Not at all. Graduate School is part of a undergraduate college, its just where you can get a masters degree. Some colleges don't offer graduate school, but no community college offers a graduate...
It comes from the word definately. So So Def Recordings, Def Leopard, Def Jam Recordings also has def in their name. Def Leppard's name comes from an intentional misspelling of the words Deaf Leopard.