A (p, q)-graph G is said to be square sum, if there exists a bijection f : V(G) → {0,1, 2,...,p — 1} such that the induced function f * : E(G) → N given by f * (uv) = (f (u))2 + (f (v))2 for every uv ∈ E(G) is injective. In this paper we initiate a study on square sum graphs and prove that trees, unicyclic graphs, mCn, m > 1, cycle with a chord, the graph obtained by joining two copies of cycle Cn by a path Pk and the graph defined by path union of k copies of Cn, when the path Pn = P2 are square sum.