Let G be a simple graph, δ and a be minimum degree and independence number of G, respectively, Faudree et al showed, in 1991, the Hamiltonian result with condition | N(x)∪ N(y) | ≥n-8. In 1993, Chen further considered the Hamiltonian with condition max |d{x) , d(y)| n/2 for each pair of non-adjacent vertices x , y with 1≤|N(x)∩NV(y)|≤a-l. In this paper a sufficient condition for a graph to be Hamiltonian graph is shown and the following result is obtained ： let G be a 2-connected graph of order n , if| N(x) U N(y) |≥ n-δ-1 for each pair of non-adjacent vertices x, y with 1≤ | N(x)∩ N(y) |α-1, then G is Hamiltonian or G∈{K_{(n-1)/2, (n + 1)/2}, K₂^* V3K_(n-2)/3}，This result generalizes some results in Hamiltonian graphs .