The time-delay characteristic concealment and the bandwidth expanding of the chaotic output of mutually coupled vertical-cavity surface-emitting lasers (VCSELs) without feedback can′t be achieved simultaneously, so an external optical feedback is added to one of the lasers of mutually coupled VCSELs without feedback. The time-delay characteristic and the bandwidth of the chaotic output from the single-ended feedback and mutually coupled VCSELs are numerically studied. Results show that feedback intensity, frequency detuning and mutually coupled intensity have effects on the time-delay characteristic and the bandwidth of the single-ended feedback and mutually coupled VCSELs. The laser with feedback can′t obtain the time-delay characteristic concealment and the bandwidth expanding simultaneously， but the laser without feedback can obtain the time-delay characteristic concealment and the bandwidth expanding simultaneously in the proper mutually coupled intensity and frequency detuning. Changing the feedback intensity does not effect the quality of the time-delay characteristic concealment and the width of the bandwidth, but can effect the position of the frequency detuning when the time-delay characteristic concealment and the bandwidth expanding are achieved simultaneously. The position of the frequency detuning region moves to the negative frequency detuning region when the feedback intensity increases.