In single relay selection is an effective technique to fulfill cooperative communications in two-way relay networks. Previous works took the minimum outage probability as the objective function in relay selection. This paper proposes a single relay selection scheme to maximize the sum rate of the network. The paper establishes the direct relationship between the tight lower bound of the sum rate and the harmonic mean of channel gains. By selecting the maximum harmonic mean among all relay nodes, the scheme maximizes the tight lower bound of the sum rate. Meanwhile, this paper proves that the proposed relay selection scheme could achieve the full diversity order in two-way AF relay networks with physical layer network coding. Simulation results show that the proposed relay selection scheme achieves significant performance improvement over the random selection scheme and performs very close to the optimal selection scheme in the sense of maximum sum rate.
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