Is it polite to ask my mentor to recommend me to Professors she knows in the graduate school? I am currently applying for the doctoral program. One thing I am keep considering is that should I politely ask my mentor who will write me the ref letter to recommend me to professors that she knows in those schools that I will be applying to? I am having a good relationship with my mentor and she knows me well and I think she had good impression on me. The reason I am considering this is because she was the alumni of two of my target schools and also used to be a faculty at one of my other target school. Could someone gives me some suggestion about whether should I politely ask her? I am a little afraid of making her feel like I am trying to rely on 'connection' but I also feel like if I don't ask and maybe she will be willing to, I might feel very regret of not asking so. I will be very appreciated for all suggestion and helps!
Yes it is polite and it is actually proper to ask a mentor for this in most cultures, I suspect. Some professors might be reluctant to do this, but I think not very many.
But what it actually gets you depends on the admissions system at the universities you apply to. In some, say in the US, it might have little effect on admissions but it places (and fields) in which professors hire their own students it would be much more valuable.
But in any case, having been introduced will be a small (at least) advantage if you wind up at one of these schools and it is a good idea to go meet the people you were introduced to. You might even be able to pick up a new mentor at the new place.