Abstract: Flocking behavior is a common phenomenon in nature. The research of flocking theory has been widely applied in economy, military and management. This paper studies the group behavior of two models with time delays under different topology structures, and obtains necessary and sufficient conditions. For the Cucker-Smale type model with directed graphs, we get the necessary and sufficient conditions of time delay .For another model with competitive relationship between individuals under bipartite graph, we also explore the influence of time delay on group behavior, and get the necessary and sufficient conditions. Finally, we verify the correctness of the conclusion by numerical simulation. 

Key words: Directed graph, Flocking-cluster , Time delay, Bipartite graph