Spatially coupled low-density parity-check (SC-LDPC) codes are a class of capacity approaching LDPC codes with low message recovery latency when a sliding window decoding is used. In this paper, we first present a new method for the construction of a class of SC-LDPC codes by the incidence matrices of a given non-negative integer matrix $E$, and then the relationship of 4-cycles between matrix $E$ and the corresponding SC-LDPC code are investigated. Finally, by defining a new class of integer finite sequences, called {\it good sequences}, for the first time, we give an explicit method for the construction of a class of 4-cycle free SC-LDPC codes that can achieve (in most cases) the minimum coupling width.