In this letter, an integrated design of joint source-channel coding scheme based on the double protograph low-density parity-check (DP-LDPC) codes is proposed. As degree-2 variable nodes (VNs) structure plays an important role in the performance of both water-fall and error-floor regions, the maximum number of degree-2 VNs for DP-LDPC codes is determined from a global perspective. By comparing different allocation schemes for degree-2 VNs, new design principles are proposed. Several DP-LDPC codes with different numbers of degree-2 VNs are proposed. The simulated results reveal the superiority of the optimized DP-LDPC codes, which are in line with the decoding threshold analysis by a joint protograph extrinsic information transfer algorithm.