General (α,2)-Path Sum-Connectivirty Indices of One Important Class of Polycyclic Aromatic Hydrocarbons

Abstract

The general ( α , t ) -path sum-connectivity index of a molecular graph originates from many practical problems, such as the three-dimensional quantitative structure–activity relationships (3D QSAR) and molecular chirality. For arbitrary nonzero real number α and arbitrary positive integer t, it is defined as t χ α ( G ) = ∑ P t = v i 1 v i 2 ⋯ v i t + 1 ⊆ G [ d G ( v i 1 ) d G ( v i 2 ) ⋯ d G ( v i t + 1 ) ] α , where we take the sum over all possible paths of length t of G and two paths v i 1 v i 2 ⋯ v i t + 1 and v i t + 1 ⋯ v i 2 v i 1 are considered to be one path. In this work, one important class of polycyclic aromatic hydrocarbons and their structures are firstly considered, which play a role in organic materials and medical sciences. We try to compute the exact general ( α , 2 ) -path sum-connectivity indices of these hydrocarbon systems. Furthermore, we exactly derive the monotonicity and the extremal values of these polycyclic aromatic hydrocarbons for any real number α . These valuable results could produce strong guiding significance to these applied sciences.