"On the edge of something" and "on the verge of something" mean the same thing. They mean "on the margin or edge of something". An example would be "Sara was on the verge of tears when she heard the news."

"On the brink of something" , on the other hand, means to be at a crucial or critical point, especially of a situation or state beyond which only success or catastrophe will occur. One might say "We were on the brink of war."

http://english.stackexchange.com/questions/77143/on-the-brink-verge-edge-of