Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAH_k

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 S_ν is the summation of degrees of all neighbors of vertex ν in G, i.e. S_ν = Σ_uv∈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

R(G)=∑uν∈E(G)1d(u)d(ν).$$\begin{array}{} \displaystyle R(G) = \sum\limits_{u\nu \in E(G)} {\frac{1}{{\sqrt {d(u)d(\nu)} }}} . \end{array}$$

A widely used connectivity topological index is geometric-arithmetic index (GA) introduced by Vukicevic [5] and it can be defined as

GA1(G)=∑uν∈E(G)2d(u)d(ν)d(u)+d(ν).$$\begin{array}{} \displaystyle G{A_1}\left( G \right) = \sum\limits_{u\nu \in E(G)} {\frac{{2\sqrt {d(u)d(\nu)} }}{{d(u) + d(\nu)}}} . \end{array}$$

The second member of this family of index was introduced by Fath-Tabar [6]

GA2(G)=∑uν∈E(G)2nνnunν+nu,$$\begin{array}{} \displaystyle G{A_2}\left( G \right) = \sum\limits_{u\nu \in E(G)} {\frac{{2\sqrt {{n_\nu}{n_u}} }}{{{n_\nu} + {n_u}}}} , \end{array}$$

where n_u is the number of vertices of graph G lying closer to the vertex u than that to the vertex ν for the edge uv∈E(G).

Bo-Zhou [7] proposed the third geometric-arithmetic index of a graph G

GA3(G)=∑uν∈E(G)2mνmumν+mu,$$\begin{array}{} \displaystyle G{A_3}\left( G \right) = \sum\limits_{u\nu \in E(G)} {\frac{{2\sqrt {{m_\nu}{m_u}} }}{{{m_\nu} + {m_u}}}} , \end{array}$$

where m_u is the number of edges of graph G lying closer to the vertex u than that to the vertex ν for the edge uv∈E(G).

The fourth geometric-arithmetic index was proposed by Ghorbani [8]. This index is defined as

GA4(G)=∑uν∈E(G)2ενεuεν+εu,$$\begin{array}{} \displaystyle G{A_4}\left( G \right) = \sum\limits_{u\nu \in E(G)} {\frac{{2\sqrt {{\varepsilon _\nu}{\varepsilon _u}} }}{{{\varepsilon _\nu} + {\varepsilon _u}}}} , \end{array}$$

where ε_u is the number of eccentricity of vertex u.

Recently, Graovac [9] defined the fifth version of geometric-arithmetic index as

GA5(G)=∑uν∈E(G)2SνSuSν+Su.$$\begin{array}{} \displaystyle G{A_5}\left( G \right) = \sum\limits_{u\nu \in E(G)} {\frac{{2\sqrt {{S_\nu}{S_u}} }}{{{S_\nu} + {S_u}}}} . \end{array}$$

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 PAH_k 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].

Fig. 1

In this paper, we compute the fifth geometric-arithmetic index of Polycyclic Aromatic Hydrocarbons.

In this section, we computed the fifth version of Polycyclic Aromatic Hydrocarbon (PAH_k).

Theorem 1

Consider the graph of Polycyclic Aromatic Hydrocarbons (PAH_k), then the fifth geometric-arithmetic index of Polycyclic Aromatic Hydrocarbon is equal to

GA5(PAHk)=9k2+k(6215+972−15)+(12−972).$$\begin{array}{} \displaystyle G{A_5}(PA{H_k}) = 9{k^2} + k\left( {6\frac{{\sqrt {21} }}{5} + \frac{{9\sqrt 7 }}{2} - 15} \right) + \left( {12 - \frac{{9\sqrt 7 }}{2}} \right). \end{array}$$

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)|d_ν = 1} and degree 3 as V₃ = {ν ∈ V(G)|d_ν = 3}. On the basis of degrees of the vertices we divide the edge set into a partition E₄ = {uv ∈ E(PAH_k)|d_u + d_ν = 4}, E₆ = {uv ∈ E(PAH_k)|d_u + d_ν = 6} and |E₄|=6k, |E₆|=9k²-3k.

Fig. 2

The sum of degrees of vertices for each edge of PAH_k is represented as follows:

There are k edges e=uν for which, S_u = 3, S_ν = 7 when u ∈ V₁, ν ∈ V₃ and uν ∈ E₆.

There are 6 edges e = uν for which, S_u = S_ν = 7 when u,ν ∈ V₆ and uν ∈ E₆.

There are 9k² − 15k + 6 edges e = uν for which, S_u = S_ν = 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,

GA5(PAHk)=∑e=uv∈E(Hk)2S(ν)S(u)S(ν)+S(u)$\begin{array}{} G{A_5}(PA{H_k}) = {\sum\nolimits _{{\rm{e}} = {\rm{u\nu}} \in E({H_k})}}\frac{{2\sqrt {S\left( \nu \right)S\left( u \right)} }}{{S\left( \nu \right) + S\left( u \right)}} \end{array}$

=6k23×73+7+627×77+7+12(k−1)27×97+9+(9k2−15k+6)29×99+9=6k215+6+(k−1)972+(9k2−15k+6)=9k2+k(6215+972−15)+(12−972)$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} { = 6k\frac{{2\sqrt {3 \times 7} }}{{3 + 7}} + 6\frac{{2\sqrt {7 \times 7} }}{{7 + 7}} + 12\left( {k - 1} \right)\frac{{2\sqrt {7 \times 9} }}{{7 + 9}} + (9{k^2} - 15k + 6)\frac{{2\sqrt {9 \times 9} }}{{9 + 9}}}\\ { = 6k\frac{{\sqrt {21} }}{5} + 6 + \left( {k - 1} \right)\frac{{9\sqrt 7 }}{2} + (9{k^2} - 15k + 6)}\\ { = 9{k^2} + k\left( {6\frac{{\sqrt {21} }}{5} + \frac{{9\sqrt 7 }}{2} - 15} \right) + \left( {12 - \frac{{9\sqrt 7 }}{2}} \right)} \end{array} \end{array}$$

which is the required result.