Computing Fifth Geometric-Arithmetic Index for Circumcoronene series of benzenoid Hk

Let G=(V; E) be a simple connected graph. The sets of vertices and edges of G are denoted by V=V(G) and E=E (G), respectively. The geometric-arithmetic index is a topological index was introduced by Vukicevic and Furtula in 2009 and defined as uv E G 2 d d G d +d u v u v GA in which degree of vertex u denoted by dG(u) (or du for short). In 2011, A. Graovac et al defined a new version of GA index as ( ) 5 2 v u uv E G v u A G S S S G S where ( ) . u u v v E G d S The goal of this paper is to compute the fifth geometric-arithmetic index for "Circumcoronene series of benzenoid Hk (k≥1)". Indexing terms/


INTRODUCTION
Let G=(V;E) be a simple connected graph of finite order n=|V| and the number of edges e=|E|, such that it has vertex set V=V(G) and edge set E=E(G). A general reference for the notation in graph theory is [1][2][3]. A molecular graph is a simple finite graph such that its vertices correspond to the atoms and the edges to the bonds.
Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. Chemical graph theory is a branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena. This theory has an important effect on the development of the chemical sciences.
In chemical graph theory, we have many different topological index of arbitrary molecular graph G. A topological index of a graph is a number related to a graph which is invariant under graph automorphisms. Obviously, every topological index defines a counting polynomial and vice versa. The simplest topological indices are the number of vertices and edges of the graph G.
Also, an important terminology of graph theory is degree of a vertex v V(G), that it is the number of adjacent vertices with v and we denoted by dv (In other words, the degree of a vertex v is equal to the number of its first neighbors.). If u,v V(G) then the distance dG(u,v) (or d(u,v) for short) between u and v is defined as the length of (number of edges in) any shortest path in G connecting u and v. An edge e=uv of the graph G is joined between two vertices u and v (d(u,v)=1). [4][5][6][7][8][9] is the first reported distance based topological index which have very chemical applications, mathematical properties and is defined as half sum of the distances between all the pairs of vertices in a molecular graph, which:

The Wiener index W(G)
One of important connectivity topological indices is geometric-arithmetic index of G. A class of geometric-arithmetic topological indices [10] may be defined as where Qv is some quantity that in a unique manner can be associated with the vertex v of the graph G.
The first member of this class for Qv=dv was considered by Vukicevic and Furtula [11], in 2009, and GA1 index was defined as The second member of this class was considered by Fath-Tabar et al. [12] by setting Qu to be the number nu of vertices of G lying closer to the vertex u than to the vertex v for the edge uv of the graph G: The third member of this class was considered by Bo Zhou et al. [13] by setting Qu to be the number mu of edges of G lying closer to the vertex u than to the vertex v for the edge uv of the graph G: The fourth member of this class was considered by Ghorbani et al. [14] in 2010 as follows: where the line graph L(G) of a graph G is defined to be the graph whose vertices are the edges of G, with two vertices being adjacent if the corresponding edges share a vertex in G.

Main Results and Discussions
The goal of this paper is to compute a closed formula of this new Connectivity index "fifth geometric-arithmetic index GA5" of circumcoronene homologous series of benzenoid Hk (k≥1).
The circumcoronene homologous series of benzenoid is family of molecular graph, which consist several copy of benzene C6 on circumference. The first terms of this series are H1=benzene, H2=coronene, H3=circumcoronene, H4=circumcircumcoronene, see Figure 1 and Figure 2, where they are shown. Readers can see a general represetation of Hk in Figure 2. In addition, this benzenoid molecular graph is presented in many papers, for further study and more historical details, readers can see the paper series .  From Figure 2, we mark the members of E4, E5 and E6 by red, green and black color and obviously the size of these three edge types are equal to 6, 12(k-1) and 9k 2 -15k+6, respectively. According to Figure 2, one can see that the summation of degrees of vertices of this benzenoid graph have four types, such that for vertices a,b V2 & ab E4 S(a)= db +3=5 (S(b)= da +3). And also, for vertices x V2 & y1,y2 V3 and edges xy1, xy2 E5 S(x)=dy1+dy2=6 and S(yi)= 2+2+3=7. It is easy to see that for all other vertices u, v from V3 and all other edges e=uv belong to E6 S(u)= dv+3+3=9 (S(v)=du +3+3). Now, by arrangement above formula, we have: Here, we complete the proof of Theorem 6.■