In this paper, we define the exterior square graph Γ E ∧ G which is a graph associated to a non-cyclic finite group with the vertex set G Z ∧ (G), where Z ∧ (G) denotes the exterior center of G, and two vertices x and y are joined whenever x ∧ y = 1, where ∧ denotes the operator of non-abelian exterior square. We investigate how the group structure can be affected by completeness, regularity and bipartition of this graph.