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The belief propagation (BP) decoding of polar codes provides better bit error rate (BER) and packet error rate (PER) performance than the successive cancelation (SC) decoding. It has been reported in the literature that the graph of BP decoding of polar codes can be permuted to provide even better decoding performance. In this paper, we theoretically prove that all permutations of the graph are equivalent in terms of the polar encoding. For BP decoding, it is shown that permutations of the layers of the graph also permute the bit channels. A principle which protects the weakest information bit channel is provided in the paper. Two parallel BP decoders, one employing the standard graph and the other selected based on the selection principle, are also proposed to work together. The simulation results show that when the total number of iterations is fixed, the proposed BP decoding can have the same performance as the existing permutation scheme while provides a higher throughput.