The General (α,3)-Path Connectivity Indices of Polycyclic Aromatic Hydrocarbons

The general ( α , t ) -path connectivity index of a molecular graph originates from many practical problems such as three-dimensional quantitative structure-activity (3D QSAR) and molecular chirality. It is defined asRtα ( G ) = ∑P t = vi 1vi 2⋯ vi t + 1⊆ G[ d ( vi 1) d ( vi 2) ⋯ d ( vi t + 1) ] α, where the summation is taken over all possible paths of length t of G and we do not distinguish between the pathsvi 1vi 2⋯ vi t + 1andvi t + 1⋯ vi 2vi 1. In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons ( P A H n ), which play a role in organic materials and medical sciences. We try to compute the exact general ( α , 3 ) -path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values ofR3α ( P A H n ) for any real number α . These valuable results could produce strong guiding significance to these applied sciences.