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Introduction
Let G(V, E) be a simple connected graph, where V and E represent the set of vertices and the set of edges, respectively. The number of elements in V and E are called the order and the size of the graph G, respectively. The distance between the vertices u and ν is the length of the shortest path connecting them. The maximum distance between u and any other vertex of the graph G is called eccentricity of u. For a vertex ν∈V, the number of vertices attached to ν is called the degree of the vertex ν. It is denoted as d(ν). The notation Sν is the summation of degrees of all neighbors of vertex ν in G, i.e. Sν = Σuv∈E(G)d(u). More details about the standard notations from the graph theory in the paper can refer to [1]–[3].
A topological index is a function (TI) from the set of finite simple graphs to the set of real numbers, with the property that TI(G) = TI(H) if both the graph G and H are isomorphic. There are hundreds of topological indices being introduced. M. Randic [4] introduced the first degree based topological index named as Randic index. The Randic index of a graph G is defined as
For history and further results on this family of topological indices, please refer to [10]- [16].
Chemical graph theory is the branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena. This theory has vital effect on the development of the chemical sciences.
Polycyclic Aromatic Hydrocarbons PAHk are obtained from the burning of organic material, and naturally as a result of thermal geological reaction. For many years, they attracted much attention, because some of them are strong carcinogens. The Polycyclic Aromatic Hydrocarbons consist of several copies of benzene on circumference. The first three members of this family are shown in Figure 1. For further details, please refer to [17]–[43].
In this paper, we compute the fifth geometric-arithmetic index of Polycyclic Aromatic Hydrocarbons.
Computation Techniques and Main Results
In this section, we computed the fifth version of Polycyclic Aromatic Hydrocarbon (PAHk).
Theorem 1
Consider the graph of Polycyclic Aromatic Hydrocarbons (PAHk), then the fifth geometric-arithmetic index of Polycyclic Aromatic Hydrocarbon is equal to
Proof. A graphical representation of Polycyclic Aromatic Hydrocarbons is shown in Figure 2. It contains 6k2 + 6k vertices and 9k2 + 3k edges. In this structure, there are two types of vertices, vertices with degree 1 and degree 3. We denote the sets of vertices with degree 1 as V1 = {ν ∈ V(G)|dν = 1} and degree 3 as V3 = {ν ∈ V(G)|dν = 3}. On the basis of degrees of the vertices we divide the edge set into a partition E4 = {uv ∈ E(PAHk)|du + dν = 4}, E6 = {uv ∈ E(PAHk)|du + dν = 6} and |E4|=6k, |E6|=9k2-3k.
The sum of degrees of vertices for each edge of PAHk is represented as follows:
There are k edges e=uν for which, Su = 3, Sν = 7 when u ∈ V1, ν ∈ V3 and uν ∈ E6.
There are 6 edges e = uν for which, Su = Sν = 7 when u,ν ∈ V6 and uν ∈ E6.
There are 9k2 − 15k + 6 edges e = uν for which, Su = Sν = 9 when u,ν ∈ V3 and uν ∈ E6.
Now, we apply these calculations to the definition of the fifth geometric-arithmetic index which can be demonstrated as follows,
The reader can find some values of the fifth geometric-arithmetic index of Polycyclic Aromatic Hydrocarbons (PAHk) for integer k = 1, 2, 3,..., 1012in following table.
Corollary 3
Consider the molecular graph Polycyclic Aromatic Hydrocarbons (PAHk) shown in Figure 1, thus Theorem 1 and Table 1 imply that for enough large integer number k, the approach fifth geometric-arithmetic index of PAHk is equal to
Computing GA5 index for Polycyclic Aromatic Hydrocarbons PAHk (∀k = 1,2,3,...,1012):