A topological index is a real number related to a molecular graph, it does not rely on the labeling or pictorial representation of a graph. The first and second Zagreb indices are defined as ( ) ( ) ( ) 2 1 v V G M G d v Î =Σ and ( ) ( ) ( ) 2 uv E(G) M G d u d v Î =Σ , respectively. Recently, Nilanjan introduced the first and second Zagreb eccentricity indices as ( ) ( ) ( ) 2 1 v V G E G e v Î Õ =Õ and ( ) ( ) ( ) 2 uv E(G) E G e u e v Î Õ =Õ ´ . In this paper, we compute the first and second Zagreb indices of Polycyclic Aromatic Hydrocarbons (PAHk).