This work studies the edge-based and node-based adaptive event-triggered control (ETC) problem for multi-agent systems. The aims are to achieve asymptotic consensus, guaranteed positive minimum inter-event times (MIET) and avoiding using global information. First, an edge-based distributed ETC strategy is proposed based on absolute state information, which can achieve the desired aims but subjects to some limitations in terms of symmetric conditions and practical implementation. Then, a node-based ETC strategy is proposed based on relative state information between agents, which can overcome limitations of edge-based control and can be applied to multi-agent systems with undirected/directed communication topology. The main advantage of the proposed ETC strategies is that positive MIET can always be guaranteed by designing an auxiliary function with time-varying coefficients, meanwhile, the implementation of ETC strategy does not depend on any global information, and the node-based ETC strategy can also be used under directed communication topology. Finally, numerical examples are given to illustrate the proposed approach.