The non-commuting graph ∇(G) of a non-abelian finite group G is defined as follows: its vertex set is G − Z(G) and two distinct vertices x and y are joined by an edge if and only if the commutator of x and y is not the identity. In this paper we prove if G is a finite group with ∇(G) ∼= ∇(Ap+3), then G ∼= Ap+3, where Ap+3 is the alternating group of degree p + 3, where p is a prime number.