The graph to considered will be in general simple and finite, graphs with a nonempty set of edges. For a graph G, V(G) denote the set of vertices and E(G) denote the set of edges. Now, let Pr=(0, 0, 0,r) ∈ R4,r ∈ R+ .The r-polar sphere, denoted by SPr, is defined by {x ∈ R4/∥x∥=1 Λ x≠Pr} The primary target of this work is to present the concept of r-Polar Spherical Realization of a graph. That idea is the following one: If G is a graph and h: V (G) → SPr is a injective function, them the r-Polar Spherical Realization of G, denoted by G*, it is a pair(V (G*),E(G*)) so that V (G*)={h(v)/v ∈ V (G)} and E(G*)={arc(h(u)h(v))/uv ∈ E(G)},in where arc(h(u)h(v)) it is the arc of curve contained in the intersection of the plane defined by the points h(u),h(v),Pr and the r-polar sphere.