ABSTRACT.Here, we define a cube sum labeling and cube sum graph. Let \( G \) be a \( (p,q) \) graph. \( G \) is said to be a cube sum graph if there exist a bijection \( f : V(G) \to \{0,1,\ldots,p-1\} \) such that the induced function \( f^{*} : E(G) \to \mathbb{N} \) given by\[f^{*}(uv) = |\,f(u)^{3} + f(v)^{3}\,| \quad \text{for all } uv \in E(G)\]are all distinct. In this paper, we developed the concept of cube sum labeling of some family of graphs like paths, cycle, stars, wheel graph, fan graphs are discussed in this paper.