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Simple and flexible as AODVjr is, it is used in Zigbee mesh networks as the routing mechanism. To reduce and balance the network's energy consumption, many improved energy-aware routing algorithms have been presented based on AODVjr. However, they paid little attention to the excessive energy consumption of low-energy nodes caused by RREQ flooding which may result in network earlier death. To deal with this problem, we propose an improved energy-aware routing algorithm, EG-AODVjr, to extend the whole network lifetime by eliminating the unnecessary RREQ overhead of low-energy nodes. In EG-AODVjr, nodes are classified into four grades according to its remaining energy and adopt different RREQ processing mechanisms separately due to its grade to prevent low-energy nodes from wasting energy on forwarding RREQ packets. Furthermore, we modify the path selected scheme by putting forward a new method to calculate the path cost. EG-AODVjr is simulated and the results show that it can prolong the network lifetime by 26.1% compared with that of original AODVjr, achieving a better performance on extending Zigbee network lifetime.