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

Let G be a simple connected graph. The geometric-arithmetic index of G is defined as GA1(G)=∑uν∈E(G)2d(u)d(ν)d(u)+d(ν)$\begin{array}{} G{A_1}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {d(u)d(\nu)} }}{{d(u) + d(\nu)}} \end{array}$, where d(u) represents the degree of the vertex u in the graph G. Recently, Graovac defined the fifth version of geometric-arithmetic index of a graph G as GA5(G)=∑uν∈E(G)2SνSuSν+Su$\begin{array}{} G{A_5}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {{S_\nu}{S_u}} }}{{{S_\nu} + {S_u}}} \end{array}$, where S_u is the sum of degrees of all neighbors of vertex u in the graph G. In this paper, we compute the fifth geometric arithmetic index of Polycyclic Aromatic Hydrocarbons (PAH_k).