Is it safe to go running or do some light jogging 1 month post op BA?

Thank you for your question. You definitely want to follow your own surgeon's post-op activity instructions. This is a general guide I give to my patients:
1) No heavy lifting or strenuous activity for 6 weeks.
2) Resume walking at a leisurely pace right after surgery (e.g. 2 mph)
3) At 2 weeks, you can walk 2 miles at 2mph
4) At 3 weeks, you can walk 3 miles at 3mph
5) At 4 weeks, you can walk 4 miles at 4mph
6) At 5 weeks, you can jog 5 miles at 5mph
7) At 6 weeks, you can resume all activities, but listen to your body and use discomfort or tightness as a guide so you don't over do it.

At one month after breast augmentation, we allow the patients to start some light low impact cardio.  After 2 months you can run.  If the bouncing is painful, then it is too much for you ta this point.

Each plastic surgeon has his or her own paradigm with reference to post augmentation physical activity.That being said, if you were my patient, I routinely recommend running at four weeks wearing a double sports bra.  I strongly encourage my patients to go on long walks at the end of one week, to go swimming at the end of two weeks, especially since I put all my implants in through the armpit and the activity of swimming gets the arm and shoulder working again.  At the end of three weeks, I recommend going back to gym except for pushups, bench pressing, or those activities that specifically target the pectoralis muscle.  With all of the above activities, I recommend taking baby steps at first, and that includes running at four weeks, possibly run-walking just to check out how you feel.  Obviously, let your body be your guide.  If it hurts, give it a rest for a day or two and then come back with more baby steps.