Independence Number, Neighborhood Intersection and Hamiltonian Properties

Let G be a 2-connected simple graph of order n with the independence number\alpha. We show here that \forall u; v \in V (G)\backslash\{u,v\} and any z \in \{u,v\}; w \in V (G)\backslash \{u,v\}; with d(w; z) = 2, if |N(u) \ cap N(w)| \geq \alpha - 1 or |N(v) \cap N(w)| \geq \alpha - 1, then G is Hamiltonian, unless G belongs to a kind of special graphs.