The Eccentricity of Atom-Bond The Eccentricity Version of Atom-Bond Connectivity Index of Linear Polycene Parallelogram Benzenoid ABC 5 (P(n,n))

Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. The atom-bond connectivity index of a connected graph G is defined as ABC(G) = where d v denotes the degree of vertex v of G and the eccentric connectivity index of the molecular graph G is defined as ξ (G) = , where ε (v) is the largest distance between v and any other vertex u of G . Also, the eccentric atom-bond connectivity index of a connected graph G is equal to ABC 5 (G) = In this present paper, we compute this new Eccentric Connectivity index for an infinite family of Linear Polycene Parallelogram Benzenoid.


Introduction
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. In chemical graph theory, there are many molecular descriptors (or Topological Index) for a connected graph, that have very useful properties to study of chemical molecules. [1][2][3][4] This theory had an important effect on the development of the chemical sciences.
A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.
One of them is Atom-Bond Connectivity (ABC) index of a connected graph G = (V,E) and defined as (1) where d v denotes the degree of vertex v of G, that introduced by Furtula et.al. 5,6 On the other hands, Sharma, Goswami and Madan 7 (in 1997) introduced the eccentric connectivity index of the molecular graph G as (2) where ε(u) is the largest distance between u and any other vertex v of G. If 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. In other words, is maximum distance with first-point v in G.
The Eccentric Connectivity polynomial of a graph G, was defined by Alaeiyan, Mojarad and Asadpour as follows: 8,9 (4) Alternatively, the eccentric connectivity index is the first derivative of ECP(G;x) evaluated at x = 1. Now, by combine these above topological indexes, we now define a new version of ABC index as: 10 (5) cal index for an infinite family of Linear Polycene Parallelogram Benzenoid.

Main Results and Discussion
In this section, we computed the Eccentric atombond connectivity index ABC 5 of an infinite family of Linear Polycene Parallelogram of Benzenoid graph, 19 by continuing the results from. 8,9,18,19 This Molecular graph has 2n(n + 2) vertices and 3n 2 + 4n -1 edges.
For further study and more detail representation of Linear Polycene Parallelogram of Benzenoid P(n,n), see. 8,9,18,19 Also, reader can see the general case of this Benzenoid molecular graph in Figure 1.