In this paper, a new fully distributed state estimation (DSE) based on weighted least square (WLS) method and graph theory is proposed for power system. The proposed method is fully distributed so that the centralized facilities, e.g., supervisory control and data acquisition (SCADA) and centralized estimators, are not required. Also, different from the existing DSE methods, the proposed method is a bus-level DSE method, in which the power system is not required to be partitioned into several areas. In order to realize the proposed fully distributed DSE method, a novel information propagation algorithm is developed in this paper. This algorithm has great potential in future applications since it is useful to broadcast the local information of the nodes to the entire system in a fully distributed network. The proposed DSE method is compared with the conventional centralized state estimation method and existing multi-area DSE method in different models in this paper. The results show that the proposed method has better performance than the traditional methods.