In my head, I always thought that the Mayor was the Apple siblings mother. It... Kind of makes sense when you think about it. You got the mayor, who's a natural leader, and running ponyville, and Apple Jack runs Sweet Apple Acres. The Mayor puts a lot of pressure and trust into Apple Jack to win enough money to fix the Town Hall. I dunno, but there's just something between the two (Apple Jack and the Mayor) that seems to connect.