Abstract: Let G=(V, E; w) be a weighted graph, and define the weighted degree d^w_G(v) of a vertex v in G as the sum of the weights of the edges incident with v. In this paper, the following theorem is proved: suppose G is a 2-connected weighted graph, where (i) w(xy)=w(yz) for every induced path xyz, and (ii) in every induced subgraph T of G isomorphic to K_1,3 or K_1,3+e, all the edges of T have the same weight and min{max{d^w_G(x), d^w_G(y)} : d(x,y) =2,x,y ∈ V(T)}≥ c/2, then G contains either a Hamilton cycle or a cycle of weight c at least. This
respectively generalizes three theorems of Fan^[5], Bedrossian et al^[2] and Zhang et al^[7].

Key words: Semi-normal weighted graph, Heaviest longest path, Hamiltonian cycle, Weighted degree