Is Eddie Adams the main reason that cause Vietnam's south become invaded by Communist ?

It is very seldom that one single event or person triggers a wholesale historical change. I had not heard of Eddie Adams before your question, though I know the photo. At that time there had already been fighting in Vietnam and considerable American involvement. The communist North Vietnamese (called the Vietminh at the time) had been fighting the French for independence since the very early 1950s, so the idea of an ideological fight was not new, and was part of the general decolonisation of former European colonies of that time in the wake of WW2. The Americans were helping South Vietnam in fear of the domino effect.