Recently, it is found that the bit error rate performance can be improved by optimizing the edge connections between the source code and the channel code ( ${B}_{L1}$ or ${B}_{L2}$ in the joint base matrix ${B}_{J}$ ) for joint source-channel coding system based on double protograph low density parity check codes. In this letter, the connection relationship in ${B}_{L1}$ from the angle of both the source code and the channel code is analyzed, where the protograph LDPC codes with various row weights used as the source code are taken into account. And a search algorithm for ${B}_{L1}$ under the constraints is proposed based on the joint protograph extrinsic information transfer (JPEXIT) and the source PEXIT to improve the performance. Both the JPEXIT analysis and the simulation results indicate that the performance of the searched optimal scheme has considerable gain compared with the conventional solutions.