On square sum graphs

Let G =( V,E )b e a( p,q)-graph and let f : V (G) →{ 0,1,2,...,p− 1} be a bijection. We define f ∗ on E(G )b yf ∗ (uv )=[ f(u)] 2 +[ f(v)] 2 . If f ∗ is injective on E(G), then f is called a square sum labeling. If f ∗ (E) consists of the first q consecutive integers of the form