 [

In this study, we call this eccentric version of the first Zagreb index by the third Zagreb index and denote by

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

lycene parallelogram of benzenoid P(n,n) is shown in

Now, we can exhibit the closed formula of the third Zagreb index M_{3}(H_{k}) in the following theorem.

Theorem 1. Considering the linear Polycene parallelogram of benzenoid P(n,n) ("nÎℕ), then its third Zagreb index is equal to

Proof. "nÎℕ, let P(n,n) be the linear Polycene parallelogram of benzenoid, as shown in

So according to

And also, the eccentric vertices with degree two are between 2n, 2n+1 to 4n−4, 4n−3 or in the number set

Therefore, by using above results and [

Mehdi Alaeiyan,Mohammad Reza Farahani,Muhammad Kamran Jamil,M. R. Rajesh Kanna, (2016) The First Eccentric Zagreb Index of Linear Polycene Parallelogram of Benzenoid. Open Journal of Applied Sciences,06,315-318. doi: 10.4236/ojapps.2016.65031